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Determination of gas effective diffusivities in porous solids, dispersion coefficients in packed beds.. 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 Thesis 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, 19̂ 5 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r - m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s , , I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i - c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f C h e m i c a l E n g i n e e r i n g T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e ^/j*****^ ftf(>(o The University of British 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 British 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 effective d i f f u s i - vities are not generally adaptable to the pellets in a packed bed, for example a catalytic reactor, An unsteady state pulse method has been developed employing simple gas chromatographic rate theory, The method is generally applicable to pellet sizes down to about 2mm, ; With homogeneous pellets reasonable agreement was obtained on comparison of effective diffu- s i v i t i e s measured by a steady state method. For aniso- tropic solids the unsteady state diffusivity can be quite different from the steady state value due to differences in diffusion path, Pulse dispersions measured in beds of non porous pellets have revealed a laminar flow regime where the . dispersion coefficient is dependent on the square of the velocity. This regime was reported for flow, in straight pipes but has not previously been demonstrated in 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 versatile technique for the determination of binary gas diffusion coefficients. The molecular di 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 in the method, GRADUATE STUDIES Field of Study: Diffusion and Kinetics Chemical Engineering Reactor Design Industrial Kinetics and Catalysis Mass3 Heat and Momentum Transfer Chemical Kinetics D , Gas Adsorption and Solvent. Extraction D. S, Scott. ' D. S, Scott S, D, Cavers E.. A. Ogryzlo G. L a James D, S, Scott S, Ds Cavers Other Studies Chemical Engineering Thermodynamics Analogue Computers Mathematical Operations in Chemical Engineering Fortran Programming Differential 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 I I 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 for 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 applicable to assemblages of such p e l l e t s , for example, as i n c a t a l y t i c reactors, and require special techniques or apparatus. A pulse • technique has been developed vhich can successfully y i e l d a reasonable value of the d i f f u s i v i t y by analysis of pulse dispersion i n terras of simple chromato- graphic rate theory. A non-adsorbing pulse gas i s necessary, and hydrogen i s nearly i d e a l . Because of the high molecular d i f f u s i v i t y of hydrogen the smallest size of p a r t i c l e vhich can be tested v i t h t h i s gas i s about 2 mm aiEuaeter. The unsteady s'tate pulse effective 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 for c a t a l y t i c studies was compared v i t h a conventional steady state method and good agreement obtained^in a spherical i s o t r o p i c p e l l e t ( ^ ) ; hovever, as may be expected agreement was poor with anisotropic p e l l e t s . A regime vas found i n a study of beds of non porous pellets vhere the dispersion c o e f f i c i e n t i s proportional to the' square of the v e l o c i t y . This regime i s reported for pipes but has not been realized as a separate regime i n packed beds. This dispersion data i s compared v i t h the limited 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 Effusion of one gas from a packed bed of knovn geometry into a second fl o v i n g gas has been evaluated as a v e r s a t i l e technique for 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 analysis method. Optimization of experimental parameters should y i e l d s a t i s f a c t o r y results (t 3/6). I l l SECTION 1 EXPERIMENTAL MEASUREMENT OF EFFECTIVE GAS DIFFUSIVITIES III POROUd PELLETS AND DISPERSION COEFFICIENTd IN PACKED BEDS INTRODUCTION II THEORY Pape A. THIELE MODULUS 1 B. DIFFUSION MECHANISMS 3 Molecular Diffusion k Knudsen Diffusion 5 Intermediate Diffusion 6 v. Effective Diffusion Coefficient 7 C. EXPERIMENTAL ESTIMATION OF EFFECTIVE DIFFUSIVITIES 8 Prediction - 8 Steady State Methods 9 Chemical Reaction Method 10 Unsteady State Methods 10 Frequency Response and Pulse Methods 11 Comparison of Methods 12 D. STATEMENT OF OBJECTIVES 13 A. DERIVATION OF VAN DEEM'TER'3 EQUATION l6 Height Equivalent to a 'Theoretical Plate (HETP) l6 Measurement of HETP • 19 Input Pulse Distribution 20 Rate Theory 21 Mass Transfer Coefficient and Effective D i f f u s i v i t y 25 External Mass Transfer Coefficient 26 Internal Mass Transfer Coefficient 26 iv Page Van Deciliter" s Equation 29 T y p i c a l Values o f the E f f e c t i v e D i f f u s i v i t y Terra(C) 50 Least Square E r r o r F i t t i n g o f Data to Van Deemter 33 Equation B. LONGITUDINAL DISPERSION COEFFICIENTS 3U V e l o c i t y P r o f i l e C o n t r i b u t i o n 36 I I I APPARATUS 39 A. DEVELOPMENT ' 39 B . DESCRIPTION OF APPARATUS 39 C. DETECTORS kk Hydrogen Flame I o n i z a t i o n Detector kk Thermal C o n d u c t i v i t y Detector kj IV EXPERIMENTAL 'PROCEDURE k8 A. OUTLINE OF EXPERIMENTAL INVESTIGATION ko B . NON POROUS PELLETS IN PULSE APPARATUS kd C. POROUS PELLETS IN PULSE APPARATUS " 51 D. INDEPENDENT EFFECTIVE DIFFUSIVITY MEASUREMENT ? 52 E. PREPARATION OF TEST COLUMNS * ' , 5 3 F. OPERATION OF PULSE APPARATUS ' $k Hydrogen Flame Detector " 55 Thermal C o n d u c t i v i t y Detector 55 V RESULTS 56 A. 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 'B. 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 88 Results t IV DISCUSSION A. NON POROUS PELLETS 89 HETP vs. Velocity Curves 89 Axial Dispersion Coefficient 90 Correlation of the Eddy Diffusivity 93 B. POROUS PELLETS 9̂ Inconsistency of the Steady State and Pulse Results 95 for Activated Alumina Porosity 96 Non Spherical Pellets "' 96 Methane Pulse . 97 Errors 97 VII CONCLUSIONS 99 VIII RECOMMENDATIONS 99 v i SECTION I I DEVELOPMENT OF AN UNSTEADY STATE FLOW METHOD FOR MEASURING BINARY GAS DIFFUSION COEFFICIENTS I INTRODUCTION 100 I I THEORY 102 A. SIMPLIFIED SOLUTION OF DIFFUSION EQUATION 102 B. MORE RIGOROUS SOLUTION 105 C. COMPUTATION OF SLOPE OF DECAY CURVE WITH A RESIDUAL 110 CONCENTRATION I I I APPARATUS 112 IV PROCEDURE 11^ A. SELECTION OF DISPLACED AND DISPLACING GAS llh B. OPERATION OF EQUIPMENT l l 6 V RESULTS - 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 V I I I RECOMMENDATIONS 1 3 6 NOMENCLATURE 137 LITERATURE CITED ikO v i i APPENDIX I DETERMINATION OF DIFFUSIVITY IN 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 Alumina P e l l e t l / U " Diameter Norton C a t a l y s t Support, l / 2 " Diameter "Alundum" CONCLUSIONS APPENDIX I I RESULTS FROM PULSE APPARATUS NON POROUS Program Results POROUS Program Re s u l t s FLOW METER CALIBRATION v i i i 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" diameter SA 20J mixture 185 A c t i v a t e d Alumina P e l l e t s Alcoa HI51 and l/8" 185 diameter POROSITY OF PACKED BEDS l88 ESTIMATION OF THE MOLECULAR DIFFUSIVITY OF THE METHANE AIR SYSTEM 191 APPENDIX IV ADSORPTION OF GASES BY ACTIVATED ALUMINA PELLETS 192 THEORY AND APPARATUS ,192 RESULTS • 191+ CONCLUSION L96 APPENDIX V RESULTS FOR SECTION I I , MOLECULAR DIFFUSIVITY APPARATUS 198 PROGRAM 198 RESULTS 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 Packing Bed 226 END 23k i x TABLES SECTION 1 1.1. The e f f e c t o f p e l l e t diameter and e f f e c t i v e d i f f u s i v i t y on the 31 e f f e c t i v e d i f f u s i v i t y term (C) i n equation I.5O. l . I I Summary o f the p e l l e t and tube t o p e l l e t diameter r a t i o s covered .50 by the experimental runs. l . I I I 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 . I V E f f e c t o f pulse s i z e (peak height) on HETP. 67 l . V Further 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 . V I Values o f the eddy d i f f u s i v i t y term constant, . 70 l . V I I P r o p e r t i e s o f porous p e l l e t samples. 80 l . V I I I 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 . Qk l . I X Comparison o f experimental 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 P o t e n t i a l E r r o r s . 98 SECTION I I 2.1 P r o p e r t i e s o f d i f f u s i o n c e l l s . 115 2.II Results f o r p a r a l l e l tube packing. ' 120 2 . I l l Results f o r porous s o l i d packing. 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 data 131 APPENDICES A II.1 Flow meter c a l i b r a t i o n 181 A IV.1 R e s u l t s f o r a d s o r p t i o n apparatus ' 197 X FIGURES SECTION I 1.1 Pore model f o r d e r i v a t i o n of e f f e c t i v e n e s s f a c t o r . 2 1.2 Model f o r d e r i v a t i o n o f p l a t e theory. 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 Mathematical model f o r the column. 22 1.5 T y p i c a l p l o t o f HETP vs. v e l o c i t y (equation 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 flame d e t e c t o r . 1+5 1.10 HETP vs. v e l o c i t y f o r run 52. 61 l . l i HETP vs. v e l o c i t y f o r runs 51, 69 and 70. 62, 1.12 Eddy d i f f u s i o n term, A, (equation 1.50) v s » p e l l e t diameter. 71 1.13 D i s p e r s i o n c o e f f i c i e n t , Dj, vs. i n t e r s t i t i a l v e l o c i t y , u. 73 l . l U Inverse eddy P e c l e t number vs. s u p e r f i c i a l Reynolds number. lh 1.15 Inverse eddy P e c l e t number v s . molecular P e c l e t number. 75 1.16 Inverse Eddy Schmidt number v s . h y d r a u l i c diameter 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 2.1 Model o f the bed f o r proposed d i f f u s i o n experiment. 103 2.2 D i f f u s i o n apparatus. 113 2.3 Results w i t h p a r a l l e l tube bed. Hydrogen-nitrogen. 121 2.1+ Results w i t h p a r a l l e l tube bed. Ethane-nitrogen. 122 2.5 Results 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. Hydrogen-nitrogen. 128 2.7 Results w i t h s p h e r i c a l packing 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 packing bed. Butane-nitrogen. 130 x i APPENDIX I A I . l Sample mounting i n steady s t a t e apparatus. ihh A 1.2 Steady s t a t e apparatus. 1̂-6 A II.1 Flow meter c a l i b r a t i o n . ' 183 A IV.1 Adsorption measurement apparatus 193 A IV.2 Gas volume vs. i n v e r s e pressure f o r a d s o r p t i o n measurement 196 •' ( x i i ) , ACKNOWLEDGEMENT Thanks are extended to Dr. Forsythe and the f a c u l t y and s t a f f o f the U n i v e r s i t y - o f B r i t i s h Columbia, chemical engineering department. P a r t i c u l a r l y the author wishes to thank Dr. E p s t e i n f o r h i s help i n Dr. S c o t t ' s absence and Mr. E. R. Rudischer and Mr. J . Baranovski f o r t h e i r a s s i s t a n c e and co-operation. j The author i s indebted to Dr. Scott f o r h i s d i r e c t i o n and support 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 the r a t e a t which r e a c t a n t s and products can d i f f u s e i n and out of the s o l i d . T h i e l e ( l ) q u a n t i t a t i v e l y described t h i s e f f e c t w i t h a mathematical treatment v h i c h i s a p p l i e d to a simple case o f an i n f i n i t e s l a b i n the d e r i v - a t i o n below. In F i g . 1.1 a s i n g l e pore o f rad i u s r and l e n g t h L i s shown. A f i r s t order gas phase r e a c t i o n w i t h r a t e = k C A moles/(sec)(cm 2) i s assumed to be t a k i n g place i s o t h e r m a l l y on the pore w a l l s , and a constant c o n c e n t r a t i o n C ^ Q moles/cm 3 i s maintained a t each face o f the s l a b a t the pore mouth. A m a t e r i a l balance around the element dx y i e l d s , -D dCA 7T r 2 - (-D) f* dCA + d 2CA Sx ITT r 2 - k C A 2 TTv fix = 0 ( l . l ) dx~ L d x dx* J which may be s i m p l i f i e d t o , d 2 C A = gk C A (1.2) dx^" rD The boundary c o n d i t i o n s , C A = C A Q a t x = 0, and dC A/dx = 0 when x = L , may be a p p l i e d to the s o l u t i o n o f (1.2) to g i v e the c o n c e n t r a t i o n i n the pore, GA = CA0 cosh[ h( 1 " L")] " (1.3) cosh h where h = L / 2k, commonly known as the T h i e l e modulus. <\ Dr The r a t e o f r e a c t i o n i s g i v e n by the r a t e o f d i f f u s i o n o f A i n t o the pore mouth, which i n t u r n i s given by, rate/pore = -D [ ~ J 7T r 2 (l.h) \ dx / x = 0 = D C A O h tanh (h) T T r 2 Figure 1.1 Pore Model For Derivation Of Effectiveness Factor - 3 - I f the whole pore contains gas a t the surface c o n c e n t r a t i o n , C^Q, the r a t e of r e a c t i o n w i l l "be a maximum, gi v e n by k . C • 2TTrL moles/ AO sec. The r a t i o of the r a t e given by equation (l.h) and t h i s maximum r a t e i s defined 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 h tanh h = tanh h (1.5) maximum r a t e 2kL 2 h The 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 of the T h i e l e modulus o n l y , and can be used to c a l c u l a t e the r a t e of 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 r e a c t i o n / p o r e = k C ^ Q 2TTr L E (1.6) A d d i t i o n a l equations can be d e r i v e d f o r other orders of r e a c t i o n , (2) other assumed pore geometries (3), or f o r cases where the s t o i c h i o m e t r y does not a l l o w equirnolar counter d i f f u s i o n to occur ( l ) ( l + ) . In p r a c t i c a l cases i t i s very d i f f i c u l t t o d e f i n e a c c u r a t e l y the pore geometry o f a porous s o l i d , and r a t e constants are more commonly based upon u n i t mass of c a t a l y s t . I t i s convenient mathematically t o t r e a t the porous s o l i d as a homogeneous medium having 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 than attempting to use 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 together w i t h the v o i d f r a c t i o n and s u i t a b l e assumptions about the pore geometry. B. DIFFUSION MEGIIANISMS There are 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 which obey F i c k ' s laws o f d i f f u s i o n , namely, molecular 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 c o l l i s i o n s , and Knudsen d i f f u s i o n , which depends o n l y upon w a l l c o l l i s i o n s . I n a d d i t i o n , a phenomenon known as "surface d i f f u s i o n " can take p l a c e , but t h i s i s not a w e l l understood process. Surface d i f f u s i o n i s b e l i e v e d to r e s u l t from m u l t i l a y e r s o f gas molecules condensed to a l i q u i d - l i k e s t a t e , which flow from the' areas w i t h s e v e r a l l a y e r s to those of lower surface c o n c e n t r a t i o n . 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 than those p o s s i b l e by c o l l i s i o n mechanisms. Gases above their critical temperature are less likely to display this phenomenon, because of the reduction in surface adsorption under these conditions. Molecular Diffusion in 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 in a two component mixture takes the form, % = - D B ^ G T + ( N A + N B ) CA (1.7) vm where the last term accounts for bulk flows xrhich may be caused by non- equimolar counter diffusion rates of the two gases with respect to stationary coordinates. In order bo apply the equation to a porous structure the flux is taken per total unit area of solid and pore, rather than unit area of pore only, N A X = N A 6 p = - [ £ 3 ^ , J ] dCA + ( N A X + Ng1) C_A (1.8) i 1 j dx Pm where X is 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 will be discussed later. Fick's second law which describes the unsteady state diffusion process is usually expressed as, b CA _ C A (1.9) At 4x 2 In a porous solid, introducing the concept of an effective diffusivity, this equation should be modified as shown below: - 5 - In u n i t area o f a porous i n f i n i t e s l a b , a mass balance over the element Sx when no chemical 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 bulk flow gives the r a t e o f change o f gas content 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 , D^, as, <fp - ^ A - Sx o + D E / h %\ Sx - a(u C A ) Sx (1.10) where u i s the 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 g i v e s , 0 C A = + ° E _ C A - __1 d(u C A ) (1.11) $t € P $ x 2 ep d x For equimolal counter d i f f u s i o n , t h i s reduces to the u s u a l form of F i c k ' s second- law, t h a t i s , equation ( l . 9 ) except t h a t the molecular d i f f u s i o n c o e f f i c i e n t i s repla c e d 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 p o r o s i t y . I f equimolar c o u n t e r d i f f u s i o n occurs then equation (1.8) f o r the.- steady s t a t e reduces to N A X = - D E d C A (1.12) dx The b i n a r y molecular 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 p r o p o r t i o n a l t o the absolute temperature to about the 1.7 power, and i n v e r s e l y p r o p o r t i o n a l t o the pressure. Knudsen D i f f u s i o n This mechanism predominates when the mean f r e e path o f the gas molecules i s gr e a t e r than the 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 to 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 the presence o f other gases. 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 case, and so F i c k ' s law i n the form of equation (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 given by, D K = 2/3 r v (1.13) vhere v i s the average v e l o c i t y o f the gas molecules, and r the pore r a d i u s . In consequence, the value of t h i s c o e f f i c i e n t i s independent o f pressure, and p r o p o r t i o n a l to the square root o f the temperature. Intermediate or Mixed D i f f u s i o n C o e f f i c i e n t I n the intermediate range between molecular and Knudsen d i f f u s i o n there i s a region where both the above d i f f u s i o n mechanisms occur. The r a t i o o f pore ra d i u s t o mean f r e e path l i e s approximately between the f o l l o w i n g l i m i t s i n the intermediate zone: Knudsen , Intermediate Molecular 0.1 < r < 10 A, 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 molecular r e f l e c t i o n from the w a l l s , S c o t t and D u l l i e n (5) derived the f o l l o w i n g r e l a t i o n s h i p f o r the f l u x i n a b i n a r y gas mixture i n the intermediate r e g i o n . N A = - _Z d y A RT dx 1 - JYA + 1 D B DKA (1.1M where j = 1 + N ^ / N - Q , and y^ i s the mole f r a c t i o n o f A . I f the term i n brackets i s considered as the d i f f u s i o n coef- f i c i e n t , i t i s obvious t h a t i n t h i s r e g i o n the 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 the form o f (1.12) and measured i n t h i s r e g i o n i s not v a l i d f o r use i n the T h i e l e modulus as def i n e d p r e v i o u s l y , as the s t o i c h i o m e t r y o f the chemical r e a c t i o n imposes a f l u x r a t i o which i s u n l i k e l y to be the same as the f l u x r a t i o • obtained i n an independent non-reactive d e t e r m i n a t i o n . - 7 - • 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 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 equation (1 . 8 ) f o r molecular d i f f u s i o n where two f a c t o r s are used to modify the t r u e or i n t e r s t i t i a l d i f f u s i v i t y . 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 f a i r l y e a s i l y d e fined and measured absolute q u a n t i t y , and i n a g r a n u l a r bed may be o f the range o f 0.3 to 0.5. However, the 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 property, e s p e c i a l l y i n non- uniform pore s t r u c t u r e s . Although a value o f around 1.5 might be expected from simple pore models, i t can vary from 1 to 100 when c a l c u l a t e d from experimental r e s u l t s . Thus, a t y p i c a l simple s t r u c t u r e may have an e f f e c t i v e d i f f u s i v i t y about It- times l e s s than the i n t e r s t i t i a l v a l u e . The l a r g e range of t o r t u o s i t y values can be a t t r i b u t e d to two sources. F i r s t , the pores are not n e c e s s a r i l y open-ended and so the 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 passages, oecond, the pore r a d i u s i s l i a b l e to vary along the l e n g t h of the pore, and i t has been shown (6) (7) t h a t the r a t e o f d i f f u s i o n i s s m a l l e r through a pore o f 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 pore o f e q u i v a l e n t volume t o 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 serve as a simple c o r r e c t i o n to the d i f f u s i o n mechanism so t h a t the d i f f u s i o n equation describes the t r a n s p o r t behaviour i n a uniform porous s t r u c t u r e . On the other hand, porous s t r u c t u r e s r can be so haphazard t h a t any o f the mechanisms d e s c r i b e d may occur a t the same time i n s e r i e s or i n p a r a l l e l i n the same s o l i d . The use of 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 to f o r c i n g the behaviour observed t o f i t one of the d i f f u s i o n equations, and so the r e s u l t cannot be used to p r e d i c t the d i f f u s i v e behaviour under other conditions'. C. EXPERIMENTAL ESTIMATION OF THE EFFECTIVE DIFFUSIVITY P r e d i c t i o n The b a s i s f o r the p r e d i c t i o n o f the 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 the previous paragraph, 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 the 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 experimental t o r t u o s i t y values are o f t e n not too s a t i s f a c t o r y due to the non-uniform nature 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 approximated by the " p i l e o f b r i c k s " s t r u c t u r e which 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 described i n d e t a i l by 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 simple pore 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 pores w i t h "ink b o t t l e " c a p a c i t i e s (10) v h i c h are used to e x p l a i n the h y s t e r e s i s i n c e r t a i n a d s o r p t i o n - d e s o r p t i o n curves. P o s s i b l y o f more general a p p l i c a t i o n t o the problems i n v o l v e d 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 pore s t r u c t u r e model proposed by Wakao and Smith ( l l ) and Mingle and Smith (12). In the l a t t e r paper, a concept o f l a r g e r macro pores i n s e r i e s w i t h micro pores i s used. In the former, three p a r a l l e l mechanisms are considered; f i r s t , d i f f u s i o n through the macro pores between the b a s i c p a r t i c l e s from which the p e l l e t i s pressed, second, d i f f u s i o n i n the micro pores o f the 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 micropores to macro pores or v i c e v e r s a . The model does not r e q u i r e e m p i r i c a l constants, o r assumptions regarding the mode of d i f f u s i o n i n any o f the pores, but p o r o s i t i e s and a pore s i z e frequency d i s t r i b u t i o n f u n c t i o n are r e q u i r e d i n a d d i t i o n to t o r t u o s i t y v a l u e s . • - 9 - Steady State Experimental Method Cor Measurement of Diffusion in Solids In this method, a cylindrical catalyst pellet i s fitt e d into a tube and two test gases of known composition arc passed contirmously across the ends. The two exit streams are analyzed, and from an appropriate solution of the diffusion equation the effective 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 calibration of the porous pellet by a gas pair of known di f f u s i v i t y allows calculation of the d i f f u s i v i t i e s of other gas pairs by making the assumption that the tortuosity is independent of 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 is a flow method, and so analysis i n situ i s not required. On the other hand, care must be taken that a narrow pore size distribution exists and that Knudsen diffusion does not occur. \Th.en used as a method for measuring the effective d i f f u s i v i t y i n porous pellets one must be sure that the correct diffusion equation has been used (e.g. eqn. (1.8) or (1.12)). The method can be applied to the mixed diffusion range i f measurements are made at varying total pressures. However, it. has the limitation of being tedious i f a representative average value is needed, because each pellet must be tested separately, and cracks and fissures have an overwhelming influence' on the result. The technique i s not convenient for use with other than cylindrical" shapes, and there- fore other shapes must be machined to cylinders. If the pellet i s not isotropic, this procedure may result in a faulty value of the diffusion coefficient. - 10 - The r e s u l t obtained by the steady 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 the d i f f u s i v i t y i n favour o f the l a r g e r pores, but i n the chemical r e a c t i o n case most o f the conversion takes place i n the micro pores. This b i a s i s f r e q u e n t l y not serious as the micropores are g e n e r a l l y s h o r t , and so a micropore 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, the d i f f u s i o n a l r e s i s t a n c e to r e a c t i o n i s i n the macropores, and the steady 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 value may be q u i t e adequate. Chemical Reaction Method I t i s o b v i o u s l y p o s s i b l e to c a r r y out a chemical r e a c t i o n o f knovn k i n e t i c behaviour a t constant c o n d i t i o n s u s i n g s u c c e s s i v e l y smaller s i z e s of p e l l e t u n t i l the r e a c t i o n r a t e becomes constant, i n d i c a t i n g t h a 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 reached. From these e f f e c t i v e n e s s f a c t o r s the 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 the k i n e t i c behaviour i s not complex. This method i s not easy to apply e x p e r i m e n t a l l y , and i s s u b j e c t t o many e r r o r s . Unsteady State 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 the t e s t gases, and then b r i n g i n g them together a t time z e r o 1 w i t h the l i g h t e r gas on top. One or both o f the c y l i n d e r s i s removed a t a given time, and the 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 then c a l c u l a t e d from the s o l u t i o n d e r i v e d from F i c k J s second law (equation (l . 9 ) ) « I t i s d i f f i c u l t to achieve accuracy i n t h i s experiment due t o the tendency f o r eddies to be created e i t h e r when the c y l i n d e r s are f i t t e d together o r by the a c t i o n o f temperature g r a d i e n t s . - 11 - For porous pellets the analog of the above experiment cannot be readily applied due to the rapidity of the diffusion process i n ,^ases. For example, i f a one cm. diameter pellet of typical pore structure is i n i t i a l l y bathed in one gas", ana at time zero the surface i s flushed with another gas, then SQ.Q^o of the f i r s t gas in the pellet is removed by diffusion i n 10 seconds i f the d i f f u s i v i t y D-g/Ep is 0.01 cm2/sec. (See Appendix III for details of this calculation.) Thus, i t is obvious that some means to extend the time scale i n experiments with small pellets would be very desirable. Currie (6) has developed a non-flow apparatus of this type which can be used only at normal temperatures and pressures for measuring d i f f u s i v i t i e s i n soils and other granular beds. Only rather complex - frequency response techniques, discussed below, are presently available for the measurement of effective diffusion coefficients by transient response methods. Frequency Response and Pulse Methods McHenry and Wilhelm (15) have described a method for measuring the eddy d i f f u s i v i t y i n packed beds, and this apparatus has been used also by Deissler and Wilhelm (l6) to measure both the effective 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 at the entrance and exit of a test section. In the same way, Van Deemter, Zuiderweg and Klinkehberg (17) have applied the delta function (that i s , an ideal pulse) to packed beds i n the form of gas chromatography columns and ion exchange beds. Hougen (l8) - 12 - has pointed out that there i s no r e a l difference between the results obtained by a delta function or by a frequency response method. In the work of Van Deemter et a l the dispersion effects due to 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 transfer c o e f f i c i e n t 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 a l l o ' . v s separation of the influence of each factor on the delta function. 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 quantity 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 may be calculated from i t . The theory on which t h i s approached i s based i s dealt with more f u l l y i n succeeding sections. Comparison of Various Methods "In porous solids there i s a basic difference 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 state, 'iliis difference i s the r e s u l t of the capacitance.effects which manifest themselves i n the unsteady state. 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 indicate. This effect i s allowed for 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 porosity) i s u t i l i z e d . 'Similarly, i f adsorption occurs on the surface of the s o l i d then the volume of gas adsorbed must be added to the porous volume or porosity 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 equilibrium 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 effective d i f f u s i v i t y in an isotropic solid determined by a steady state or unsteady s+ate method. If bulk diffusion is the transport mechanism there is no d i f f i c u l t y i n correctly defining the effective coefficient for either the steady state or unsteady state methods. However, this i s not true when Knudsen diffusion predominates. Consider a simple model of dead end pores of eg.ual length i n parallel i n which Knudsen diffusion i s taking place. The total 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 yield the major flux, but after a time the lower flux i n the smaller pores w i l l result i n larger concentration gradients, which w i l l eventually result i n the flux from the smaller pores equalling or exceeding that from the larger pores. Hence, an unsteady state experiment i n the Knudsen regime may yield a d i f f u s i v i t y which varies with time. An interesting aspect of this latter conclusion arises because the majority of 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 effective. The steady state method i s insensitive to the' resistance which may occur in dead end pores, while the unsteady state method i s potentially capable of allowing for this resistance. The unsteady state method w i l l give a d i f f u s i v i t y which is some average value cf a l l pore resistances, and the a b i l i t y of this value to describe the rate of a diffusion 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 application to the data of simple solutions of the diffusion equation applicable at longer times. Thus, i n this case, the - i n - d i f f u s i v i t y obtained from such experiments may b e expccLed to be weighted in favour cf the small pores i f Knudsen diffusion predominates. D. OBJECTIVES OF 'THE PRESENT WORK On the basis of the foregoing comparison of methods for obtaining a value of the effective d i f f u s i v i t y , i t is apparent that, i n most cases, a diffusion GoSffieiint Obtained from an unsteady-state experiment i n v h i G h a l l the pores contribute to the diffusional process may well b e a better value for use i n chemically reacting systems. In many instances, steady- state experiments may also give suitable values, but this cannot b e assumed without considerable knowledge of the particular porous structure. It would be useful 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 frequency- response measurements, while giving the advantages of an unsteady-state method and which could be applied to a representative sample of pellets vithout requiring special shaping. It might be possible to make use of such a technique to follow changes in catalyst diffusional behaviour with age. Recent advances in the theory of transport processes in chromatographic columns suggest that i t might be possible to interpret pulse dispersion results in such a way as to yield an' effective diffusion coefficient. The primary objective of the present work was to attempt the development of a pulse method as a means of measuring effective d i f f u s i v i t i e s of gases i n porous pellets, a technique not previously reported. A secondary objective was to be the investigation of the use of unsteady state flow methods for measuring the binary diffusion coefficient 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 unit tortu- osity 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 effects would aid i n making possible measurements at widely varying temperatures and pressures, as does the freedom i n choice of concentration measurement. - 16 - I I THEORY A. DERIVATION OE VAN DEEMTER EQUATION Height Equivalent to 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 "height equivalent to a t h e o r e t i c a l p l a t e " (HETP) v I n a gas chromatograph column a narrow band' of sample gas i s i n j e c t e d i n t o a stream of c a r r i e r gas which passes through the column to a d e t e c t i n g d e v i c e . The components o f the sample have d i f f e r i n g r e t e n t i o n times i n the column depending upon the p r o p e r t i e s o f the gas component and the l i q u i d s t a t i o n a r y phase i n the column. I t i s obvious t h a t a column which r e s u l t s i n a broadening o f the pulse i s d e t r i m e n t a l to the s e p a r a t i o n d e s i r e d , and the height equivalent, to a t h e o r e t i c a l p l a t e (HETP) which i s def i n e d below i s a measure o f the 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 obtained by p o s t u l a t i n g t h a t the mechanism o f pulse broadening i s caused by equilibration o f the s t a t i o n a r y m a t e r i a l i n a given p l a t e w i t h the mobile gas phase which then passes on to the next p l a t e . A l i n e a r a b s o r p t i o n isotherm W C = C T (where C = mobile phase c o n c e n t r a t i o n , = s t a t i o n a r y phase concentration) i s assumed and a m a t e r i a l balance around the nth p l a t e (see Figure 1.2) w i t h an increment o f gas flow dU y i e l d s , du(cn_1 - cn) = (v +v;v)dcn (1.15) from which i s obtained, d C n = Cn-1 - C n (l.l6) dU V + Wv - 17 - + '8 Figure 1 .2 ' Model For Derivation Of Plate Theory where V = volume of mobile phase in plate and v = volume of plate stationary phase. Assume that a l l the pulse gas is i n i t i a l l y i n stage 1 yielding an i n i t i a l gas concentration C . Applying n = 1 to equation ( l . l 6 ) , C n = Ci, and C n _ i _ = 0, - d C i iU (1.17) V-p where Vp = Volume of plate = (V + Wv). On integration, (1.17) gives, C i = K exp / - U \ (1.18) V. - 18 - When U = 0, = C , and. t h e r e f o r e K = c ' i n ( l . l 8 ) . Hence, C i = C exp I - _U_ \ (1.19) \ V J Wow app l y i n g the above r e s u l t to equation (l.l6) w i t h n = 2, dCg. + C ^ = C _ e x p / - U _ \ (i.20) * U V P V P I w Equation (1.20) can be solved by use of the i n t e g r a t i n g f a c t o r , exp^ + U C s e x p / y V exp - U_ + U_ dU = C ' U_ + K (1.21) l V j V V V V * ' P P P P When U = 0, C2 = 0, and so K = 0, y i e l d i n g the r e s u l t , C 2 = c ' U_ exp / - U_ \ (1.22) Kence by co n t i n u i n g t h i s process to the nth sxage V P c„ = c ' i/ 1" 1 , exp /- y_ \ (1.23.) ' n (n-£)!V p-l ~ " * ( ~ 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 of p l a t e s t h i s d i s t r i b u t i o n approaches a Gaussian or normal d i s t r i b u t i o n . The mean o f the above d i s t r i b u t i o n i s U_ and the varia n c e i s U_ (th a t i s , V P V P t h a t the ( M e a n ) 2 / ( s t a n d a r d d e v i a t i o n ) 2 =/U_\ / U_ = U_ 1V ] V V \ VI P p Now U i s the t o t a l volume o f gas which has flowed, and i s the volume o f a t h e o r e t i c a l p l a t e , so t h a t when the mean approaches the end o f the column the (mean)2/©*2 = no. o f t h e o r e t i c a l p l a t e s . By d e f i n i t i o n , t h e r e f o r e , EETP = L \ 2 (1.2U) \ mean / where L i s the column l e n g t h . 19 Measurement of RTLTP For a large number of stages the output can be assumed to be a Gaussian distribution, and the mean and variance may be read directly from the record of the output at the end of the column by using the properties of the Gaussian distribution shown in Figurel.3. This represent- a t i o n i s not s t r i c t l y c o r r e c t , i n t h a t the output is Gaussian w i t h respect to position in the column, while the recorded profile at the end of a column is with respect to time. However, i f the time of purge of the pulse i s small relative to the time of the mean, the error i n reading this time distribution compared to the distance distribution is negligible. Figure 1.3 Gaussian Distribution 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 inequality are squared and multiplied by L, on rearranging 16 Lcr2 « L or L » 16 HETP mean2" 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 record. Input Pulse D i s t r i b u t i o n The derivation of the HETP assumed that 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 effect would be noticed i n the output. I t has been shown by Van Deemter (17) that the effect 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 , A r v,7H < °'5 ( 1 ' 2 5 ) where Ac i s the volume of gas i n the i n i t i a l pulse and n i s the number of theoretical plat.es. Rate Theory The theoretical plate model does not attempt to explain the rate processes occurring i n a chromatograph column, out r e l i e s on the fact 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 is molecular d i f f u s i o n i n the mobile phase. Longitudinal d i f f u s i o n i n the - 21 - s t a t i o n a r y phase can g e n e r a l l y be ignored as the s t a t i o n a r y phase i s discontinuous i n a packed bed, and, i n a d d i t i o n , the d i f f u s i o n c o e f f i c i e n t i s s m a ll i n t h i s phase. There i s a group o f l i t t l e understood processes which, cause a , pulse to d i s p e r s e due to the flow p a t t e r n i n the packed bed. F o r t u n a t e l y , i n a deep "bed these a b e r r a t i o n s arc of a s t a t i s t i c a l nature v h i e h tend te r e s u l t i n a Gaussian d i s t r i b u t i o n as obtained f o r molecular d i f f u s i o n , so tha t they can be grouped together i n a term described as the eddy d i f f u s i v i t y . In the work of Van Deemter et a l (17) the 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 considered to be caused by the d i f f e r e n c e m flow paths between p a r t i c l e s . These concepts are discussed i n the f o l l o w i n g s e c t i o n s . A pulse broadening mechanism analogous to the t h e o r e t i c a l p l a t e mechanism described e a r l i e r can a l s o occur i n the chromatograph column. I f a r e s i s t a n c e e x i s t s preventing e q u i l i b r i u m between the mobile and s t a t i o n a r y phase, then the degree o f pulse broadening caused by the capa c i t a n c e o f the s t a t i o n a r y phase i s increased due to the f a c t t h a t although l e s s m a t e r i a l enters the s t a t i o n a r y phase the time taken to get out again causes the pulse to broaden more than would be the case f o r the e q u i l i b r i u m s i t u a t i o n . Lapidus and Amundson (2) have derived an expression based on a d i f f u s i o n model to d e s c r i b e the c o n c e n t r a t i o n p r o f i l e f o r the c o n d i t i o n s where a pulse gas passes through 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 isotherm between the gas and s t a t i o n a r y phase. 'The pulse i s not assumed to be i n e q u i l i b r i u m w i t h the s t a t i o n a r y phase, due to 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 molecular and eddy c o n t r i b u t i o n s i s -22 - characterized by a dispersion coeff ic ient , D , and i s assumed to occur i n ii the mobile phase, but not i n the stationary phase. The model i s shown i n Figure l.k. A material balance around the element 6x yields F i iCx = F 2 D L * 2 C I - FiU dC± + OC (WC2 - Ci) (1.26) at d x 2 d x F 2 JCZ = o « ( C i - WC2) (1.27) <H VThere W is the equilibrium constant between the mobile and stationary phases. If the stationary phase is a porous sol id V/ can be replaced by 1 . Fi C. Fa dx I ' * t d p i - -• WCs a x uCi MOBILE STATIONARY PHASE PHASE Figure l.k Mathematical Model for the Column - 2$ - For a small pulse i n j e c t i o n time, t , and an i n i t i a l pulse c o n c e n t r a t i o n C , Lapidus and Amundson (20) obtained the f o l l o w i n g s o l u t i o n to the above equations, C_i = x t 0 exp / -(x - u t ) 2 - o t t \ + Xh t n exp / - ( x - u t 1 ) 2 \y' C 2t^|YruLt ^ h D L t F i J 2titf TH^t* \ 4 D]_t / F ( t * ) d t 1 (1.28) where F ( t ) 1 =/ 2 Wt 1 V^exp/ o*W ( t - t 1 ) - c c t i \ l a / 2 / 2 V / o 1 \ \ F 1 F 2 ( t - t 1 ) J 1 F 2 F i / \ / F i F 2 J W (1.29) where t i s time, t , time o f i n i t i a l pulse w i t h c o n c e n t r a t i o n C , x i s dis t a n c e along the column, and I i i s the h y p e r b o l i c Bessel f u n c t i o n . I t has been shown by Van Deemter et a l (17) t h a t the above s o l u t i o n can be reduced to a Gaussian 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 are th a t the height o f a t r a n s f e r u n i t Fj.u^CL, the height oc o f the bed, and the l o n g i t u d i n a l mixing stage 2 DL « L. E s s e n t i a l l y , u these requirements s t a t e t h a t the 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 which case the co n c e n t r a t i o n p r o f i l e reduces t o , C i = fltp exp / L / u - Bt \ where 1 = 1 + Fg_ W, o*i = 2 DLL and C^ 2 = 2 F 2T, (1.30) £ F i ~ 3 FiW^u This i s a Gaussian d i s t r i b u t i o n w i t h mean L/u or Bt and varia n c e C i 2 + C 2 2 . As mentioned above, the varia n c e o f a Gaussian d i s t r i b u t i o n i s composed o f the sum of the i n d i v i d u a l v a r i a n c e s , so equating the r a t i o cf2 f o r the above s o l u t i o n y i e l d s the f o l l o w i n g which can be combined mean2" w i t h the HETP d e r i v a t i o n o f equation (1.2*0. - 2k - 2. tfi2 + c ^ 2 = 2 D| L / u 2 \ + / 1 \ 2 2 F 2 L / u 2 \ = 0 ' ( L / u ) 2 ~u^ \ L 2 j [ l + _ l 2 _ I <*FiW2u ( L 2 ] rnearr2" (1.31) \ F i W / = 2 D L +/ ' 1 \ 2 F i u = cr 2 x L = HETP (1-32) — ' { J - ( l + WFi ] oc mean 2 \ F 2 y Trie d i f f u s i v i t y i n equation (1.32) r e f e r s to any a x i a l mixing Mechanism which O Q G U K ' H i n the mobile phase,so t h a t i t can 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 the eddy d i f f u s i v i t y . I t was pointed out by Van Deemter t h a t i n the laminar r e g i o n the eddy d i f f u s i v i t y i n a packed bed i s probably created by the d i f f e r e n c e i n flow patterns i n the bed. A p e r f e c t l y uniform bed thus conceivably has no eddy term. The molecular d i f f u s i v i t y c o r r e c t e d f o r the path lengthening i n a packed bed by a t o r t u o s i t y f a c t o r , and the eddy d i f f u s i v i t y D^'are commonly assumed to be a d d i t i v e , so tha t the d i s p e r s i o n c o e f f i c i e n t Dv i s g i ven by, D L = + DL* (1.33) where DjJ^ depends on the a x i a l d i s p e r s i o n caused by the flow p a t t e r n s . This assumption i s discussed by Klinkenberg and S j e n i t z e r (19) and they 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 the theory adequately describes the r e s u l t s . 'The abundant work on gas chromatography appears to lend support to the assumption 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 high flow r a t e s , the molecular d i s p e r s i o n becomes n e g l i g i b l e compared t o the t u r b u l e n t d i s p e r s i o n , so t h a t the o v e r a l l d i s p e r s i o n i s the same as the flow d i s p e r s i o n and can be c a l l e d the eddy d i f f u s i v i t y . At low flow r a t e s , e.g. p a r t i c l e Reynolds numbers 1, the eddy d i f f u s i v i t y can be represented according to Van Deemter et a l by the expression D L* = u d . Thus equation (I.32), a f t e r i n t r o d u c i n g the - 25 - concept of additive coefficients stated in (1.33)> takes the form, HETP = 2 #a + 2 D-n + f 1 "* 2 1 + V/Fx E 2 2 Fm (l.3>0 The quantity i s reported bo decrease with larger diameter pellets, having a value of about 8 for 200 mesh, and practically zero for ~$0 mesh, particles. ' Mass Transfer Coefficient and Effective Diffusivity Diffusional resistance to mass transfer from the mobile phase to the interior of the pellets (stationary phase) is made up of two parts, the f i r s t being due to resistance in the mobile phase and the second to the resistance within the pellets. The solution of Lapidus and Amundsen (2) used by Van Deemter et a l (17) (equation 1.28) treats the resistance in terms of a mass.transfer coefficient. Van Deemter et a l treated the two resistances as separate mass transfer coefficients which could be combined by the resistances-in- series rule. (A mass transfer coefficient i s really a conductance rather than a resistance hence the reciprocals•are the additive property.) 1 = 1 + W = 1 + W (1-35) OC e<x <XZ k ^ A p k 2A P Where oft. is the mobile phase coefficient/unit vol. of bed, ofe i s the stationary phase coefficient and ocis the overall coefficient with k i and k 2 being the corresponding surface mass transfer coefficients. W i s the partition coefficient, which is necessary in gas chromatography because the diffusion i n the stationary phase occurs i n a liquid arid the liq u i d - phase concentration gradients are expressed i n terms of equivalent equilibrium gas phase concentrations i n order to make equation (1.35) - 26 - consistent. In diffusion in porous solids, the effective d i f f u s i v i t y is defined on the basis of the i n t e r s t i t i a l gas concentrations and so the partition coefficient becomes a quantity relating i n t e r s t i t i a l concentrations to stationary phase concentrations, that is 1 , where 6 is the pellet r» porosity. External Mass Transfer Coefficient Based on the work of Ergun (21) Van Deemter et a l suggested the use of the following correlation for the mass transfer coefficient in the mobile phase, k i = 2_5_ Dp. A cm. /sec. (lo6) 6 F i 9 \Taeve k i is the mass transfer coefficient per unit area and A is the P surface area per unit volume of bed, «*i = Ap k i sec." 1 (1.37) In a bed of spherical particles of diameter d , the surface area per unit volume, A^, is given by the following, i f the bed porosity is F i , A p = 6(1 - F i ) / d p (1.38) Internal Mass Transfer Coefficient In this work i t is desired to obtain the effective d i f f u s i v i t y in the porous pellet, and so i t is necessary to find a relationship between the mass transfer coefficient, cfe = k 2 A^, and the effective d i f f u s i v i t y . Such an expression was given by Van Deemter et a l without derivation, but i t can be obtained in a pellet of radius R as shown below. Let * ± ) = Ka (C s - C ) (1.39) \ ar / r=R vhere C s and C a v g are the surface and average concentrations in the pellet respectively. 27 - Crank (22) (page 2J3) has obtained solutions of the diffusion equation for a spherical pellet 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 + 2 RC„ V (^lV^ sin ( nTTr \ exp / - Dr:n 2TT 2t \ ~7f~r~ 2- n \ R J [ R / n=l (1.1*0) Similarly the average concentration i s given by: cavg - C g - 6 C exp R 2 (1A1) n=l If t is large then only the f i r s t terms of the series solutions need be considered. This amounts to suggesting that C g approaches C a V g and in view of the rapidity of gas diffusion as demonstrated i n the example given i n the introduction, this assumption would appear to be reasonable. From (1.1*0), CA_ - Cs From (l.4l), 1 = C A - C s = -2_R_ C S 7Tr 1 = c avs; C s i n | 7 £ r j e x p ^ - D F. rr 2 t j • 6 exp / - D W T 2 t \ T P I " " i ^ 2 " ) Divide (1.42 by (1.4-3) = 2 R TT 3in 77 r = C S Cavg c s CA (1.1*2) (1.1*3) (1.44) C s - C a v g Let C s - C = A r 6 r H , thensubstitute i n equation (1.39) R D E R 7 T Sin TTr 3r R Taking the limi c as A.? Cc - C a v g . - ^2 (C s - C a Vg) limit 3k2 bin A r 7Tr " R TTr R A r (1^5) - 28 - o r 2 TT 2DE = 1 where d p = 2 R kad p therefore k 2 = 2 ]-p2 £E (l.t6) 3 d p The mass transfer per unit volume of bed efe i s obtained by multiplying k 2 by the surface area per unit volume A p given by equation (1.56). 'The two mass transfer coefficients could be combined using equation (l.35)• However, at this stage, the conditions of the present work differs from that of Van Deemter et a l . If k i is large compared to k 2, then when the inverse is summed in equation (1.35)* ma7 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 less, and so i f reason can be found to neglect k j in the laminar region, i t need not be considered i n evaluating mass transfer i n the turbulent flow region. If we as.sume the effective d i f f u s i v i t y in the porous pellet is l/5 of the molecular d i f f u s i v i t y as suggested in the introduction, and assume a bed porosity of O.k, then the ratio of k i / k 2 from equations (I.36) and (1.1)5) is around 28. Tims, at most, the resistance outside the pellet makes a yjo contribution and can be ignored. The derivation for k 2 was made on the assumption of a step change in the surface concentration, but in this work a Gaussian curve is expected to describe the surface concentration. It would be desirable to have a derivation applicable to other surface functions, or at least to the Gaussian function. An attempt to obtain an alternate expression for k 2 using a ramp surface function could not be made to reduce to the expression of (1.46), because the exponential functions would not cancel -29- out as i s tne case i n the step y i e l d i n g equation (1.4'-:). Thus, a further degree of approximation results from using (1 . 4 6 ) , but i t i s prooable chat an experimental constant other than 2/577"2 can be found which would y i e l d a satisfactory 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 ) . Ignoring k i , and combining (1 .35) ( l « 3 0 ) and (1 .46) OC = 2 / 3 7 1 2 D£ 6 ( I - F i ) 3 p d p Substituting (1 .47) into (1 . 3 4 ) , HETP = 2 tfdp + 2 D 3 + (1.47) 1 +VfFi TP J ? 2 2 2 2 ? id,, u Tff* U E ( l - F i t (1.43) Making the substitutions, F i = eB F 2 = 1 - F i = (1 - 6 3) w = - l — HETP = 2 ̂ d + 2 DT P T 1 1 + JL 2 6* ^ u 4 7T D „ U - i v ) , ( l > 9 ) £p(l - 6 B ) This i s the equation derived by Van Deemter et a l (17) and i t may be observed to be of 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 of the behaviour of th i s equation i s shown i n Figure 1.5 indicating the physical significance of the constants A, B and C. - 30 - Trie magnitude of the A r,erm, vhich may he called the eddy cilfCusivity term, depends largely on the value of As poin'cea out by Van Deemter et a l , % is expected to be quite small for lar^e packing sizes, e.g. 30 mesh diameter or larger. Therefore, i t is l i k e l y that the eddy di 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 effective d i f f u s i v i t y , or Cu, term w i l l predominate. The effective d i f f u s i v i t y , or Cu term, is of primary interest and so this term w i l l be considered i n more detail. It was mentioned earlier that a lower mass, transfer coefficient or effective d i f f u s i v i t y causes the pulse to broaden, and in the effective d i f f u s i v i t y term a lover effective d i f f u s i v i t y does indeed result i n a larger HETP, which is a measure of the amount of pulse dissipation. Similarly, a non porous pellet w i l l have zero porosity (£p), and so the pellet capacity term becomes zero. 'This implies that for a bed of non porous pellets the following equation should apply: HETP = A + B/u The influence of the bed porosity ^ on the magnitude of the effective d i f f u s i v i t y term is very small over the range of porosities commonly found i n random pellet packings. On the other hand, the pellet diameter which has an exponent of 2 has a strong influence on the value of the effective d i f f u s i v i t y term, and suggests that this term w i l l be much more easily evaluated for larger pellets. •Typical Values of the Effective Diffusivity Term (C) In Table 1 . 1 following,some values of the effective d i f f u s i v i t y term are calculated for some typical porous pellet properties and dimensions, - 31 - and f o r a range o f p o s s i b l e e f f e c t i v e d i f f u s i v i t i e s . The v e l o c i t i e s where the molecular d i f f u s i o n term equals 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 (1.4-9) ( i . e . the minimum shown i n Figure 1.5) a-re a l s o c a l c u l a t e d , 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 cm 2/sec, an assumed molecular d i f f u s i o n c o e f f i c i e n t o f 0.2 cm 2/sec, and a t o r t u o s i t y f a c t o r o f 1.33. I t can e a s i l y be shown t h a t , 2DR ^ i n 1 1 + 6 D € . 3 dp 2 p v _ . ^ 2 2 D e ( I - eB) v 2 B ^ G Because the v e l o c i t y a t the minimum i s p r o p o r t i o n a l to l / d the p a r t i c l e Reynolds number, d p up / j i , a t each o f the minima obtained 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 the same, and has a value o f around 6 i f a value o f 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) (% = 0.4, E.p = 0.33) * ____d cms. E f f . D i f T r ^ - - ^ . ^ ^ 1 • 5 .25 .1 .1 cm 2/sec .0375 .00938 .00235 . . O O O 3 7 5 .01 • 375 .0938 .0235 •00375 .001 3-75 .938 .235 .0375 V e l o c i t y a t minimum from u m i n = fB^cS/sec Icj .895 1.79 3.16 8.95 From the f i r s t three l i n e s o f c a l c u l a t i o n s i n Table 1 . 1 , i t i s obvious 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 diameters a pulse d i s p e r s i o n method may break down because the e f f e c t i v e d i f f u s i o n E D D Y D I F F U S I O N C O N T R I B U T I O N INTERSTITIAL VELOCITY u - y - * — Figure 1.5 Typical Plot of 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 is 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@ia" *aeael (CUsQusflwet i n the fellowiny seQtien ) applies, ae suggested by McHenry and Wilhelm (15)* then the eddy d i f f u s i v i t y Djf i s ^iven by DTJ* = l/2 ud p for superficial Reynolds numbers from about 10 to 4 0 0 . 'This expression compares favorably with that suggested by Van Deemter et a l ( D L ' = % ud ). However, the constant given as l / 2 actually varied with Reynolds number from l / l . 8 to l / 2 . 2 i n McHenry and Wilhelm's experimental work, and this d r i f t would tend to cause errors i n determining the effective d i f f u s i v i t y term i n equation (l.50) i f pellets with high d i f f u s i v i t y and small diameter were used. As mentioned earlier, the velocities at the bottom of the 'fable 1.1 show the location of the minimum i n Figure 1.5* a^d correspond to a constant particle Reynolds number of about 6 which is well above the flow range considered by Van Deemter et a l . Nevertheless, i t is apparent that i f the effective d i f f u s i v i t y term is to be maximised relative 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 relating H to u: for 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 equations i n v o l v i n g A, L) and C can "bo obtained i n the usual way, that i s , 1. Sum a l l the n aata p o i n t s : nA + \ + c£u = £ l (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 the c o e f f i c i e n t o f C: A£U + nB + c£u 2 = £liu (l.5Id) E l i m i n a t i n g A and B from (1.51b), (l.51c), and (l.51d) T(H) - n X(Hu) \ / Z H - n 2 § ) C - S - ^ 1 g / V^^-/ (l.5ie) ' I(u) - nj££ \ /2u - n l Hence, (l.51f) m~ir J \WW u 4tr) n£(|) , N2 v w Z(*) A = Il-i - L i u - D i g (l.51g) n 3. LONGITUDINAL DISPERSION COEFFICIENTS 'There are two models g e n e r a l l y used to d e s c r i b e the l o n g i t u d i n a l d i s p e r s i o n i n packed beds o f non porous s o l i d s . The d i s p e r s e d plug flow model superimposes co-ordinates moving a t the average stream v e l o c i t y , u, on the 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 equation. 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 per u n i t area of 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 plane - 5 5 -a t ;c = 0 i s given D y ( 2 2 ) CA = M exp / - x 2 \ (1.52) 2*/ tfDLt \ 2 ( 2 D L t ) y which i s a Gaussian 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.ft. With plug flow a t a v e l o c i t y u t h i s becomes CA » M . . exp / - ( L - u t ) 2 \ (1.53) JHTSSZT [ 2(2D Lt) J w i t h mean L and vari a n c e 2 D j t . A second model i s the "pe r f e c t mixers i n s e r i e s " which 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 e a r l i e r , a g a i n y i e l d i n g a Poisson d i s t r i b u t i o n , w i t h a mean o f tL and vari a n c e U , where U i s the volume o f gas which has passed through, and V the v L L volume o f each mixer. The number o f p e r f e c t mixers i n s e r i e s i s equivalent t o the number o f stages and i s g i v e n by U / V ^ as bef o r e . Equating the r a t i o (mean) 2 = number o f mixers, N, f o r both v a r i a n c e models, VL) = N = L 2 = uL (1.5*0 2 D L t 2 D L I f one mixer i s assumed to correspond to each l a y e r o f p a r t i c l e s , then the number o f mixers = L/dp, and t h e r e f o r e , D L = 1/2 u d p (1.55) 'The o n l y work, (apart from a few data p o i n t s i n the laminar flow regime obtained by Carberry and B r e t t o n (23)) which has been c a r r i e d out w i t h gases f o r the purpose 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 by McHenry and Wilhelm (15), u s i n g a frequency response technique.' They found t h a t over 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 the above r e l a t i o n s h i p held reasonably w e l l . Other facto'rs which i n f l u e n c e the 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 are buoyancy e f f e c t s which may be expected when flow r a t e s approach laminar c o n d i t i o n s , and w a l l e f f e c t s which I-Iiby (2^) has shown g r e a t l y increase the apparent d i s p e r s i o n c o e f f i c i e n t . V e l o c i t y P r o f i l e C o n t r i b u t i o n Taylor (25) has separated the v e l o c i t y p r o f i l e c o n t r i b u t i o n 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 f l o w . Taylor found t h a t f o u r d i s p e r s i o n regimes e x i s t i n pipe f l o w . The f i r s t i s due t o molecular d i f f u s i o n which predominates a t low flow r a t e s . As the v e l o c i t y i n c r e a s e s the p a r a b o l i c p r o f i l e c o n t r i b u t e s 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 p u l s e , but the molecular d i f f u s i v i t y i s a b l e to l a r g e l y remove the r a d i a l c o n c e n t r a t i o n p r o f i l e s . This y i e l d s an eddy d i f f u s i v i t y , Dv* = K u a R 2 (1.56) where u i s the mean v e l o c i t y , D-g the molecular d i f f u s i v i t y , and R i s the pipe r a d i u s . I t may be noted t h a t h i g h molecular d i f f u s i v i t y gases reduce the eddy d i f f u s i v i t y i n t h i s r e g i o n . 'The constant K i s i j _ g f o r p i p e s , but A r i s (26) has shown t h a t the constant depends upon the geometry o f the system. The range o f a p p l i c a t i o n o f the above regime i s described by Turner (27) as, 7 DB « u < k D B Lx (1.57) R R 2 where L 1 i s the 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 the p u l s e . Within the above l i m i t s the molecular d i f f u s i v i t y c o n t r i b u t i o n i s n e g l i g i b l e so DL* = D^. Since a Gaussian d i s t r i b u t i o n i s assumed we can say t h a t 95c/> of the pulse e x i s t s i n fou r standard d e v i a t i o n s . - 5 7 I f L i s defined as k<J then, L = K J 2D Lt S u b s t i t u t i n g f o r D L from (I.56) and s e t t i n g t = ^ , where L i s the l e n g t h o f column (or mean), the upper l i m i t becomes, u « 2 162 K X L , D B " I 2 For pipes K i = 1 g i v i n g an upper l i m i t o f SB" u « 10 LD B R 2" Turner (27) i n h i s d e r i v a t i o n obtained 5 LD/R 2 f o r the R.H.S. of (1.58), apparently because o f the omission o f a f a c t o r o f 2 i n d e f i n i n g L . The residence time i s introduced i f the v e l o c i t y i s r e p l a c e d by L then, t L = u « 10 *fs_ or t » _Rf_ (I.58) t R 2 10 D B In other words, a pulse must be allowed t o flow f o r some f i n i t e time a f t e r the i n j e c t i o n before the eddy d i f f u s i v i t y i s d e f i n e d by equation (1.56). I f the d i s p e r s i o n i s measured very s h o r t l y a f t e r i n j e c t i o n (a time l e s s than R 2/l0 Dg), then the eddy d i f f u s i v i t y i s g i v e n by some undefined f u n c t i o n . 'This l a t t e r f u n c t i o n does not i n c l u d e the molecular d i f f u s i v i t y , and so resembles the f u n c t i o n f o r the 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 the t u r b u l e n t flow regime i n pipes has been d e a l t w i t h by Taylor by use o f the u n i v e r s a l v e l o c i t y p r o f i l e . This approach y i e l d e d DL* . = 7.1̂  B u ^ f " (1.59) where f i s the Fanning f r i c t i o n f a c t o r . - 38 - The a p p l i c a t i o n o f the Taylor d e r i v a t i o n t o packed beds has been somewhat l i m i t e d , although B i s c h o f f and Le v e n s p e i l (28) have considered the o v e r a l l p r o f i l e i n a packed bed. Inasmuch as the v e l o c i t y p r o f i l e i n packed beds approaches plug flow the c o n t r i b u t i o n o f the o v e r a l l p r o f i l e to a x i a l d i s p e r s i o n i s s m a l l . Saffman (29) has developed 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 are j o i n e d i n a random manner. Assumptions must be made regarding the l e n g t h and diameter o f the c a p i l l a r i e s , Saffman de r i v e d the f o l l o w i n g expression t o cover the t r a n s i t i o n from laminar to an eddy regime and by assumming a c a p i l l a r y l e n g t h t o diameter r a t i o o f 5 the experimental r e s u l t s o f Hiby (2k) f o r l i q u i d were f i t t e d , 2 * uL [ L o g e 5 uL - 17 - 1 uL 6 ~ L 2 D 3 1 2 8 I2 D J + Dg + k_ D 3 + 0 X 9 (I.60) 'The value o f the t o r t u o s i t y X obtained by Saffman from t h i s model approaches 3 t f l k, c o n s i d e r a b l y higher than the t o r t u o s i t i e s normally encountered i n beds o f spheres. Saffman's model appears to show the most p o t e n t i a l a t present i n d e s c r i b i n g the a x i a l mixing i n packed beds, but as i n Taylor's work on pipes assumptions made concerning the nature o f the flow l e a d t o d i f f e r e n t s o l u t i o n s . Hence, the b a s i c flow mechanism must be understood before one can apply the ap p r o p r i a t e s o l u t i o n from the model. - 39 - I I I APPARATUS A. DEVELOPMENT The i n i t i a l work to t e s t the c a l c u l a t i o n o f e f f e c t i v e d i f f u s i v i t i e s by means o f the Van Deemter equation was c a r r i e d out on an apparatus based on a gas chromatograph as shown i n Figure 1.6. A cyclopropane pulse was i n j e c t e d by a chromatograph sample v a l v e i n t o a helium c a r r i e r gas and was detected on a GOV/ MAC model 9238-D thermal c o n d u c t i v i t y c e l l . The flow r a t e was measured w i t h a soap bubble flow meter a t the c e l l o u t l e t , and the d e t e c t o r output was recorded on a Leeds and Northrup -1 to 10 mv rec o r d e r . 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 plane, although i n i t i a l l y a set of r e s u l t s were taken w i t h a h o r i z o n t a l bed, but the s e t t l - i n g of the packing r e s u l t e d i n a channel along the top o f the bed. The f i r s t v e r t i c a l apparatus s u f f e r e d from the 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 operate w i t h i n a l i m i t e d gas flow range (about 50 mls/min.). 2. The small ports i n the sample i n j e c t o r r e s t r i c t e d the flow o f gas. 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 the recorder 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 the pulse output was not broad enough to make accurate measurements o f standard d e v i a t i o n s p o s s i b l e . B. DESCRIPTION OF APPARATUS The apparatus w i t h which the b u l k o f the r e s u l t s were taken i s shown i n Figure 1.7. The shortcomings o f the e a r l i e r apparatus were el 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 CYCLOPROPANE PULSE jGAS THERMAL CONDUCTIVITY DETECTOR SOAP BUBBLE FLOW METER PACKED TEST COLUMN HELIUM CARRIER GAS MOORE* FLOW CONTROL Figure 1 .6 Apparatus For Exploratory Tests - U l - F i g u r e 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 the column e x i i to maintain a s l i g h t p o s i t i v e pressure i n the column ( l / 2 to 2 inches of mercury). This pressure made i t p o s s i b l e to d i r e c t a s i d e stream through a c a p i l l a r y J. o supply the d e t e c t o r a t a f i x e d flow r a t e . At gas flow r a t e s l e s s than the amount needed f o r the de t e c t o r the manostat sup p l i e d a d d i t i o n a l gas, thus r e v e r s i n g the flow d i r e c t i o n between the manifold and the manostat. 2. A sample o r pulse i n j e c t i o n v a l v e was constructed having l a r g e p o r t s as shown i n Figure 1.8. This Figure a l s o shows an experimental pulse i n j e c t i o n system which was used to t e s t the e f f e c t o f v a r y i n g pulse s i z e . J . A Bausch and Lomb 0- 10, 100, 1000 mv recorder w i t h c h a r t speed adjustments from 0.05 t o 20 inches/min. allowed the pulses to be recorded i n such a way t h a t good accuracy could be obtained i n measuring the d i s p e r s i o n o f the p u l s e . The apparatus was set up w i t h the t e s t bed mounted i n a v e r t i c a l plane, and r e s t i n g on a manifold block a t the discharge end. The s i d e stream f o r the d e t e c t o r was taken from the manifold,, and the main column e f f l u e n t gas discharged through the manostat. A p o r t connected to a mercury manometer i n d i c a t e d the manifold absolute pressure. A i r c a r r i e r gas was taken a t e i t h e r 3O70 RE from the b u i l d i n g supply or from a c y l i n d e r o f d r y a i r . The a i r passed through a r e g u l a t o r s e t f o r 22 p s i g . downstream pressure, and then to a flo w meter c o n s i s t i n g o f a sp. g r . I o i l manometer and c a p i l l a r y tube. A 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 bubble flow meters o r a wet t e s t gas meter, so t h a t a wide range o f flows could - 43 - - I r H r l - I" POLYETHYLENE / BLOCK ELLINCER I'D. SILVER STEEL ROD WITH SLOTS FOR jfe 'O'RINGS. _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 covered. No attempt was made to s i z e the c a p i l l a r i e s so that they remained i n t h e i r l i n e a r range. From the flow meter, the c a r r i e r gas passed through a Moore Constant D i f f e r e n t i a l gas flow c o n t r o l and by-pass loop to the pulse i n j e c t o r . The pulse 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 could be adjusted over 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 . Polyethylene tubing was used to supply the c a r r i e r gas, as w e l l as the pulse gas to the i n j e c t o r . The f l e x i b i l i t y o f the polyethylene t u b i n g allowed the i n j e c t o r to be adjusted anywhere on the r a i l without the neea o f p i p i n g a l t e r a t i o n s . A microswitch mounted on the i n j e c t o r was e i t h e r opened or clo s e d a t each movement of the i n j e c t o r s . This a c t i o n operated an event marker on the recorder t o i n d i c a t e the s t a r t o f each run. C. DETECTORS Hydrogen Flame I o n i s a t i o n Detector In equation (1.49), i t was evident t h a t l o v e r e f f e c t i v e d i f f u s i v i t i e s increased the magnitude o f the C term. To take advantage o f t h i s , the hydrogen flame 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 allowed the use o f a i r and hydrocarbons o f any convenient molecular weight, as opposed to the need f o r hydrogen o r helium (which have hig h d i f f u s i v i t i e s ) as one of the gases i n thermal 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 , the hydrogen flame d e t e c t o r i s l i n e a r over a 5 decade range, and the h i g h s e n s i t i v i t y allows the use o f extremely small pulse volumes. The d e t e c t o r was constructed from the c i r c u i t d e scribed by Harley, Nels, and P r e t o r i u s (30) and i s shown i n Figure 1.9. A power supply was a l s o constructed to supply the d e t e c t o r , however, the AC f i l a m e n t supply was found to create excessive n o i s e i n the output so the 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 Detector - 1,6 - Wo output could be obtained i n i t i a l l y from the 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 the vol t a g e s on the 6 SN 7 tube were found t o be outside the range i n which a response could be expected. To c o r r e c t t h i s problem i t was necessary to change the two load r e s i s t o r s from 10 Ki"L to 100 KSL . I t i s concluded "chat a m i s p r i n t has occurred i n the o r i g i n a l p u b l i c a t i o n . The a c t u a l i o n i s a t i o n or combustion cheuribar was constructed t o minimize the holdup time of the primary a i r c o n t a i n i n g the t r a c e s o f pulse gas from the m a n i f o l d . The a i r - e n t e r e d through the annular space i n the g l a s s tubes and j o i n e d w i t h the hydrogen before passing through the s t a i n l e s s s t e e l o r i f i c e which 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 flo w c o n t r o l l e r and a rotameter a t a r a t e o f about 150 mis/min. Lower flow r a t e s increased the d e t e c t o r output but i n the extreme, the flame became uns t a b l e . A i r and pulse gas a r r i v e d through the c a p i l l a r y a t the r a t e o f about 0.7 mis/sec. The volume of the d e t e c t o r a i r s ide and supply tubes from the manifold was estimated a t 0.2 mis, g i v i n g a time l a g o f about 0.3 seconds. I n i t i a l l y , the flame o r i f i c e was made f l u s h w i t h the metal e l e c t r o d e , but the heat from the flame caused the glassware to crack and so the o r i f i c e was modified by adding about 1 l/2" of l/8 i n c h s t a i n l e s s s t e e l tube. ' The whole assembly was held on a rubber bung, thus supplying the i n s u l a t i o n f o r the platinum e l e c t r o d e which was supported by a heavy wire i n s e r t e d i n the rubber. Shielded cable connected the d e t e c t o r t o the e l e c t r i c a l system, and a grounded copper chimney s h i e l d e d the flame from draughts. Other m o d i f i c a t i o n s to the reference c i r c u i t ( a l s o shown i n Figure 1.9) i n c l u d e d a f o u r t h p o s i t i o n on the 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 zero 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 runs. 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 the hydrogen tune and the manifold to reduce the noise i n the d e t e c t o r caused by dust. The de 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 output, probably caused by dust i n the secondary a i r , but no attempt was made to c o r r e c t t h i s . Thermal C o n d u c t i v i t y Detector A "Gow mec" model 9238D tungsten w i r e thermal 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 the recommended conventional a u x i l i a r y c i r c u i t s . A 6v. b a t t e r y s u p p l i e d the curr e n t f o r the d e t e c t o r and the event marker on the recorder. 'The output to the recorder was f i t t e d t o an attenuator having 1, 2, 5, 10, 50, 100 and 500 r a t i o s , but o n l y the 1, 2.and 5 p o s i t i o n s were needed i n the pulse apparatus. 'The reference s i d e o f the d e t e c t o r was s u p p l i e d through a needle valve from the 22 p s i g a i r l i n e , and a s m a l l bleed maintained. The c a p i l l a r y s u p p lying the measuring s i d e o f the d e t e c t o r from the manifold was s i z e d t o g i v e approximately 46 mls/min. o f a i r a t l / 2 " Hg gauge manifold pressure. - 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 pel le t diameters and higher flow rates; (b) measure the effective d i f f u s i v i t y i n some samples of porous pellets using the pulse method, and (c) compare the effective 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 carried out by injecting methane and hydrogen pulses i n beds containing non-porous pe l le t s , while (b) was an obvious extension of (a) to porous p e l l e t s . The well-tested steady state method was selected to obtain an independent effective d i f f u s i v i t y value. It was convenient, however, to develop a specific solution of the diffusion equation to f i t pellets with curved faces. The details of this section of the work are recorded i n Appendix 1. B. NON POROUS PELLETS IN PULSE APPARATUS A simple gas chromatograph assembly was used for some early exploratory runs with a cyclopropane pulse i n an helium or a i r carrier gas flowing through beds of 2 mm. glass spheres. These results were discarded due to limitations of the apparatus, which included a limited supply of the 2 mm. glass beads necessitating short beds, as well 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. - k9 - Because the value of the quantity " C " i n equation (l.50), HETP = A + B + Cu u was not found to he zero i n the exploratory work with non porous pe l le t s , runs 5 0 to 5 5 were designed to determine the magnitude of this term, and to investigate ways of minimizing i t . To check the p o s s i b i l i t y that this effect was caused by a high velocity "by-pass" flow at the w a l l , run 5 0 was made with a 5 cm. diameter column packed with the 0 . 2 0 8 cm. lead shot, and having a maximum particle Reynolds number of 2 . Run 5 0 w a s different from the other runs i n that a higher pressure was used, giving a lower d i f f u s i v i t y . Run 5 1 w a s made with a 2 . 5 cm. column packed with the lead shot to see i f part ic le to tube diameter ratio had much influence on the wall effect . Run 5 2 duplicated run 5 0 , but used a higher Reynolds number range, and normal column pressure. A run designated 51D w a s also made on the 2 . 5 cm. bed, but f ive doughnut rings were distributed evenly down the column i n an attempt to eliminate the wall effect . 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 inject ion system using a solenoid valve, which allowed varying pulse sizes , was introduced i n this 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. balls to see i f a tube/pellet diameter rat io < 2 could eliminate the wall effect . This la t ter case has been designated as a "single pel le t " bed. In a l l the foregoing runs, the test system used was a methane pulse i n a i r as a carr ier gas. Following these tests , runs 5 6 to 6 2 with porous pellets were carried out. One of the porous pel le ts , an activated alumina, gave - 50 - abnormally low values for the effective d i f f u s i v i t y . Further investigation, which i s summarized i n Appendix IV, showed that methane was adsorbed to a s ignif icant degree on the activated alumina. The need for a non-adsorbing system resulted i n further runs using the non porous pellets 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 single pel let diameter. Runs 69 to 72 were a similar 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 of 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 > N s Tube/Pel le t Ratio Pel let X . Diameter 1 3 6 12 25 Methane .208 cm. 5 k * 51 51D 50 <52 .568 cm. 6k 65 1 .0 cm. 72 55 69 53 Hydrogen .567 cm. 63 66 1 .0 cm. 71 70 *pel le t diam. 0 . 2 9 cm. - 51 - 0. POROUS PELLETS IN PULSE APPARATUS Three samples of porous spherical pellets were acquired for test ing. These included l/8" and l / U " diameter KL51 Alcoa activated alumina pe l le t s , and l/2" diameter Norton Alundum catalyst supports. The physical characteristics of these pellets are summarized i n Appendix III . 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 pellets could not Le adequately dried i n the steady state apparatus, as the epoxy resin holding the sample could not stand the necessary drying temperature. In view of this problem, the pulse investigation was attempted on the "wet" pel le ts , because i t was found that the moisture content of the pellets 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 pel le ts , but the manufacturer's l i terature (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 pel le t porosity and the moisture content. The detai ls of these calculations and other confirming experiments with respect to the porosities of the pellets are included i n Appendix III . 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 pellets i n a four foot long single pel le t diameter bed. The pellets were i n equilibrium with a i r at room temperature. Unexpectedly high dispersion of the pulse (HETP) caused some doubt about the number of transfer uni ts , so the bed was lengthened for 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 the long bed and the s h o r t . As shown l a t e r i n th« "R e s u l t s " , the short column was found to c o n t a i n insuff ic ient t r a n s f e r units f o r a Gaussian dis t r ibut ion . In run ^Q, xhe same bed as t h a t used i n run 57 was employed but the pellets were previously dr ied . At this 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 ignificant extent under these 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 p e l l e t s . In run 6 l , the use of a hydrogen pulse was tested on the same dry l /4 " diameter HI51 activated alumina pellets 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 pellets and using a hydrogen pulse. D . INDEPENDENT EFFECTIVE DIFFUSIVITY MEASUREMENT A conventional steady state method was selected for a second determination of effective d i f f u s i v i t y , but the technique was adapted for use with spherical pe l le ts . This modification consisted of mount- ing the pellets with epoxy resin 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 resin had dr ied . The solution for the d i f f e r e n t i a l di f fusion equation with this geometry is included i n Appendix I, along w i t h the results and detai ls of this experiment. Only the l /4 " and l /2 " pellets - 53 - were tested. On the basis of the manufacturer's data Knudsen di f fus ion was expected i n the l/h" alumina pel le ts , while molecular diffusion was expected i n the l /2" Norton p e l l e t s . The major problem with this part of the investigation was the moisture content of the pel le ts . The activated alumina could only be dried i n s i t u , hut the epoxy resin would not survive the drying temperature. Since the moisture content of the "wet" pellets remained re la t ive ly constant, as mentioned previously, i t was decided to test the pellets wet and correct the porosity accordingly. E. PREPARATION OF THE TEST COLUMNS The packed beds (columns) were constructed from glass tubing with rubber bungs or tubing i n the ends. The dimensions of the beds were generally obtained with a metric rule except for small diameter tubes, where a caliper rule was used. The bed porosities were obtained either by weighing the beds f u l l and empty i f the pel let density was known, or by addition of water and weighing. For the single pel le t diameter beds, the porosity was calculated by counting the number of pellets i n a given length of bed, and calculating the pel let volume from the mean pel le t diameter. The mean pel le t diameter was measured by placing a known number of pellets i n l i n e and measuring the overall length. For the porous pel let beds, the porosity was calculated as for the single pel le t beds above, or from the weight of pellets i n the bed with the characteristic 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. OPERATION OF PULSE APPARATUS 1. One of the four calibrated flow meter capi l lar ies was selected and f i t t e d . 2. The column was assembled, (after taking the necessary data for the porosity calculations) , and f i t t e d to the apparatus. 3. The a i r supply was turned on with the flow capi l lary bypass open, and the column pressure was set at a convenient level (usually around 0.5" Hg), using the cartesian manostat. h. 'The column was tested for leaks with soap solution. 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 exit to estimate the flow. 7. A suitable a i r flow rate was passed through the column using the flow meter and control . The flow meter manometer reading was recorded. 8. The recorder chart was started at any speed (unless previous experiments suggested a specific chart speed), and a pulse injected. 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 scale) , and a pulse width on the chart of at least 1.5 cm. 9. A series of pulses were injected, each at a different gas flow rate, to give about ten results covering the flow range desired. 10. During the course of each run the room temperature, atmospheric pressure and column pressure were recorded. - 55 - Hydrogen Flame Detector 1 . The detector was connected to the manifold with the correct capi l la ry . 2. Hydrogen flow was started at around 150 mls/min (using rotameter) and the flame was igni ted. 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 amplifier was always set at the No. 4 position for 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 positive flow (e.g. by bubbling i n water). 3. The filament current was adjusted to 100 ma. after connecting to 6v supply along with event marker leads. h. The recorder was set to zero and the detector to zero s ignal . - 56 - v RESULTS A. NON POROUS PELLETS Treatment of Data for Non Porous Pellets For each pulse input the primary data consisted of : the Clow rate of carrier gas, Q mis/sec, at 3'J?P, which i s actually recorded as a manometer reading and transformed using the calibration charts i n Appendix II to a f lov rate, the width of the pulse at half the height (WIDTH) taken from the recorder chart and also the "mean" or distance from the pulse injection 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 details of the columns pertinent to the individual run. These include a "Run number" which starts at 50 for the sophisticated apparatus, but a run (No. l ) from the preliminary results obtained on the i n i t i a l simple apparatus is included. The column length (L), diameter (d^) and porosity (Eg) are included i n the heading along with pel let porosity (Ep) and diameter (dp), and the carrier gas temperature (T°K), molecular weight and pressure (P). The pulse gas-carrier d i f f u s i v i t y i s also printed for the run temperature and pressure. The values for the molecular d i f f u s i v i t y of the pulse-carrier 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 results of Currie (6) . Currie found a temperature dependence of - 57 - d i f f u s i v i t y to the 1.715 power f o r this system, and t h i s was used to interpolate from the experimental results a d i f f u s i v i t y of 0.755 em 2/sec 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 the Hirschfelder equation using the force constants tabulated i n Bird, Stewart and Lightfoot (52). The computation, which i s shown i n Appendix III , yielded a d i f f u s i v i t y for methane i n a i r of 0.212 cm2/sec at 298°^ and 1 atmosphere. values corresponding to the table headings were fed di rec t ly 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 carrier gas viscosi ty , but i n the computations shown i n Appendix II the viscosi ty value read i n has been over-ruled i n the program by a viscosity for a i r computed from the Sutherland equation (53)t = 0.01709 f 275 + 114 1 / T \ 2 ^ 3 ( l . 6 l ) [ ) ( \ T 1-114 J \ 273 j The carrier gas density was calculated assuming the perfect gas law P = Mol. Wt. 2J3_ x P (1.62) 22400 T F i n a l l y , the hydraulic diameter was calculated from the following equation, h D = 4 Free Volume = d T € B (I.63) Wetted Area / 3_ <*T ( l - €B) + 1 \ I 2 dp I As mentioned earl ier the primary data of flow rate at STP, WIDTH (= 2.360) and TOTAL (mean) are given i n columns 8, 6 and 7, - 58 - respectively, i n Appendix II. l ' 1 - . . . ' ' , ' I K 1 . - -LU-T, ane. ..V >. - I . i the heading of each table, the Collo^ i . i g calc i •.let.-o L E K are p;-_* • . In column 1 the intere ..j. ij.nl velocity was calculeu. • -rom tube ,:.lameter d-jj, and the flow rate 4, correct, a for temperature T an. pressure ?. u = Q J L 1 f _ „ 1 1_ (1.6*0 273 p [nvL*J E B 'The HETP was calculated as defined by equation (1.26) HETP = Lcr = L I" WIDTH*] 2 f 1 ] 2 (I.65) mean2 [2.7;6 J [ TOTALJ 'Thi-ee Reynolds numbers were calculated for comparing the ax ial dispersion data with data of other workers and are defined as follows: the part icle Reynolds number shown i n column 4 of the table of results i n Appendix II is given by u d pp , 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 coefficient D^, was obtained from equation (1.34), which for non porous pellets reduces to , HETP = 2 D L (1.66) u so D L = HETP u 2 and this value is printed i n column 9 under the heading of "eddy d i f f u s i v i t y " . In fact , i t is 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 2 D 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 coefficient D^. Inspection of the term shows that the KTU i s smallest at low veloc i t ies , and since low velocit ies 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 test . The use of "long" beds has generally eliminated the NTU as a l imi t ing c r i te r ion i n this work. To make possible comparisons between the eddy d i f f u s i v i t y computed from this work and the correlations and results of other workers, the Peclet and Schmidt numbers were also calculated. The molecular and so-called "eddy" Peclet numbers are recorded i n columns 3 and 10, respectively, and were computed from the following def ini t ions , Molecular Peclet number u d p Eddy Peclet number u d p This eddy Peclet number should probably be called 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 coefficient D L i n this work the eddy Peclet or dispersion Peclet are interchangeable. The Schmidt number based on the dispersion coefficient i s recorded as the inverse Schmidt number i n column 12, that i s , Schmidt At the base of each table the least square error f i t of the HETP vs . 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 certain runs was restricted to the eddy di f fus ion 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 re la t ive ly small quantity in this range. To offset this problem a second least squares computation is carried out on the data to f i t the equation, HETP = AA + C'Cu (1.67) where H E TP ' = HETP - B . u Tne value of B is set at 2 x .75 x Dj}, where 0.75 represents the inverse of the tortuosity l/X i n equation (1.49). From the value of B derived from the three constant equation (1.50), the inverse of the tortuosity has been calculated for each run. Since ^ varies from 1 to eo as discussed i n the introduction, then the inverse ranges from 1 to 0. The usual value expected i n a packed bed is about O.67 to 0.8. The result printed on the computer sheet (Appendix III) is i n the nomenclature originated by Van Deemter (l7)and so the inverse tortuosity computed from equation 1.49 as 1 = 2 D3 j . s given under the X 3 heading GAMMA. Similar ly , the value of the constant characteristic of the eddy d i f f u s i v i t y which has been designated is computed from the value of the eddy dif fusion 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 for 200 mesh particles to about zero, for , say, l/8 inch par t ic les . The computer has printed the values of under the heading LAMDA (from Van Deemter et a l . ) i n Appendix III . Results for Beds of Non Porous Pellets Some typical curves of the HETP vs. velocity 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 dif fusion regimes while Figure 1.11 shows the results for runs 51, 69 and 70. The five straight l ines shown  0 5 . 10 15 20 25 ' INTERSTITIAL VELOCITY (u) CM/SEC. Figure 1.11 HEPP Vs. Veloci+y For Runs 51, 69 and 70. - 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 part ic le diameter ratio 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. It may be noted that both sets of data have the same intercept, indicating that Van Deemter's def ini t ion 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 velocity exponent of 2 must be added to account for 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 pel let diameter, due to the decrease of the coefficient # . In Figure 1.12 i t may be noted that with the larger pellets and generally higher flow rates i n this work the trend has been reversed, and "A" increases with pel let diameter. If a straight l i n e i s put through the points i n Figure 1.12 a slope around unity is 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 carried out on a 134 cm. bed with a cyclopropane pulse i n an a i r carrier gas stream i s included i n the data. The results of this run together with additional results from i n i t i a l tests with methane pulses i n beds of non porous pellets (runs 50 to 55) are summarized i n Table l . I I I . The most s ignif icant feature of these results i s that over a range of pel let diameters from 0.2 cm. to 1 cm., with tube to pel le t diameter ratios from 1 to 25, the wall effect or "C7 term, which might mask the dispersion effect due to pel let porosity, gave results 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) is 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 veloci ty i n the dispersion coefficient is less than 2. The plots of the dispersion coefficient vs. u given i n Figure 1.13 would appear to substantiate the lat ter 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 effect . 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 fa i lure of equation (I.50) when results i n the eddy regime only are used i n the least squares evaluation of the three constants A, B and C. Values obtained i n run ^k also demonstrate this point. The large diameter pellets i n runs 53 and 55 show a large intercept, or A term, compared to the other runs which show essentially zero intercept. Inasmuch as A is approximately proportional to d^, this difference is to be expected. It i s rather interesting that a bed with a single pel let diameter (run 55) has essentially the same or less slope ( i . e . C value) at high Reynolds numbers as the bed six particles i n diameter of run 53. This , as well as other results given i n Table l . I I I , indicate that the dispersion due to the wall effect i s not a function of tube diameters. TABLE l . I I I DISPERSION RESULTS WITH BEDS OF NON POROUS PELLETS Run Pellet Diameter Column Length Column Diameter 1 0.22 134.6 2.6l 50 0.208 111.8 5.0 51 0.208 118.1 2.6 51 D 0.2C8 118.1 2.6 52 0.208 111.8 5.0 53 1.03 186.3 6.27 54 0.297 185.4 O.415 55 1.005 121.0 1.15 Column to Pellet Diameter Ratio A B C 11.9 0.13 0.18 O.07 24 -0.07 0.35 O.07 12.5 0.050 0.31 O.052 12.5 -0.27 0.87 0.071 24 0.001 0.37 0.041 .6.1 0.68 O.36 0.071 1.4 -0.22 3.88 0.079 11.1 0.601 0.16 0.060 Range of Reynolds . AA CC Number 0̂.28 -0.150 C5 - 2.4 0.04 0.053 0.29 - 31.3 -O.O37 0.06h 2.6 - 32.6 0.032 O.O37 C8 - 33.0 0.72 0.064 5.0 - u-.o 0.177 O.069 16.0 - 79.0 0.260 0.3 22 3.0 - 48.0 Continued.. . TABLE l . I I I (Continued) Number of Run Points Gamma 1 10 50 9 0.76 51 15 0.73 51D 8 2.C7 52 30 0.87 53 13 0.85 54 10 s.2U 55 10 0.37 Remarks Doughnt rings i n column 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 six part ic le diameter bed containing 1 cm. spheres. Methane pulses are used and the effect of pulse size (as measured by peak height) at a particle Reynolds number of 62.k i s shown i n Table l . I V . Over a 13 fold range different pulse sizes resulted i n essentially 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 part icle Reynolds number of 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 error. This l imitat ion was later discarded, and the four points i n Table k were included with those of Run 53* However, they were found to change seriously the constants of the least square equation (1.50), indicating an inconsistency. Run 60 repeated the conditions of Run 53> but employed the normal pulse injec t ion , and these data were consistent with the results 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 injector at high flov; rates because of the fa i lure of the pop valve i n the injector to close cleanly. The requirements suggested by Van Deemter to ensure that the feed pulse size does not influence the exit dis t r ibution (equation 1.25) were easily sat isf ied i n this work, part icularly 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 Particle Reynolds number of 62.k Run 53 HETP PEAK HEIGHT 1.65 27.5 units 1.66 59 1.75 56 1.59 19 TABLE l . V FURTHER DISPERSION RESULTS WITH BEDS OF NON POROUS PELLETS Column to Pellet Pellet Column Column Diameter Run Diameter Length Diameter Ratio Pulse 63 O.568 421.0 0.66 1.16 H 2 64 O.568 421.0 0.66 1.16 CH 4 65 O.568 119.5 2.175 3.83 CH4 66 O.568 119-5 2.175 3.83 H 2 69 1.03 186.3 6.27 6.1 CH 4 70 1.03 186.3 6.27 6.1 H 2 71 1.005 122.0 1.15 1.1 H 2 72 1.005 122.0 1.15 1.1 CH 4 A l l Dimensions , cms. 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 C12 0.37 0.901 0.057 0.18 0.C49 0.12 O.87 0.59 0.027 -0.13 0.039 0.7C - O.32 O.76 O.O63 0.703 O.O63 0.68 0.86 0.57 O.C£3 0.54 0.030 0.31 1.14 0.79 0.028 0.34 0.C27 0.64 0.17 0.41 0.06 0.59 0.062 Continued Table l . V (Continued) Run Range of Reynolds Numbers Number of Points 63 6 - 3 5 10 6 - 33 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 Remarks OA vo - 70 - Table l . V shows the 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 allows comparison o f the hydrogen pulse technique with methane pulse results . Once again the w a l l dispersion effects (C value) f o r the methane vary only from 0.057 to 0.08l with pel let 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 pel let diameter. Runs 63 and 6h show h i g h B values (inverse tortuosity) , indicating that insuff ic ient 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 pel let diameter i n Figure 1.12, which indicates, i n spite of considerable scatter, the approximately l inear dependence of the packed bed eddy d i f f u s i v i t y on pel le 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 variation of the constant If In Van Deemter's eddy di f fus ion expression. However, the data from this work aligns i t s e l f well with the typical values of If quoted by Van Deemter (17)> as shown i n Table 1.6, except that the trend is reversed with larger pel le ts , and X increases with pel le t diameter. TABLE l . V T VALUES OF THE EDDY DIFFUSIVITY TERM CONSTANT, if = u d Pellet diameters cms. if Van Deemter (17 .003 - .0074 8 " .015 - .025 3 .035 - .C83 Z0 This work .2 0.06 .6 0.13 1.0 0.37 > A A A H. 1 X A CH4 O V • van Deemter O -O-ll o o : C t . / A - 9 A X 4 0-1 _ 0-3 0-5 0-7 0-9 10 O PELLET DIAMETER CMS Figure 1.12 Eddy Diffusion Terra, A, (Equation 1 . 5 ° ) V s . Pellet Diameter B. - 72 - LONGITUDINAL DISPERSION COEFFICIENT The data obtained 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 plus molecular c o e f f i c - i e n t s ) and are compared w i t h the c o r r e l a t i o n s and t h e o r i e s o f other workers i n Figures 1.15, to 1.17. I n Figure 1.13 a l l the' data except those from run 1 are 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 . the i n t e r s t i t i a l v e l o c i t y (u). Tne data p o i n t s form smooth curves but the slopes i n the t u r b u l e n t r e g i o n vary, showing an ex 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 the 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 an exponent o f 2 f o r the s m a l l e r packing s i z e s . At low v e l o c i t i e s , the d i s p e r s i o n c o e f f i c i e n t s approach the value o f the molecular d i f f u s i v i t y . I n Figures 1.11+ _ 1.17, the smoothed, data from Figure 1.13 has been used, and i s shown as a continuous curve w i t h i d e n t i f y i n g symbols marking the s t a r t and f i n i s h o f the l i n e . r e s u l t s w i t h those o f McHenry and Wilhelm (15) obtained by the frequency response method i n a bed o f 0.3 cm diameter spheres. The data from t h i s work are not 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 probably due to a d i f f e r e n c e i n the Schmidt number s i n c e McHenry and Wilhelm used a JQfo hydrogen stream w h i l e the pulses 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 thus 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 transit ion flow regimes as pointed out by Hiby (2l+). The data o f Cairns 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 Figure 1.14. Hiby (2k) suggested that a t low flow r a t e s (approaching the molecular 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 the molecular Peclet numbers as ,• shown i n Figure I.15. U n f o r t u n a t e l y the molecular P e c l e t numbers could not be c a l c u l a t e d from McHenry and Wilhelms'publication without access to 111 Figure 1.1-- 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 Dj^ i s ud_p p l o t t e d vs . the s u p e r f i c i a l Reynolds number u€ 3 d_ f* /A / to 'compare the •I 10 T — 100 ~ 1 — PELLET DIA.CMS TUBE/PELLET DIAMETER RATIO 10 1 A 6 v 0-6 IX 4 + 0-2 I2« 25 o 0-2 12 DOUGHNUT 6 0-2 25 HIGH PRESS <D 03 2o HYDROGEN J IN AIR METHANE IN AIR • • » Ol I 10 INTERSTITIAL VELOCITY CMS/SEC. Figure 1 . 1 3 Dispersion Coefficient, D L Vs. Inl ers' i :- .ial Veioci H y, u . 100 t t Hydrogen pulses in dark points PELLET TUBE / PELLET DIA. CMS DIAMETER RATIO 0-3 2X 10 1 A 67 0-6 la 4 0 0-2 12 O 25® aims 8 Prausnitz. Mc Henry SWilhelni. I 10 100 PARTICLE REYNOLDS NUMBER (based on superficial veL) Figure l.lk Inverse iuj.uy Peclet 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 1 A 6 V 0-6 ID 4 0 0-2 120 25 0 0-3 2X 0-1 I 10 100 INTERSTITIAL VELOCITY,HYDRAULIC DIA. REYNOLDS NO. Figure 1.16 Inverse lkldy Schmidt Huraiber Vs. Hydraulic Diaueter Reynolds 1,'ur.iber - 77 - M O M Eb &4 w o o o M cn i 10 E-t 10 PELLET DIA. CMS TUBE/ PELLET DIAMETER RATIO 10 0-6 0-3 • 0-2 IA 6V ID 40 2 X l 2o 25 rf Hydrogen pulse: Dark points 01 a e A « 4 01 I 10 CALCULATED DISPERSION COEFFICIENT Figure 1.17 Empirical Dispersion Coefficient Correlation - 78 - primary data and so a comparison could not be made, but i t may be n o t i c e d t h a t i n Figure 1.15 the r e s u l t s from t h i s work are not so s c a t t e r e d as i n the previous i l l u s t r a t i o n s . Tne data f o r l i q u i d s p u b l i s h e d by Hiby (2h) are a l s o i n c l u d e d i n Figure 1.15* but the values are lower than the r e s u l t s from t h i s work. This decrease was t o be expected because Hiby took pains to e l i m i n a t e the 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 . I t i s a l s o s i g n i f i c a n t t h a t Hiby considered t h e , r e s u l t s o f McHenry and U i l h c l m to show lower values of the eddy d i f f u s i v i t y or d i s p e r s i o n c o e f f i c i e n t than would be expected i n a bed w i t h w a l l e f f e c t s . I n Figure 1.16, the c o r r e l a t i o n suggested t y B i s c h o f f and Lev e n s p i e l (28) i s examined by p l o t t i n g the data as i n v e r s e d i s p e r s i o n Schmidt number vs. the Reynolds number based on h y d r a u l i c diameter. 'The covergence o f the data i s no b e t t e r than i n the other p l o t s . The Saffman model (29) could not be t e s t e d because the boundarie of the d i s p e r s i o n regimes i n packed beds are not known as they are 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 the Saffman model may have the 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. In the 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 developed below, which i s an ex t e n s i o n o f the simpler form proposed by B i s c h o f f and L e v e n s p i e l (28). DT = 0.75 D w + 0.6 u h,, + 0.02 u 2 h n 0 , 6 (1.68) ' 0.75 D B + 0.022u hD - 79 - This c o r r e l a t i o n i s p l o t t e d i n Figure 1.17 as experimental v s . c a l c u l a t e d values 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 accurate to a l l o w a c o r r e c t i o n to be c a l c u l a t e d f o r the "C" term i n equation (I.50), which w i l l c o r r e c t the value 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 of eddy d i s p e r s i o n are not a l l o w -:A for i n the eddy d i f f u s i o n , or " A " , term. C. POROUS PELLETS Porous P e l l e t Samples The p r o p e r t i e s of the three porous p e l l e t samples t e s t e a are summarized i n Table l . V I I . However, there was o r i g i n a l l y some question about 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 which are discussed 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, but the manufacturers' data s u p p l i e d w i t h the p e l l e t s seemed to be somewhat i n c o n s i s t e n t . The data on the l / 2 " Norton c a t a l y s t support p e l l e t were g e n e r a l l y s a t i s f a c t o r y . However, i n the trade l i t e r a t u r e a 4l$ p o r o s i t y was quoted f o r these p e l l e t s . In a p r i v a t e communication, a value o f 36-40$ was g i v e n , and a 'simple experimental measurement described i n Appendix I I I found a 36$ p o r o s i t y . A value of 38$ has thus been accepted as a reasonable 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 data, the amount o f moisture contained i n the p e l l e t presented a problem. As discussed e a r l i e r , the epoxy r e s i n s used t o mount the t e s t p e l l e t i n the steady s t a t e d i f f u s i o n apparatus ' A - J o t . l . V I I PROPERTIES OF P0R0U3 PELLE1'1 SAMPLES Manufac :urers' Trade Description l / 2 " H o r t o n Catalysi suppor'" SA 203 mixiure Pellet Pellet Diameter Porosity 1.30 cm. l/k" Alcoa ac:iva!.ed O.597 cm. P J V , . _ 1: 151 1/8" Alcoa activated O.32 cm. r.lumina II 151 O.38 0.30 0.50 Pellet Moisture Cont ent i n 6 0 $ RH Air negligible 12$ 12$ Porosity of Moist Sample O.38 O.31 at 12$ wet 0.34 at 10$ wet 0.31 a+ 12$ wet Pore Diame ter 9 0 $ of por JS 2-IiC microns c 50 A o 50 A Solid Densi iy gm/ml 3.5 3.2 3.2 C o o - 8 1 - could not stand the drying temperature necessary, and so i t was decided to make the diffusion tests on the w< t pe l le ts . The moisture content of the wet pellets was found to he stable, and not sensitive to atmospheric humidity, remaining between 10-lU$ by weight. In addition, the manufacturer's l i terature (31) suggested that adsorbed water existed i n l i q u i d form, so that i f the dry pel let porosity could he found, the porosity of the wet pellets could be calculated. • The suppliers quoted a dry pel let porosity of 60-65$, while the manufacturer's l i terature stated 50$« The moisture content i n equilibrium with 60$ R.H. a i r was given as 20-24$, but at no time could more than 15$ water actually be found i n the pel le ts . Examination of some of the manufacturer's drying data indicated that after 6 months a 12-15$ moisture content was normal. In order to obtain a better value of the dry pel let porosity, special measurements were carried out. One of the experiments for this purpose described i n Appendix III involved putting pellets under vacuum and then flooding them with water. This test suggested that the 50$ porosity was correct, and this value was later v e r i f i e d more exactly by placing dry pellets i n a chromatograph sample loop, and measuring the resulting reduction i n sample volume of the loop. Hydrogen gas was used at the sample gas i n the loop. This experiment gave a 50$ porosity for the dry pellets and yielded 28$ and 33$ porosities for wet pellets having a 12$ moisture content. A porosity of 31$ corresponds to a 50$ dry porosity i n a pel le t containing 12$ by weight water i n the l i q u i d state. In addition, the alumina pellets are not homogeneous i n that they are apparently manufactured by seeding a c o l l o i d a l solution. Examination of a s l i c e of pel let on a microscope s l ide showed pores up to 0 150 microns i n the centre core, compared to a pore diameter of 50A i n the - 82 - outer s h e l l . These pellets 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 result for use i n catalysis work, while the unsteady state method would yield 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 effective 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 2/sec. at 23°C and 76O.7 ram. Hg. The effective dif fusion coefficient of hydrogen in l/k" diameter Alcoa HI51 activated alumina pellets containing 12$ by weight of water was found to be O.OO67 cm 2/sec. at 26°C. "reatment of Data for Pulse Apparatus For the porous pel le ts , 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 velocity rather than the molecular Peclet number which was used with the non porous pel let 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 las t term of the empirical correlation 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 , 6 6 (I.69) du (O.75 D B + 0.2 u* hD) where u* i s the mean velocity from a l l the data points and allows the correction to be made i n the middle of the velocity range of interest . The correction i s subtracted from the slope C and the corrected slope C applied i n the calculation of the effective d i f f u s i v i t y from equation I.U9 and I.50 using the form, Porous Pellet Results In Table l . V I I I , runs 56 to 62 were made with l / V Activated Alumina pel le ts , except run 60 which was made with the l/2" Norton Catalyst support, and run 75 i n which the l / 8 " Activated Alumina pellets were used. A l l the results were taken i n single pel let diameter beds, except run 73 which used a bed having a 7si diameter r a t i o . 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 pellets were dr ied . Run 59 w & s essentially unsatisfactory, but i t shows the results of an attempt to eliminate the adsorption effect with extremely high flow rates. Run 6l 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 pel let diameter bed packed with the l/2" Norton Catalyst supports. The results for porous pellets are summarized i n Table l . V I I I . It may be noted that there appears to be an end effect i n comparing runs 56 and 57. However, under Table l . V I I I the values of the c r i ter ion for Gaussian dispersion, Fiu 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 solution (17) "to obtain equation (1.31) would not hold. The effects of adsorption on a catalyst pel let 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 pellets was TABLE l . V I I I DISPERSION RESULTS FOR POROUS PELLETS Moisture Column Column Content Pellet Length Diamet er Pulse Pellet Run Pellet v t . , # Porosity Cm. Cm. Gas Diameter A B C 56 l/4" activated .12 O . J l 129 0.66 CH4 0.597 -0.26 -0.64 1.32 Alumina O.58 1 .6l 57 l/k" activated 12 0.31 k21 0.66 CH4 0.597 -0.22 Alumina 58 l/k11 activated 0 0.50 421 0.66 CH4 0.597 -2.2 4.65 1.699 Alumina 68 59 l/k" activated 0 0.50 421 0.66 C H 4 0.597 -592 -C.33 Alumina 0.50 0.44 60 1/2" Norton - O.38 420 1.6 CH4 1.5 0.29 Catalyst Support 0.64 0.87 0.22 61 l/k" activated 0 0.50 421 0.66 H 2 0.397 Alumina 62 l/k" activated 10 0.34 421 0.66 H 2 0.597 0.38 1.5 0.237 Alumina 1.43 O.O96 75 1/8" activated 12 0.31 119.4 2.17 H 2 O.32 -0.015 Alumina Height of Transfer Unit FJU < L at max. velocity l66 cm. 3.38 cm. 20.2 cm. 5.58 cm. Run Number 56 60 61 73 TABLE l . V I I I (Continued) Dispersion Results for Porous Pellets D i f f u s i v i t y Assuming Run Slope Correction Corrected Slope D i f f u s i v i t y Equilibrium Adsorption Bed Porosity Reynolds Range 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 0.0045 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 - 8 6 - measured, and the procedure, which involved taking pressure and volume measurements of a gas pel let sample trapped i n the leg of a mercury manometer i s described i n Appendix I V . The results of this experiment, which showed methane adsorbed to the extent of 1.J7 mis/ml of pe l le t , are u t i l i z e d i n run 5 8 to calculate the effective 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 pel let 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 pellets (after correcting for the different 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 dif fusion are compared for a l l the runs. The effective 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) assum- ing equilibrium adsorption. This result would indicate that the methane probably does not approach equilibrium adsorption closely 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 of the theory and/or an inconsistency with the results from the eddy diffusion runs with non- porous pel le ts , but i f the fact that the C terms are extremely large due to the adsorption of methane i n runs 56 to 58 is considered, tnen the A and B terms are negligible , 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 D i f f u s i v i t i e s cm2/sec units Pulse Method Puis e Experimental Factor to Convert to Hydrogen or Result as a Hydrogen Assuming Adsorption at A,from Steady Run Gas Result H 2 - N 2 D i f f u s i v i t y Diffusion Equilibrium Column 5 State 56 CH* O.OOO85 A T6 0 0.0024 O.OO67 57 CH4 0.00069 fl6 2 0.00195 3.02 O.OO67 58. CH 4 0.00127 J r f O.OO36 0.017 2.64 60 6l CH 4 H 2 0.0193 0.0102 0 . 7 5 0 0.208 1 0.0694 0.0102 4.14 0-93 O.O67 62 H 2 O.OO56 1 O.OO56 I.05 O.OO67 73 H 2 0.0045 1 0.0045 1.31 O.OO67* Co —J •For 1/4" pellets I n t e r s t i t i a l Diffusivities Dg Nitrogen and Hydrogen = Dj£ Hydrogen in 50 A pores at 296 °K O.755 cme/sec. at 1 aim. and 296°K = 2 x 25_ x 1.84500 x 29_6 = 0.019 cm2/sec. 3 108 273 - 88 - Comparison of Steady State and Fa l s e Apparatus PL-suits In order to compare' r e s u l t s , 'Cable l . I X presents the pulse data converted to the equivalent hydrogen d i f f u s i v i t y (or hydrogen-nitrogen f o r bulk d i f f u s i o n ) . If the r e s u l t s i t h the l/2" Norton p e l l e t s i n run 60 are examined, i t i s seen t h a t the pulse method agrees v i l l i th.- •„. tency state value within 4$. Run 62 should give the same d i f f u s i v i t y f o r hydrogen i n the l/k" Alcoa activated alumina pellets as the steady s t a t e apparatus, '.JUT. the la t ter result i s 20$ higher than the pulse r e s u l t . I f the d i f f u s i v i t y i n the l/8" pellets could he expected to be the same as t h a t i n the l/k u s i z e , the steady state result i s 52$ higher than the result from run 73- In view of the lack of homogeneity of the alumina pellets these results are not surprising. The pellet tortuosity 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 pel le t porosities shown below Table l . I X also indicate that the Alcoa activated alumina pellets are not homogeneous, as the tortuosities are much lower than would be expected for this type of material . The steady state results should be even more influenced by the macroporous pel let centre or seed because of the removal of part of the microporous s h e l l , and i f the tortuosity i s calculated from the steady state result an impossible value of 0.88 i s obtained. 'The reason for this anomalous result i s because the pore size has been assumed to be 5OA i n the calculation of the Knudsen dif fusion coeff ic ient , 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 Results The HETP vs. v e l o c i t y curves shown i n Figures 1.10 and 1.11 appear to f i t Van Deemter's equation ( l . 5 0 ) w e l l . However, a v e l o c i t y dependent, o r Cu term, was not to be expected w i t h non porous p e l l e t s on the b a s i s o f Van Deemter's a n a l y s i s . From Figures 1.10 and 1.11, as w e l l as from the r e s u l t s i n Tables l . I I I and l . V , the 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 diameter, but i n v e r s e l y p r o p o r t i o n a l to the molecular d i f f u s i v i t y of the gas system. I f the Cu term (vhich 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 p e l l e t s i s caused by the higher v e l o c i t y annulus v h i c h r e s u l t s from th/- h i g h packing p o r o s i t y a t the w a l l , then by analogy w i t h Van Deemter 'o treatment, the r e l a t i v e v e l o c i t y between the flow i n the w a l l annulus and i n the packing core could c r e a t e a term which 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 molecular d i f f u s i v i t y . This reasoning i m p l i e s t h a t t h i s a d d i t i o n a l 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 e f f e c t . On the other hand, the 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 diameter beds are considered, so i t would appear t h a t another but s i m i l a r mechanism occurs i n s i n g l e p e l l e t beds, or t h a t the above p h y s i c a l e x p l a n a t i o n i s qu e s t i o n a b l e . The i n t e r c e p t , o r A term, (which i s a d i s p e r s i o n due to the mixing e f f e c t o f the packing) o f equation (1 . 5 0 ) depends on p e l l e t diameter a t Reynolds numbers l e s s than 1, according t o Van Deemter et 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 the mixing stage model has been obtained by McHenry and Wilhelm. 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 is an approximately l inear (t 50$) function of the packing diameter for diameters from about 0.2 to 1 cm., and for a wide range of tube: pel let diameter ra t ios . Axial Dispersion Coefficient I f the same data as above are considered i n terms of the dispersion coefficient ( i . e . the data for non porous pellets 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 effect i s not the major contribution to the mixing due to packing geometry. Figure 1.15 shows that the smaller pellets tend to y ie ld a dispersion coefficient proportional to the square of the velocity which jould correspond to the Cu term i n equation 1.50, but the larger pellets show a lower exponent of 1.5.Hiby (2h) obtained the following empirical correlation for l i q u i d s , D L = 0.67 D B + O.65 (u d p ) 1 , 5 7 j D ^ + J u T p and at low flow rates where iJ^B ^ J U d this expression has a velocity exponent to the 1.5 power. In the same work (2k), the results of other workers with l iquids are summarized. In general, the ax ia l dispersion coeff ic ient found by other workers i s a l i t t l e larger than that obtained by Hiby, who eliminated the wall effect , but Hiby points out that the data of McHenry and Wilhelm, who worked with gas systems, gives the appearance of having the wall effect removed. This effect may be due to the fact that end corrections were applied to the bed data by McHenry and Wilhelm, because i n their work re la t ive ly short beds were used ( l 1 , 2* and 3' long), with only the largest being comparable to the bed lengths i n the present work. - 91 - I n Figure 1.14, i t may be seen t h a t the data from t h i s work shows higher d i s p e r s i o n c o e f f i c i e n t values than does t h a t of McHenry and U i l h e l m (15), so t h a t the data from t h i s work would appear to be c o n s i s t e n t w i t h those o f Hiby (24). The use o f the h y d r a u l i c diameter to d e s c r i b e the system as proposed i n B i s c h o f f and Levenspiel's work ( 2 8 ) , and as shown i n Figure l . l 6 , does not appear to improve the c o r r e l a t i o n . The h y d r a u l i c diameter would o n l y be expected t o account f o r the w a l l e f f e c t , and i f the w a l l e f f e c t i s not predominant, as suggested by Hiby, then 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 to r e s u l t . As mentioned i n the "Theory", Saffman's model (29) o f a s e r i e s of interconnected 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 the 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 bed. Since the r e s u l t s presented here were g e n e r a l l y obtained between p a r t i c l e Reynolds numbers o f 1 and 100 ( i . e . i n the intermediate r e g i o n between laminar and t u r b u l e n t f l o w ) , then i t i s q u i t e conceivable 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 Taylor (25) occurs, r e s u l t i n g i n regions where 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 squares o f the 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 the molecular d i f f u s i v i t y . The upper l i m i t o f the r e g i o n was found to be, from (i.57) u « 10 L D B/R 2 where u i s , the gas v e l o c i t y , L the tube l e n g t h , R the r a d i u s and D B the molecular d i f f u s i v i t y . Let the c a p i l l a r y l e n g t h be K i d p and r a d i u s K2d p i n the Saffman model, which should be a reasonable assumption f o r packings o f u n i f o r m l y s i z e d spheres. - 92 - Then, u « 10 Kiclp D B or u <̂  K 3 D£ dP where K 3 - Kj . /K 2 Thus, the smaller the pel le t diameter, (dp), the larger the right hand side of the above equation. This model would explain therefore, why the smallest pellets showed a velocity exponent of 2 as compared to 1.5 or 1.7 for the larger p e l l e t s . 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 Wilhelm1s results at a superf ic ia l Reynolds number of about 100 i n Figure 1.1k. It would appear that at least two mechanisms are operating here; 1.) the velocity 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 veloci ty p r o f i l e (equation I.56) i n the individual channels, which y i e l d a velocity exponent of 2 within the flow l imi ts derived by Taylor, given i n equation (1.57). Thus, the resultant dispersion coeff i c - ient has a velocity exponent between 1 and 2. As pointed out above, a high molecular d i f f u s i v i t y would result i n a higher upper l i m i t of significance for the velocity p r o f i l e range. Nevertheless, McHenry and Wilhelm1s results for eddy d i f f u s i v i t y using hydrogen approach a velocity dependence of 1, possibly because although the molecular d i f f u s i v i t y is high, the magnitude of the contribution to the dispersion due to the veloci ty p r o f i l e i n the capi l lar ies i s smaller with higher d i f f u s i v i t y gases (equation I.56), and so the mechanism of equation (l.55) would predominate. In pipes, when the flow becomes turbulent, the prof i le contribution changes from the velocity squared dependence of equation (I.56) to a function of velocity and f r i c t i o n factor. In this 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 Axial Dispersion Coefficient As may be seen from Figures 1.13 to l . l 6 , several attempts were made to obtain a correlation for the dispersion coeff ic ient . In addition to these effor ts , dimensional analysis and a least square calculation based on the resulting expression using a l l the non porous pel le t results yielded the following correlation, u hpl fu2 1 ^ (1.70) ,DBJ IS hDJ l h D u PJ where y andp are the carrier gas viscosi ty and density. The above correlation shows an exponent for the hydraulic diameter of nearly unity, and a velocity exponent of I.67, which i s an average of the values shown i n Figure 1.13. Equation (1.70) does not provide a par t icular ly good f i t to the data, which i s not surprising because the velocity exponent i s obviously not constant, a fact c learly evident i n Figure 1.13. Of the correlations of the above type, that of Hiby recommended for the transit ion region and shown i n Figure 1.15 seems to be most satisfactory, but due to a dependence on the packing diameter squared, the degree of correlation i s less satisfactory than that given by equation (1.70). Bischoff and Levenspiel (28) suggest the following expression, which does have the virtue of allowing for the experimental fact that the - 9k - veloci ty dependence i s 2 for small pellets and approaches 1.5 for larger ones. The expression is based on the Taylor transit ion regime i n which veloci ty p r o f i l e effects are s ignif icant , but the molecular d i f f u s i v i t y is replaced by a radial d i f f u s i v i t y which includes a veloci ty dependent term. D L = D B + D B + K a u dp Although better than equation (l.70), a further considerable improvement i n f i t was achieved by reducing the packing diameter exponent from 2 to 1. However, the equation then becomes dimensionally inconsistent. The correlation f i n a l l y u t i l i z e d essentially makes the longitudinal dispersion coefficient a summation of a molecular term, a mixing stage term as suggested from McHenry and Wilhelm's work (15) and a velocity 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 g h n ° ' 6 (0.75 D B + 0.0212 u hn) The above expression is plotted i n Figure 1.17 as experimental vs . calculated results . B. POROUS PELLETS The effect of gas adsorption on the measured d i f f u s i v i t y presents interesting features of significance i n any type of unsteady state di f fus ion measurement. The method used to measure the degree of adsorption, described i n Appendix III , has been developed since this work was done and reported as a technique for determining adsorption isotherms for gases on solids (40). If 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 pellets i n this work, but as the adsorption data were derived for large concentrations ( l atm.) of methane, while the pulse apparatus uses trace concentrations i n the presence of a i r , the state of the - 95 - equilibrium cannot rea l ly be claimed to be conclusively known. Inconsistency of Steady State and Pulse Results for Activated Alumina Pellets The results of runs 6 2 and 7 3 with l/h" and l/8" activated alumina i l lus t ra tes the potentially serious errors possible with non homogeneous pelleted materials i n measuring the unidirectional d i f fus ion 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 dif fusion occurs towards the centre and out again. There are, of course, other potential reasons for differences i n the results from steady state and pulse methods, which have already been discussed. The pulse method i n this work maintains either bulk equimolar counter dif fusion or Knudsen diffusion i n the pel le t so that equation 1 . 1 3 i s val id no matter what mechanism occurs. In the case of the alumina pel le t s , the outer s h e l l has a uniform structure with 50°A pores so that Knudsen di f fus ion occurs, and settles the choice of equation for the steady state apparatus. Thus, the discrepancy between the steady state and pulse apparatus must he caused largely by the macroporous seed which carries a disproportionately large portion of the dif fusion flux i n the steady state apparatus. The l/k" alumina pellets 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" pellets was not v i s i b l e by eye and i t i s possible that these pellets either had an extremely small seed or none at a l l . This would account for the lower d i f f u s i v i t y of the l/8" pellets compared to the l / U " ones. - 9 6 - 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 for 50 A c y l i n d r i c a l pores, the extremely large pores i n the seed would account for the tortuosity value of less than unity obtained by the steady state method and given i n Table l . I X . Another factor which could account for 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 pellets were prone to break down i n annular layers. With caps ground o f f each side i n the steady state apparatus, the strata of these layers are exposed and may represent a low resistance dif fusion 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 is an accurate knowledge of the pel let porosity. A lfo change i n porosity can result i n a 4$ variation i n d i f f u s i v i t y . 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 pellets were placed i n the sample loop of the chromatograph and a hydrogen pulse injected i n an a i r carrier 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 sol id volume of the porous p e l l e t . 'The sample gas of hydrogen had to be diluted with a i r to keep the detector i n the l inear 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 this type would be very useful , so that the porosity of the pellets as tested i s measured. Non Spherical Pellets There should be no reason why the effective d i f fus ion coefficient of granular pellets of almost any form could not be measured by applying an - 97 - appropriate shape factor, and a surface-to-volume pel le t diameter as used for effectiveness factor charts ( 3 ) . A derivation was attempted to express the mass transfer coefficient for cylinders i n terms of an effective d i f f u s i v i t y , (as i n equation (1.46)) hut no simplified approximation could be made, due to the presence Of Bessel functions i n the solution. Thus, for shapes other than spheres, a constant based on experiment would seem to be required to relate the mass transfer coefficient and effective d i f f u s i v i t y i f the simplified form of 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 correction to the dispersion measured for a bed of porous pellets which is due to eddy di f fus ion effects i s no higher for hydrogen than the correction term for methane. The desirable amplification of the pel let capacity dispersion term can be achieved by a high veloci ty , 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 this work. Errors The errors i n the result caused by the mathematical manipulations are not readily estimated, however, the effects of inaccuracies i n the measured values are considered below. The effective d i f f u s i v i t y i s given 2 €33 ( d j g f _1 • 1 + 6 B I (1 -1 s where C i s the term from equation (I.50). kit* c ( ! - « B T - 98 - The overall potential error may be estimated, by adding the effects of individual errors for a given typical set of values. In the following table typical variable magnitudes are given along with the estimated error and the effect on the resultant effective d i f f u s i v i t y . TABLE l . A POTENTIAL ERRORS Variable Magnitude Degree of Percentage e r r o r i n Uncertainty Effective D i f f u s i v i t y c 0.375 + 5$ 5$ cm/sec dp 1.0 + 2$ 4$ € 0.33 - 10$ l6$ € B 0.40 + 5$ _2$ 27? It i s f a i r l y obvious that more accuracy i n the pel let porosity values would radical ly improve the results , 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 yield the above overall error. 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 pel le t diameters could show 50$ variation between individual pe l le t s , the mean of 20 to 40 pellets 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 adequately measured usi n g a hydrogen pulse technique. A 27$ random e r r o r i s conceiv- a b l e due to e r r o r s i n the measured v a r i a b l e s ; however, t h i s can b e halved w i t h b e t t e r methods of measuring the p e l l e t and bed p o r o s i t i e s . In a d d i t i o n , a probable e r r o r e x i s t s from the mathematical d e r i v a t i o n s . This l a t t e r e r r o r should be a r e l a t i v e l y constant percentage, thus l e n d i n g i t s e l f to e l i m i n a t i o n by c a l i b r a t i o n . 2. An eddy d i f f u s i o n mechanism e x i s t s i n the t r a n s i t i o n r e g i o n between laminar flow and t u r b u l e n t f l o w i n packed beds such t h a t the 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 to the square o f the v e l o c i t y . V I I I RECOIvlENDATIONS The method f o r the measurement of the 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 pulse o f hydrogen which has been purged from the sample loop o f a chromatograph c o n t a i n i n g the t e s t p e l l e t s , should be developed f u r t h e r and i n c o r p o r a t e d i n t o the p u l s e apparatus. The main problem to overcome i s t h a t o f m i n i m i z i n g the i n t e r p a r t i c l e volume by packing i n as many p e l l e t s as p o s s i b l e . By extending the f l o w ranges covered i n t h i s work, the range o f the 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 to the square o f the v e l o c i t y may be determined. The r e s u l t s may then be compared w i t h the r e s u l t s obtained i n empty pipes by Taylor ( 2 5 ) . - 100 - SECTION I I DEVELOPMENT OF AM UNSTEADY STArf1E FLOW METHOD FOR MEASURING BINARY GAS DIFFUSION COEFFICIENT'S I INTRODUC ?ION The "bulk, or molecular diffusion coefficient o f hinary gas mixtures is not readily measured experimentally. One o f the o l d e s t techniques is the Loschmidt method which is based on b r i n g i n g two cylinders containing the gases (lighter on top) together and measuring c o n c e n t r a t i o n variation with time. However, this method is sensitive to convection or thermal eddies. In Stefan's method the rate of diffusion-of a vapour i n a ver t i ca l glass capi l lary tube is measured by following the drop i n level of a l i q u i d meniscus as evaporation occurs. The open top end of the glass tube i s flushed with the second component. This method obviously cannot be used for gases above the c r i t i c a l temperature, and, i n practice, i s limited to narrow ranges of temperature and pressure. 'The longitudinal dispersion coefficient i n a straight tube, within the l imits 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 similar to that described i n Section I, the molecular d i f f u s i v i t y may be obtained. Chromatography apparatus can also be used for this type of work. Good results can be obtained, although the apparatus i s not simple, and experimental conditions feasibly are limited ( 5 5 ) ( 5 6 ) . - 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 is fed through a capi l lary which is mounted i n the centre of a tube i n which the second gas i s flowing. The prof i le of the trace gas i n the bulk stream is 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 solution of the dif fusion equation. Very careful experimental technique i s required to obtain accurate values by this method, although wide temperature ranges can be covered. Other methods, such as measurement of di f fus ion rates through porous barr iers , have been employed by numerous workers, but these do not give absolute values, and require cal ibrat ion, and a correct interpretation of resul ts . In part icular , there appears to exist no absolute methods which can be used to give acceptable values of the binary dif fusion coefficient over wide ranges of both temperature and pressure, and which w i l l allow some investigation of concentration effects a lso . The present work i s an attempt to develop a measurement technique which w i l l sat isfy a l l these requirements. An unsteady state flow method similar to the Stefan technique was selected, as offering the p o s s i b i l i t y of analysis of an effluent stream remote from the dif fusion 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 desired. 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 capi l lary channels would also be a part of the construction. - 102 - II THEORY A. SIMPLIFIED SOLUTION OF A DIFFUSION EQUATION It has been sh6wn ( 6 ) that for equimolar di f fus ion i n a porous s o l i d F ick 's second law of dif fusion takes the following form, dCA - - 2 E & a CA (2.1) * t «I b x £ where D E i s the effective d i f f u s i v i t y , 6 B the porosity, C the concentration and t the time. The absence of significant surface adsorption is also implied by the above equation. The relationship between the effective d i f f u s i v i t y D E and the true binary dif fusion coefficient D-g of the free gas D i s given by, D E = DB E B (2.2) X where X . i s the tortuosity with values varying from 1.0 for straight p a r a l l e l pores to about 100 for a structure containing dead end pores. If Dg from equation (2.2) i s substituted into (2.1), then for a bed with tortuosity 1.0 the solution of (2.1) would yield the molecular d i f f u s i v i t y D B of the gas. A simple solution of the dif fusion equation (2.1) for the model shown i n Figure 2.1 i s obtained i f the assumption is 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 are: x = 0, dc = 0 dx C A = C 0 for a l l x when t * 0 C A = 0 when x = L for t > 0 DISPLACING GAS Y DISPLACED 8 DISPLACING GAS A TO ANALYSER END ZONE CONTAINING "WELL MIXED FLUID" PACKED SECTION X « 0 Figure 2.1 ibdel oi" f?he Bed For Proposed Diffusion Kxporinen-t - 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 J h e s e 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„ . - ~ L to +!• °°, n C A = L̂ Q. > L-lj exp - D£ (gn + 3 ) ^ T r £ o - 0 , (2nn)7Tx Tf (2n + l ) B 1| L ^ ~£ L h=0 (2.3) In order to find the f lux from the end of the v e s s e l ~ho above solution must be differentiated v i t h respect to x, and the resulting expression solved to give the concentration gradient a t the end (x = L ) . This gradient may then be applied i n conjunction with Fick's f i r s t law of d i f f u s i o n , dx where is the flux of gas A i n moles/(sec)(cm2) under conditions of equimolar counter d i f f u s i o n , which must exist i n the model of Figure 2.1. The series solution for the concentration gradient given by (2.3) when x = L i s , f&cA = 2 C n y_ exp f - DE (2nH) 2 TT 2 t (2.5) L D X J x=L L n=0 L B * ̂  T This solution can be simplified by taking into account only times greater than the time when the second term of the series i s l e s s than 1$ of the f i r s t , or i n other words, Ln 0.01 - Dg 7T2t = - 9 DE ¥ 2 t (2.6) e B klF k~e^~ ~Tr If a molecular d i f f u s i v i t y of 0.75 cm2/sec (e.g. hydrogen- nitrogen) i s assumed, and a dif fusion path'of unit tortuosity and length of 10 cm., then solving 2.6 gives t = 31 seconds. Similarly i f the gas d i f f u s i v i t y i s taken as 0.1 cm2/sec then the time before the second term can be ignored becomes 232 sec. - 105 - I f now the boundary condition requiring that the end of the bed at x = L should be at zero concentration i s achieved by sweeping the end rapidly with a second gas, then the concentration of the displaced gas i n the exit stream w i l l be proportional to the flux at the end of the bed. I f , i n addition, suff ic ient time as calculated above is allowed to elapse before concentrations are recorded so that the second and higher terms become negligible , then the flux equation from (2.5) reduces to the form, Ln C E X I T = Ln Z - DTP,7T2t (2.7) £B h L 2 where Z i s a constant including the unwanted terms from the material balance, and from equations S.k and 2.5. Z = A B D E . 2_CQ (2.8) Q L Q is the displacing gas flow rate i n m i s / s e c , and A-g the area of bed at x = L . A semi-logarithmic plot of exit concentration vs. time for a constant flow rate should yie ld a straight l ine with slope DE IT2. I f 1 U 2 the bed i s packed with para l le l tubes the tortuosity should be 1, and the slope of the plot becomes D33 TT 2 , thus providing a means for measuring k L the free gas molecular d i f f u s i v i t y without cal ibration of the apparatus. B. MORE RIGOROUS SOLUTION A problem with the above experimental model ar ises , i n that i f a displacing gas flow rate high enough to sat isfy 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 sensitive analyt ical method i s required. Possibly this very high gas flow rate could be used - 1 0 6 - anyway, "but a second problem due to turbulence caused by the high veloci t ies entering above the packed section could arise and cause eddies i n the dif fusion zone. In order to minimize the displacing gas flow rate, the end zone through which the displacing gas flows must be as small as possible, but too narrow an end zone would result i n pressure drops which could cause bulk flow i n the dif fusion section. Experimentally, i t was not found possible to achieve the boundary conditions described above, but a solution for the di f fus ion equation with a well mixed f l u i d at the end of the d i f fus ion zone ( i . e . a f i n i t e end zone) has been obtained by Carslaw and Jaeger ( 3 8 ) for heat conduction from a s o l i d . Essential ly , the solution expressed i n terms of the mass diffusion case discussed above i s for the following boundary conditions, dC A = 0 at x = 0 for a l l -t dx CA = C 0 for a l l 2 at t = 0 and a material balance around the well mixed end zone y i e l d s , " D E A B f OCA") - Q = £ A B ftCAc (2.9) I a x j X=L at where Agis the area of the end of the bed, & is the height of the end zone, and C A O i s the well-mixed end zone concentration. As before, Q is the gas flow rate, 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 rate. The solution 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 . At x = L , contact i s assumed with a mass of well s t i rred f l u i d M* per unit area of contact, and , - 107 - specific heat c vhich is cooling hy a radiation mechanism at H times i t s temperature. The i n i t i a l temperature of the f l u i d i s taken as zero. The la t ter boundary condition i s not compatible with the apparatus proposed for this work but this does not influence the solution at large times which i s the region of interest i n this work. v e 2 V 21 earn C K <*.8 t K h - k o t w 3 ) eo6 (»yx ) (a . io) n=l ( L ( h - k<*n 2) +c*nZ ( L + k ) + h) cos ) where h = H/K* , k = M*c and of n are the consecutive roots of o c t a n o C L = h - kod 2 (2.11) 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 2 / s e c . The thermal conductivity i s K cals/sec cm2 ( ° K ) / c m . The above solution has been transposed to the equivalent diffusion case, and the tabulation which follows may assist i n explaining the di f fus ion parameters. Heat Transfer Mass Transfer vp c cals/cm C A concentration moles/cm3 (>c cals/cm 3 ° K 1.0, unless i n a porous bed when equals porosity € B K cm2/sec D E \ effective d i f f u s i v i t y cm2/sec i g " ) or D B i f €jj= 1.0 K ' = K ^ c cals/sec cm 2 (°K /cm) H v cals/sec cm2 Q, C A o / A B moles/sec cm 2 , where Q i s the gas flow rate, cm 3/sec, and A B the bed area, cm2 Other quantities appearing i n equation (2.10) when written for the diffusion 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. - 1 0 8 - h = 3 ( 2 . 1 2 ) A B » E K = A B t filj = A ( 2 . 1 3 ) 6 B A B £ B where £>M i s the molar density and i i s the length of the end zone. Rewriting equation ^ ( 2 . 1 0 ) and setting x = L y ie lds , C A O = 2 CQ (h - k<) exp [- Hf fr* f t ] ( 2.U) L (h - k f r ^ 2 ) 2 + e t n 2 (L + k) + h If the time, t , i s large then the second term i n the series becomes negligible compared to the f i r s t , and equation (2.1k) becomes, cAo = 2 C (h - k < * i 2 ) e x p l ~ J ( 2 . 1 5 ) L (h - k ^ ) * + 'ac-f (L + k) + h or L n ( C ^ ) = L n "(Z) - D E _ « I 2 t ( 2 . 1 6 ) E B where Z = 2 C J h • k<*i 2) _____ I T T f c k S I 2 ! (L + k) + h Thus a plot of Ln ( C A O ) versus t f o r large times should yie ld a straight l ine of slope - B_o£i 2/6 .g . It is also of interest to note that an absolute value of the concentration i s not needed. For example, the peak height of a chromatograph is proportional to the concentration at low concentrations, and so the logarithm of peak heights rather than concentrations may be plotted versus time. Equation ( 2 . l 6 ) must be solved simultaneously with the auxil iary equation ( 2 . 1 1 ) i n order to obtain a dif fusion coefficient from a set of exit gas concentration versus time data. Examination of the equations shows that an analyt ical solution is not possible. In order to obtain a t r i a l and error solution the following i terat ive procedure was applied using the Newton-Raphson method (39). - 109 - The f i r s t root of equation (2.11) must l i e between c< = 0 and 77*/ 2L. Selection of an i n i t i a l value approaching zero could result i n a break down of the iterative operation because the second approximation f a l l s outside the zero toTT/2 L range. A further reason for 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 implified 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. Also , due to the i terative nature of the solutions to (2.l6) and (2.11) , the time when the second root can be ignored cannot easily be derived, but as the more rigorous solution approaches the simplified solution i t i s reasonable to suppose that the time calculated from the simpler solution (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 vs. time plot one gets, Slope = DE OC i 2 (2.17) «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 (2.11) i s corrected by a term for the resulting error, , A = h - k o C i 2 +04 tan<*-iL (2.l8) Differentiating (2.l8) with respect to d A , = - 2 k<*i - oC\ L „ - tan <*jL (2.19) dc<j (Cos<*iL) The second approximation for 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 (2.l8) and (2.19). 0^1 = oCx ( f i r s t approximation) - A (2.20) d A dot, - 110 - With the second approximation the process can.be repeated from equation (2.17) u n t i l a satisfactory result is obtained. C 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 easily purged, a plot of experimental data according to equation (2.7) may yie ld 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 ie ld a straight l i n e . It was found that on a log plot a s l ight change i n the steady state value caused a large change i n the slope, and hence uncertainty i n the resulting value of 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 level a least squares solution —Bt was prepared for the equation Y-C = A . e " where A, B and C are the constants to be determined, Y represents 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 solution i n favour of data at short times. However, as there is evidence that the so-called steady state value is dependent on the gas flow rate i n this apparatus, weighting i n favour of shorter times where the steady state value is negligible would seem to be j u s t i f i a b l e . The derivation of the expression for evaluating B is given below. E 2 =2[Y: - C - A e " B t t ] 2 (2.21) - I l l - vhere C represents the value o f Y approached a t i n f i n i t e time and A and . B are constants o f the system when equation (2.l6) i s f i t t e d t o (2.21), w i t h E "being the e r r o r t o be minimized. D i f f e r e n t i a t i n g 2.21 by A, B and C y i e l d s . dE£ =12 (Y ; - C - Ae " B t i ) (-e"Bt<) (2.22) dA dEf = £ 2 f Y. - C - A e " B t i J ( - A f y f ^ ) (2.23) dB dEf = £ 2 [ Y, - C - A e ~ B t i ] (-1) (2.24) dC S e t t i n g (2.22), (2.23) and (2.24) equal to zero and e l i m i n a t i n g A and C y i e l d s the f o l l o w i n g , where A represent the e r r o r r e s u l t i n g from assuming an i n c o r r e c t value o f B. . _ - B t T j , -Bt r :Bt r v - B t ^ - 2 B t ^ ~ B t r - v r " 2 B t /X = - lie 2 te 2e + ZYe Zte + 2_te j Y l e u ' n <r v " B t V - 2 B t V v - B t ̂  -2Bt v- -rBtAr - B t ) 2 o c N - l i e .̂e ~l_Y e > te + 2.*te ( 2 s 7 (2.25) n n To apply the Newton-Baphson method (39)> - 5 V -Bt f /T+ - E t v 2 -Bt ̂  , - B t l V , -Bt-r -Bt-r Vj_ -Bt dA = Z Y e I (Z^e ) +Ze 2. te J +2 te 2.e 2. Yte dB n n 2 2 Y e - B t I t 2 e " 2 B t - ^ e ' ^ ^ Y t e ^ - 2 I Y X t e " 1 * 2 t e - 2 B t + n l Y l e " 2 E t I t 2 e - B ^ + 2 l Y t e - B t I t ' e - 2 B t ^ e ^ I * t e " 2 B t n 2.t e +2.Y2,te 2.te - 22.Y t e Z e 2. t e n n n - ( 2 e ' B t ) 2 l Y t 2 e - B t (2.26) n Hence B 2 = B i - A (2.27) d A In (2.27), B 2 represents a b e t t e r value o f B than the previous assumed v a l u e , t h a t i s , B i . - 112 - III APPARATUS A constant temperature a i r bath was f i t t e d w i t h the 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 the bed f o r the d i f f u s i o n measurement. The t e s t v e s s e l s were soldered from pieces o f brass or copper pipe and were f i l l e d t o the brim w i t h the packing m a t e r i a l . A rubber gasket was used t o provide the spacer f o r the " w e l l mixed end zone" as shown i n the sketches i n Figure 2 . 2 . Two entrance flow p a t t e r n s were used i n the beds, a t a n g e n t i a l e n t r y i n the 5 cm. d i a . v e s s e l , and a d i r e c t sweep across the bed i n one d i r e c t i o n i n the 2 . 5 cm. c e l l . A schematic diagram of the apparatus i s shown i n Figure 2 . 2 . Moore constant d i f f e r e n t i a l flow c o n t r o l l e r s were used t o ma i n t a i n constant gas f l o w r a t e s , w h i l e a soap bubble meter was used f o r measurement o f the e f f l u e n t stream f l o w s . In order to reduce the hold up o f the apparatus due t o valves and f i t t i n g s , the two gas feed systems were connected t o the d i f f u s i o n c e l l w i t h l/k" polyethylene t u b i n g , and s w i t c h i n g from one gas to the other was done by d i s c o n n e c t i n g one tube a t the entrance to the constant temperature zone and connecting the second. A bypass va l v e a t the entrance t o the constant temperature bath allowed gas to flow d i r e c t l y to the flow meter. The use o f l / 8 " t u b i n g to connect the t e s t v e s s e l to the chromatograph sample v a l v e provided s u f f i c i e n t r e s i s t a n c e to flow t o make the bypass va l v e e f f e c t i v e without s h u t - o f f v a l v e s . Test 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. 99.9$ Ethane CP " ,99-0$ Hydrogen P r e p u r i f i e d " 99-9$ Butane CP " 99$ BY-PASS VALVE SOAP BUBBLE 'FLOW METER 3 TOLYETHYLENEN TUBING MOORE FLOW CONTROLS GAS I GAS 2 CONSTANT TEMPERATURE AIR BATH RIGHT ANGLE V NOZZLE ON C ' FITTING / / PACKED BED / CHROMATOGRAPH SAMPLE VALVE -GASKET GASKET FOR CROSSFLOW PATTERN AS IN ABOVE BED DIMENSIONS! MILLIMETERS 20 f— - J 20 GASKET AND POSITION OF PORTS FOR TANGENTIAL FLOW PATTERN Figure 2 . 2 Diffusion Apparatus The d e t a i l s o f the packed beds t e s t e d are shovn i n Table 2.1. The chromatograph columns were packed w i t h 25$ N u j o l on chromasorb. The s e p a r a t i o n of n i t r o g e n and ethane was accomplished w i t h a 9' x l/k" diam. column u s i n g a helium c a r r i e r . Hydrogen and n i t r o g e n were analyzed on the same column but 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 analyzed b y an 18" column w i t h helium c a r r i e r gas. Ten p s i g c a r r i e r gas pressure was used i n the long columns but the short column needed o n l y 2 p s i g . IV PROCEDURE A. SELECTION OF THE DISPLACED AND DISPLACING GAS I n the s e l e c t i o n of d i s p l a c e d and d i s p l a c i n g gas from a gas p a i r two f a c t o r s must be considered. The t a i l normally 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 rious when the columns are 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 time. 'Thus, i t was necessary to make the d i s p l a c e d gas the f i r s t peak t o appear on the chromatograph. The second e f f e c t to be considered i s t h a t the l i g h t e r gas should be placed on top, and i f the end zone i s a l s o a t the top of the bed then t h i s l a t t e r requirement i s c o n t r a d i c t o r y t o the 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 chromatographic t r a c e . TABLE 2.1 DIFFUSION CELL PROPERHES Bed Length, cms. Bed Diameter, cms. Length o f "End Zone", cms, Porosity- P r o p e r t i e s o f Packing m a t e r i a l P a r a l l e l 'Tube Packing 10.0 5.0 0.27 0.52 "Kimax" m e l t i n g p o i n t tubes 10 cm. long x 1.2 mm O.D. x 0.8 mm I.D. Porous S o l i d Packing 7.0 2.6l 0.27 0.59 Solas 01 Microporous s y n t h e t i c ceramic average pore s i z e h.5 S p e c i f i c s urface area 0.577 m 2/cm 3 o r 1.10 m 2/cm 3 by B.E.T. Ref. (5) S p h e r i c a l Packing Spheres 7.0 2.6l 0.27 0.39 H H v n 1 B o r o s i l i c a t e Glass - 1 1 6 - Three gas systems were tested on each bed, hydrogen-nitrogen, ethane-nitrogen and butane-nitrogen. The problems described above were overcome for the f i r s t pair by using a hydrogen carrier gas so that the hydrogen peak was lost completely. For ethane-nitrogen, i t was hoped that because of the identical molecular weights density effects would not be s ignif icant , however, this system does represent a more d i f f i c u l t separation i f chromatography is used for analysis. If the bed packing is f irmly held then obviously an inverted bed can be readily used also with gas chromatography for the analysis . Butane and nitrogen were readily separated i n the analysis, providing butane was used as the displacing gas. B. OPERATION OF EQUIPMENT To start a run the constant temperature a i r bath was brought up to i t s control temperature, ( 9 5 ° E ) > the carrier 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. After purging, the bypass was opened, and the displacing gas l ine connected and put on stream. The displacing gas was allowed to purge for about 1 0 minutes while the flow rate was measured on the soap bubble meter and adjusted to the desired range. The stop watch was started at the same time as the bypass valve was closed. Samples were taken and injected into the chromatograph at convenient times, u n t i l the displaced gas peak had become too small to give a satisfactory analysis , or u n t i l suff ic ient results 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 calibrations of the chromatograph indicated a l inear 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. If any discrepancy from the i n i t i a l value was found, the later measurement was u t i l i z e d because the Moore flow controls were found to d r i f t for the f i r s t few minutes after 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 for 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 results are recorded i n Appendix V for each run. The value of the d i f f u s i v i t y recorded is actually the Dg/^_ value which is obtained by this experiment. The d i f f u s i v i t y value i s for the temperature of the bed, but i s corrected to one atmosphere assuming no pressure drop i n the vent l i n e s . The effective d i f f u s i v i t y i s computed for the same conditions. The data for each bed are printed along with the constants and sums for the least mean square l ine computed from the data. Ten iterations were used for this least square calculation, but k or 5 were generally suff ic ient to obtain four figure accuracy. The number of iterations for the d i f f u s i v i t y calculation was set by a test of the - 1 1 8 - magnitude of the error, and this number i s recorded. Certain data points were rejected as described i n the following. These points are recorded, but they were not used by the computer. The results were calculated by a two-part computer program. A subroutine used the Wewton-Raphson (39) i terat ion described i n the "Theory" to compute the least mean square f i t of the equation Y - C = A exp(-Bt) to the data of peak heights (Y) vs . time, ( t ) . Then using the solution of the dif fusion 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 least squares l ine with a second Newton-Raphson i tera t ion . The least squares f i t of the equation i n the form Y-C = A exp (-Bt) weighs the l ine i n favour of the small time (large Y) points. Thus, i f the f i r s t or second point was inconsistent with the rest of the results , the computed slope showed this inconsistency i n spite of a l l the other points. From the plot of Log Y vs. t , points which appeared to be inconsistent when plotted have been discarded before arr iving at the values i n the following tables. The residence time of analysis gases i n the chromatograph was extremely short for the butane-nitrogen system, with the result 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 for rejecting data points are discussed where applicable. In order to compare the data, the tortuosity of the beds as calculated from each data point offers a convenient parameter. This calculation requires a knowledge of the value of the molecular d i f f u s i v i t y - 119 - for each gas pair used, and the values i n 'Cable 2.V show that available published results are not re l iable beyond t 5$. Because of this discrepancy the tortuosity only gives a good indication 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 tortuosity is 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 for the ethane- nitrogen system. These ratios may then be compared with the same ratios of the published experiments and calculated values, and give a comparison less dependent upon experimental error. 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 tortuosity p a r a l l e l to the tube bundle. The details of this bed are given i n Table 2.1, while the dif fusion results are summarized i n 'Table 2.II and shown graphically i n Figures 2.3, 2.4 and 2.5 as plots of the log peak heights vs. time. In some of the runs shown a Mil l ipore Type HA f i l t e r (80$ porosity) was placed over the bed of tubes to prevent eddy currents i n the dif fusion channels due to the flowing displacing gas. The results shown suggest that such currents are not s ignif i cant . An inspection of the tortuosities i n Table 2.II shows that the results scatter over a t 9$> range. Turning the bed on i t s side so that gravity effects became i n f l u e n t i a l increased the d i f f u s i v i t y by 5O70. The reason for the scatter can be seen i n the run with the hydrogen-nitrogen system at a flow rate of O.563 ml/sec. Three data points had to be discarded because the recorder automatic standardization operated and thus TABLE 2.II RESULT FOR PARALLEL TUBE BED Bed Temp. 306°K D i s p l a c i n g Gas N i t r o g e n Hydrogen Ethane Nitrogen Displaced Gas Hydrogen Nitrogen Nitrogen Butane Flovr Rate cm 3/sec. 0.510 0.544 O.565 2.81 Slope sec" 0.485 1.461 2.27 2.94 5.08 0.460 O.903 2.05 o.oo4o6 0.00434 0.0115 Average 0.00200 0.00295 0.00300 O.OO357 O.OO329 Average 6.00143 O.OOI58 0.00202 DB/X- cm 2/sec. 0.00364 O.55I 0.715 0.873 0.788 0.135 0.148 0.140 O.165 0.149 0.1505 O.0817 0.0766 0.0905 M o l e c u l a r D i f f u s i v i t y Used f o r X Computation cm 2/sec. 0.82 0.82 0.82 0.82 0.151 0.151 0.151 0.151 0.151 0.095 O.O95 O.O95 Average 0.084 T o r t u o s i t y Remarks X* 1.49 - ' Bed on Sid 1.15 0.940 I.05 M i l l i p o r e 1.11 1.02 M i l l i p o r e 1.08 M i l l i p o r e 0.915 1.015 M i l l i p o r e 1.16 1.24 1.016 M i l l i p o r e M i l l i p o r e M i l l i p o r e TIME ( S E C O N D S ) Figure 2.J> Results With. P a r a l l e l ,nube Bed. Ilydrog ,-n-!Iitrog-'n T I M E ( S E C O N D S ) Figure 2.k Results With Paral le l Tube Bed. Ethane-Nitrogen 0 1000 2 0 0 0 T I ^ E ( S E C O N D S ) Figure 2.5 Results With Paral lel fubo Bed. Butane-Nitrogen - 124 - caused a s h i f t i n the peak height p r o p o r t i o n a l i t y w i t h c o n c e n t r a t i o n . Removal o f these p o i n t s caused the l e a s t square l i n e slope to change from 0.00446 to 0.00434 w i t h a r e s u l t i n g change i n the D-Q/\ value from 1.008 to 0.873 cm 2/sec. The three discarded p o i n t s are not i n e r r o r by more than 4$, yet i n a t o t a l o f 15 p o i n t s these three cause a 3$ v a r i a t i o n i n the s l o p e , whieh i n turn 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 . C . POROUS SOLID PACKING The r e s u l t s o f the runs u s i n g p a r a l l e l tubes were i n i t i a l l y c a l c u l a t e d by hand from slopes obtained by g r a p h i c a l means. The s e n s i t i v i t y o f the method to s l i g h t e r r o r s was not appreciated a t the time, and e r r o r s were t o l e r a t e d i n the 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 those caused by the 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 set 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 these e r r o r s could a l s o be due i n some measure to eddy d i f f u s i o n w i t h i n the r e l a t i v e l y coarse-pored t u b u l a r packing, a d d i t i o n a l experiments were c a r r i e d out 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 Selas 01 ceramic f i l t e r medium s o l i d rod was f i t t e d t i g h t l y i n t o a 2.6l cm. diameter v e s s e l thereby h a l v i n g the former bed diameter, but the pore diameter was a l s o reduced from 0.8 mm (800 microns) to 4.5 microns. The d e t a i l s o f t h i s bed are g i v e n i n Table 2.1, and the r e s u l t s are summarized i n Table 2. I I I . Because o f the s m a l l e r diameter end zone, the flow p a t t e r n was changed from the former tangential i n l e t arrangement to one having f l o w i n one d i r e c t i o n across the chamber. - 125 - TABLE 2 . I l l RESULTS FOR POROUS SOLID PACKING Bea Temp. 306°K Molecular D i s p l a c i n g Gas Displaced Flow D B _ DE D i f f u s i v i t y x Gas Rate Slope A _ ,«B cm 2/sec cm 3/sec - sees 1 cm 2/sec Hydrogen Nitrogen 0.562 0.01J9 0.565 0.82 1.115 0.795 0.0159 0.596 0.82 1.37 0.928 0.0171 0.533 0.82 1.5U 1.25 0.0191 0.538 D o 1.52 1.85 0.0245 0.651 0.82 1.26 Average 0.577 1.4.3 Ethane Nitrogen "'0.39 0.00450 0.114 0.151 1.3̂ 0.82 0.00490 0.108 0.151 1.40 1.32 O.OO525 0.112 0.151 1.35 1.90 0.00523 0.108 0.151 1.39 Average 0.1105 1.37 n Butane Nitrogen 0.594 O . O O 3 3 7 0.0747 O.C99 1.32 1.14 O . O O 3 8 O 0.0802 0.099 I.23 2.06 0.00400 0.0823 0.099 1.20 Average .0791 1.25 An examination o f the r e s u l t s i n Table 2 . I I I . shows t h a t the ' t o r t u o s i t i e s are 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 the mean. However, the butane-nitrogen system t o r t u o s i t i e s are lower than those obtained from the other gases, i n d i c a t i n g t h a t a tru e d i f f u s i v i t y value higher than t h a t used would be a p p r o p r i a t e . I n order t o avoid the " 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 the d i s p l a c e d gas. The f a c t t h a t butane i s almost double the d e n s i t y o f n i t r o g e n would probably l e a d t o g r a v i t y e f f e c t s and could cause an apparent increase 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 the average t o r t u o s i t y o f the hydrogen-nitrogen and ethane-nitrogen systems i s not s i g n i f i c a n t as i t - 126 - depends upon the assumed value of the d i f f u s i v i t y . For example, i f a value o f 0.80 em 2/sec i s assumed f o r the hydrogen-nitrogen d i f f u s i v i t y r a t h e r than 0.82 cm 2/sec, both systems g i v e an average t o r t u o s i t y o f 1.37 "to I.38. The r e s u l t a t h i g h flow r a t e f o r the hydrogen c o n t a i n i n g system i s i n cluded i n the averages. I f t h i s r e s u l t i s ignored i t would appear that t h i s system i s showing about 5$ lower d i f f u s i v i t y r e l a t i v e t o the ethane system. The.average pore diameter of the Selas Bed i s 4.5 microns, w h i l e the mean f r e e path of hydrogen a t NTP i s 0.18 microns. I t i s u n l i k e l y t h a t the pore 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 percentage of the pores are not s m a l l e r than, say, 1.8 , a t which pore s i z e the r e s u l t a n t of the mixed Knudsen and b u l k d i f f u s i o n r a t e s could be 5$ l e s s than the b u l k d i f f u s i o n alone. 'Thus, i n s p i t e o f the f a c t t h a t the r e s u l t s l o o k f a i r l y good, use "- of the Selas 01 bed i s questionable 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 packing a l s o s u f f e r s from the need to c a l i b r a t e the bed t o f i n d the t o r t u o s i t y before i t can be used on gases o f unknown d i f f u s i v i t y . 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 published (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 provides an obvious means of overcoming the need t o c a l i b r a t e a porous s o l i d type o f packing to f i r s t determine i t s t o r t u o s i t y . The bed v e s s e l was the same as t h a t which h e l d the Selas 01, but i t was packed w i t h hi y diameter g l a s s spheres. However, the p o r o s i t y obtained w i t h the s p h e r i c a l packing was c o n s i d e r a b l y l e s s than f o r the porous s o l i d , and the r e s u l t i n g reduced bed c a p a c i t y l e d to a decay curve t h a t r a p i d l y decreased below the range o f a n a l y s i s by chromatography. The hydrogen-nitrogen r e s u l t s were most i n f l u e n c e d by t h i s e f f e c t . - 127 - TABLE 2.IV RESULTS FOR SPHERICAL PACKING Bed Temp. 306°K Displacing Gas Displaced Gas Flow Rate cm3/sec Slope s e c - 1 D_=DE A " B ~ cm 2/sec. Molecular D i f f u s i v i t y cm2/sec X Hydrogen Nitrogen 0.1)56 0.832 1.25 0.0164 0.0218 0 . 0 2 4 2 0.682 O.687 0.66l 0.82 0.82 0.82 1.20 1.19 1 . 2 4 Average 0.677 Ethane Nitrogen 0.604 0.919 I.36 0.00524 0.00521 0.00520 0.117 0.112 O.IO9 0.151 0.151 0.151 1.29 1.35 1.39 Average 0.113 Butane Nitrogen O.596 0.979 1 . 2 4 0.00364 0.00370 O.OO369 0.0795 O.0784 0.0775 0.099 0.099 O.O99 1.25 1.26 1.28 Average 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 diffusion flux measured i s the resultant of two decay processes, one due to the di f fus ion from the bed and the other due to dif fusion from stagnant portions of the piping. This la t ter contribution would normally be negligible for a d i f fus ion 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 this bed (Appendix V), the least square computation has shown the decay curve to approach a value higher than the data for larger times. For this reason, the slope and hence the d i f f u s i v i t y (see Table 2.Ill) i s higher for these runs, giving a lower tortuosity . - 128 - Figure 2.6 Results With Spherical Packing Bed. Hyeiogor.-Litrog,.n - 129 - 0 2 0 0 4 0 0 6 0 0 8 0 0 1000 T I M E ( S E C O N D S ) Figure 2.7 Results With Spherical Packing Bed. Ethane-Nitrogen The lover graph shows the above points after the steady state constant has "been subtracted - 1 5 0 - 100 1-23 PARAMETERS : FLOW RATE mis/sec. I I • • • • . • . . n . 0 200 400 600 800 1000 Tl^E (SECONDS) Figure 2.8 Results With Spherical Packing Bed. Butane-Nitrogen TABLE 2 .V COMPARISON OF RESULTS PUBLISHED DIFFUSIVITIES REF 40 Temp. Calculated D i f f . Experimental Ratio Temp. D i f f . °K Ratio EXPERIMENTAL RESULTS FROM THIS WORK Melting Point D 3 Ratio Selas 01 42 Micron Spheres D B Ratio D B Ratio H 2 - N 2 273.2 288.2 293.2 306* O.656 0.718 0.739 0.790 273.2 288.2 293.2 0.674 0.743 0.76 0.814 5.28 O.788 ^2k O.577 5.22 0.677 5.99 C 2 H 6 - N 2 298.2 0.144 306* 0.1̂ 98 1.0 298.2 0.148 0.154 1.0 0.1505 l i P _ 0.1105 1̂ 0 0.113 1.0 n C 4 H j 0 - N 2 298.2 O.O986 306* 0.1025 O.685 298.2 O.O908 0.0944 0.615 o.o84 0.567 0.0791 o.7i6 0.0785 0.695 •Extrapolated Values - 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 re la t ive ly somewhat higher than that of the ethane-nitrogen system, similar to the effect apparent i n Table 2 . I l l a lso . IV DISCUSSION The overal l potential error of the method cannot be estimated by the conventional methods due to the i terat ive nature of the solution. 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 potential accuracy of the method may be aided by examining the f i r s t root of the auxil iary equation, tanocL = h - k = Q, - _ _ = Q. PL - _ _ £ at A B D E X € B A B D B € B o c e B If the right hand side (RHS) of the equation is large, then «- L approaches 77/2 and oc becomes independent of the flow rate (Q), bed porosity (6 B), bed area (A B ) , end zone length ( t ) and gas d i f f u s i v i t y (D B ) . Thus i n order to reduce the present 10$ scatter of the experiments, i t 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 inversely proportional to the porosity, so i f h/tx,>? \LOC} a porosity decrease w i l l increase the RHS, however, the reduction of bed capacity which results , decreases the time available for - 1 3 3 - analysis of effluent concentrations. It 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 limitations as a decrease of the porosity, except that the benefits do not depend upon the relat ive 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 error. 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 results . 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 effective d i f f u s i v i t y and making the result flow dependent. Coupled to this i s the effect of pressure gradients from f r i c t i o n losses, or changes i n kinetic energy at the entry port, 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 effect of increasing k and hence decreasing the R H 3 . In the experimental apparatus used i n this work, k was negl igible , so that the end zone depth could probably be doubled without too much influence on the magnitude of " C . This depth increase might assist i n minimizing another potential source of error, 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 mill ipore f i l t e r used to discourage eddy penetration does not show any influence on the results , but this i s probably to be expected because the added resistance would not amount to more than 0.3$ of the to ta l while the results scatter to t lOJa. The mill ipore 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 large. On f i r s t inspection this i s an obvious improvement, however, there are l imita t ions . The dead time, before the second term of the series may be dropped, i s increased four fold by doubling the bed length. In the case of the 0.1 cm2/sec d i f f u s i v i t y gas the dead time was found to be 232 sees for a 10 cm bed (see introduction). In a 20 cm bed a 15 min dead time would be required. At the same time the effluent gas concentration must be considered. From equation 2.7, the effluent 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 after 232 sees would have doubled the concentration of the 20 cm bed after 928 sees. The longer bed thus has the effect of lengthening the time scale, 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 start 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 for the butane-nitrogen system as shown i n Figure 2.5 for - 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 points, and since the least 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 of a i r i n the gas systems appeared as nitrogen, resulting i n a curve -Bt _Bt / Y = A.e + C instead of Y = A,e (where Y and t are the variables) , which can be plotted as a straight l i n e . The use of high purity gases might simplify the interpretation of results . 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 indication of the presence of the last of these errors i n the large diameter p a r a l l e l tube bed (see Table 2.II). Examination of the data i n Tables 2.II, 2.Ill and 2.IV shows that for the p a r a l l e l tube bed (Table 2.II) there was no significant increase i n d i f f u s i v i t y with increasing gas flow rate. The other beds used, which were more isotropic i n structure, do tend to show such an increase with flow rate. As the range of flow rates used i n a l l beds was comparable, there appears to be a s l ight effect 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 for this la t ter 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 is satisfactory 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 readily achieved. VIII RECOMMENDATIONS On the basis of this work, i t i s apparent that a bed of the following dimensions could minimize the potential sources of error encountered i n the present experiments. Paral lel 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 off of the tubes and prevention of bulk flows through the bed is 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, part icular ly i f the dead time i s increased to 15 mins. or half an hour by the larger bed. Further Study The advantages of the larger bed should be experimentally ver i f ied and the magnitude of the flow effects , l i k e bulk flow and turbulence, should be investigated i f the larger beds are used to reduce the scatter. The effect of the end zone length on the mixing should also be investigated. - 137 - HOI gSKrjLA'i'TJIHil A,B and C,Constants i n least square equation. AQ Area of bed, cm2 Ap Specific surface area/unit volume of bed, cm" 1 A g . Sample or pulse volume, mis. C Q I n i t i a l gas concentration i n Section II , molcs/cm 3 C n Concentration i n stage n, moles/cm 3 C i Concentration i n mobile phase, moles/cm3 C 2 Concentration i n stationary phase, moles/cm C^ Concentration of component A, moles/cm C^Q End zone concentration, molcs/cm 3 C s Concentration at pel let surface, moles/cm3 C a V g Average concentration, moles/crru C I n i t i a l concentration, molep/cm. D Diffusion coeff ic ient , cm2/sec D-Q Molecular d i f fus ion coeff ic ient , cm2/sec Dĵ  Knudsen coeff ic ient , cm2/sec Dg Effective di f fus ion coeff ic ient , cm2/sec D-̂  Longitudinal dispersion coefficient (Overall including molecular term contribution), cm 2/scc D L * Eddy d i f f u s i o n coefficient (excluding molecular diffusion) cm2/sec E Effectiveness factor Fa Area fraction of mobile phase = 6-g F2 Area fraction of stationary phase = ( l - €3 ) H HETP, cms. HETP Height equivalent to a theoretical plant, cms. K Constants L Length of bed, cms. - 138 - lip Molar flux, moles/sec of component A per cm2 11^1 Molar flux per unit geometrical area, moles/(cm)2 sec. P Pressure, atm. Q Total flux, moles/sec, or gas flow rate, mis./sec. or pellet volume in Appendix IV K Gas constant or radius dimension T Temperature, °K U- Volume of gas, mis. 3 V Volume of gas phase i n theoretical plate, cm V Volume of theoretical plate cms3 c W Adsorp-ion or partition coefficients. Z Coefficient cf exponential. d Pellet diameter, cms. P d'n Column diameter, cms. f Fanning f r i c t i o n factor. h D Hydraulic diameter, cms. h Thiele modulus or i n section II h = Q/A^ D̂ , j 1 + % / H B k Fi r s t order rate constant, sec. k i Mass transfer coefficient i n mobile 'phase, cm/sec. k_ Mass transfer coefficient i n stationary phase, cm/sec. JL End zone rlength, cms. n Number of theoretical plates, or number of term i n series solution, r Pore radius, or radius variable i n di 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 velocity, cm/sec. v Volume of liquid sido of theoretical plate, cm3 v Average velocity of a gas molecule, cms/sec. - 139 - Distance i n direction of flux or flow, eras. Mole fract ion or peak height. Mass transfer coeff ic ient , s e c . - 1 (Section l ) Consecutive roots of equation (2.11) (Section Error i n equality of equation. Pellet porosity Bed porosity. Molar density, moles/ml. Density, grams/ml. Viscosity , cps Eddy d i f f u s i v i t y coeff ic ient . Tortuosity Standard deviation. - 140 - LITERATURE CITED 1. T h i e l e , E.W.,.Ind. Eng. Chem., _1, 9l6, (1939). 2. Wheeler, A., i n "Advances i n C a t a l y s i s " , V o l . 3, 249-327, Academic Pr e s s , New York, (1951); Wheeler, A., " C a t a l y s i s " , V o l . 2, IO5-I67, P.H. Emmett, Ed., Reinhold Pub. 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Bluh, 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 . , Edinburgh, (1955). - 14£ - 4-3. Kipping, P . J . , Jeff t ry , P.C. and Savage, C.A., Keccarch. and 0-y l-jp- ment for Industry, Wo. 39, P. l8, Feb. - March, (iS '^y). - lUp - APPENDIX I DETERMINATION OF THE EFFECTIVE GAS DIFFUSIVITY IN A POROUS SPHERICAL PELLET BY A STEADY STATE METHOD INTRODUCTION In order to evaluate the r e s u l t s obtained by the pulse technique an apparatus was constructed to measure the e f f e c t i v e gas d i f f u s i v i t y i n the 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 steady s t a t e method described by 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 regimes, Knudsen and b u l k , were a n t i c i p a t e d i n the two samples which were examined and so two s o l u t i o n s are needed f o r the d i f f u s i o n equation. Knudsen D i f f u s i o n The molar f l u x % i s g i v e n 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 Dg and the co n c e n t r a t i o n g r a d i e n t by F i c k ' s f i r s t law, N A = - Djj d C A at any plane x, moles/sec cm 2 dx R e f e r r i n g to Figure A 1.1, 'The t o t a l f l u x Q i s given by Q A = N A (Area o f plane) = N A TT (R 2 - x 2 ) moles/sec Q A = D E 7T ( R 2 - x 2 ) dCA dx dx R i x 2 P^TLdCA _1 2 R Ln R + x R - x QA +t Q A D E = QA_ Ln R + t 2 7 T R C A 2 - C A L U - • Since C A = f y A where y i s the mole f r a c t i o n and the molar d e n s i t y D E = _QA 2 TTR Pm Ln fR + t f LR - t j 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 but w i t h a c o r r e c t i o n f o r the bu lk flow caused by non-equimolar counter d i f f u s i o n . % = " D E D °A + ( I JA + N B ) m ° l e s / s e c c » 2 dx QA = % (Area) = - D_TT(R2 - x 2 ) d£ A + (Q A + % ) y A dx D E TT ( R 2 - x 2 ) Pro. djy_ = - Q A + Q A y A + Q _ y A dx _ £ A . l( i + Sa\ y A - i Q A J Q A dx DETrV>m ( R 2 - x 2 ) 1 l + Q B \ ^A) D B - Ln JAf 1 + 0B\ - 1 \ ^A) 7A, +t D_TTA m 2 TT R pm i'+ 9_a\ Q A ; Ln' R + t R - t Ln 'yA 2A + Q B \ - l ' \ ^ y A l / i + _ B \ - l \ ^A) 2R " n R ^ T cm 2/sec APPARATUS The apparatus shown i n Figure A 1.2 was assembled around a" p a i r o f brass b l o c k s between which a t h i r d b l o c k (shown i n Figure A I . l ) c o n t a i n i n g the sample p e l l e t was b o l t e d . The brass b l o c k s were constructed i n such a way t h a t two d i f f e r e n t gases could be f l u s h e d through m i r r o r image passages across the two faces o f the sample. 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 flowed through t h e i r r e s p e c t i v e c y l i n d e r pressure r e g u l a t o r s and "Moore" constant d i f f e r e n t i a l f low c o n t r o l s t o the reference sides 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 reference c e l l s the gases could be - 146 - POROUS PELLET a COUNTING f MOORE FLOW CONTROL CONNECTORS FOR CALIBRATION NITROGEN' CYLINDER GO W-MAC MODEL h 9220 "I THERMAL CONDUCTIVITY DETECTORS INCLINED MANOMETER BYEPASS VALVE r- NEEDLE -su VALVES FOR PRESSURE ADJUSTMENT \ f MOORE FLOW CONTROL HYDROGEN CYLINDER SOAP BUBBLE FLOW METERS Figure A 1.2 Steady State Apparatus - Ikl - diverted either across the sample faces, or via 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 inclined o i l f i l l e d manometer. Polyethylene l/V diameter tubing was used for connecting the apparatus, and this allowed flushing of dead end lines by loosening of the f i t t i n g s . The test samples were mounted by bathing them i n epoxy res in , 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. After the resin had set the faces of the pellet were ground by means of sand paper oh a glass plate . Two "DORION" potentiometers were used to measure the output of the ce l ls which formed part of a conventional bridge c i r c u i t . PROCEDURE Calibration of Thermal Conductivity Cells These "diffusion-type" ce l l s have the property of being re la t ive ly independent of flow rate, and at low concentrations a l inear 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 - 1 4 8 - side of the nitrogen c e l l . 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 levels 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 outlet 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 intervals 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 Calibration of thermal conductivity detectors Nitrogen content i n hydrogen c e l l Nitrogen flow: 25 mis i n 72.0, 72.2 seconds = 0.546 mis/sec. Hydrogen flow: 50 mis i n 8.2, 8.2, 8.6 seconds = 6.1 mis/sec. Mole $ nitrogen = 0 . 5 4 6 x 100 = 5*37$ 6.1 + 0 . 3 4 6 Output of 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, 7.5 seconds = 6.66 mis/sec. Hydrogen flow: 25 mis i n 6 8 . 0 , 68.5 seconds - O.367 mis/sec. Mole $ hydrogen = 0.567 x 100 = 5.22$ 6.66 + O.367 Output of detector 11.205 x 5 m i l l i v o l t s or 10.72 mv/l$ hydrogen - Ik9 - , It i s of interest to compare the above result with the cal ibrat ion of Cox (41)who obtained several points with a similar apparatus and ver i f ied 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 Pellet l A " Diameter Pellet Characteristics "Alcoa HI51 Activated alumina sphere" having h2 A mean pore diameter. Diameter of pel let used i n test = 0.255", 0.262", 0.262" Average Dia . = 0.66 cms Thickness of mounting plate, i . e . across f la ts of pel le 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 dif fusion w i l l be the predominant mechanism. The amount of nitrogen which diffused into the hydrogen stream i n this experiment was so small that with the lower sens i t iv i ty of this detector the output was of the same order as the zero d r i f t during the course of the experiment. For this reason the d i f f u s i v i t y i s calculated from the hydrogen f lux , Hydrogen flow rate: 50 mis i n 20.5, 20.5 sec. before test 22.0, 22.2 sec. after test Nitrogen flow rate : 50 mis i n 18.8, 18.8 sec. before test 19.0, 19.0 sec. after test 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 yielding 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 yielding 0.12 mv Subscript A refers to hydrogen °-A = 50 x 1.575 <°m = .00382 p moles/sec 19-0 IO.85 x 100 yA, = 1.0 - .125 x .01 = O.9995 mole fraction 1.78 V A 2 = 1.575 = 0.147$ = .00147 mole fraction 10.72 . 2 27TR p 2R + 2t 2R - 2t m 2 = .00582P_ Ln 2 ( # * = O.OO67 cm2/sec y i - y2 .66 + .476' \ \ . 6 6 - . 4 7 6 J 0.9993 - .00147 Knudsen d i f f u s i v i t y of hydrogen i n pel le t = O.OO67 cm2/sec at 26°C "Norton" Catalyst Support l/2" Diameter (Alundum) Pellet Characteristics Maximum diameter of pel le t = O.55" Minimum diameter of pel let = 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 dif fusion w i l l be the predominant mechanism. - 151 - Nitrogen flow rate 50 mis i n 28.0, 27.0, 27.5 seconds before test 28.1, 28.0 seconds after test Hydrogen flow rate 50 imLs in. 27.2, 27.1 seconds before test 29.8, 50.0 seconds after test Room temperature 25°C - , • Atmospheric pressure 160.1 mm Hg Analysis of streams Hydrogen i n nitrogen m i l l i v o l t s Nitrogen i n hydrogen m i l l i v o l t s 17.4-95 x 5 17.485 17.^91 17.1+6 17.^7 17.U5 17.^5 17.39 17.44 x 5 17.46 17.^5 17.44 17.37 17.35 17.39 U.h6 3-91 3.95 3-97 4.00 4.02 4.04 4.07 4.10 4.10 4.09 4.10 4.10 4.10 .̂13 4.13 4.15 Average = 17.421 mv Zero d r i f t : add 0.0 17.421 x 5/ 10.72 = 8.11$ Subscript A refers to hydrogen y A a = 0.081 QA = 50 . x .0811 p m = 28.05 QB = _5_9_ x .02425 C m = 29.9 - 4.103 m v add 0.22 4.32/ 1.78 = 2.425$ y A j = 0.97575 0.1445 P m m o l e s / s e c D Effective = 2TT R ? m = 0.0405 Pm moles/sec (1 + QB\ f 2R + 2 t 12R - 2t J 2 Ln 1 a + Q B \ • v QA/ • 1 = 0.1445 p} m 2TT l^pm 2 . = 0.0667 cm2/sec 0.0405_ 0.1445 Ln /I.565 + 0.?0\ 2 I 1.565 - 0.90 / 9̂7575 (.7191) ~ ) ,565 - 0.90 _ Ln /0.0811 (.7195) - 1 l o . - 152 - Effective 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 cm2/sec at 23°C and 76O.7 E ™ 1 Hg pressure. Scott and Dullien (5) pointed out that the ratio of fluxes of two gases diffusing at constant pressure i n capi l lar ies should be inversely proportional to the ratio of the square root of their molecular weights. In this experiment a rat io of 3«57 was obtained as compared with a value of 3.74 for the square root of the molecular weights. The difference is probably caused by the d i f f i c u l t y i n keeping the pressures identical across the p e l l e t . No absolute pressure measurements were taken i n the test 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 different pressure drops caused by the difference i n viscosi ty of the gases. Results on other equipment at similar flow rates'indicate 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 dif fusion coefficient for the Knudsen dif fusion of hydrogen i n a l/k" d i a . Alcoa H 151 activated alumina spheres was found to be O.OO67 cm2/sec at 26°C. The moisture content of the pel le t i s taken to be 12$ by wt from analysis of similar pel le ts , however, the actual moisture of the test pel le t during the test was not obtainable. The diffusion coefficient for the bulk di f fus ion of hydrogen and nitrogen i n l/2" d i a . Norton SA 203 Alundum catalyst carr ier spheres was found to be O.O667 cm2/sec at 23°C and 760.7 mm Hg. No moisture adsorption was found i n these p e l l e t s . I 1 f •111- S z * 9 S Hi e i : & i ii i S l I B W H i ! » a i » ^ liiii§ililiiiiilihlitii st^ i ^ a i a^iiiiiiiai^ i i i n i I oSS 3 * » S ? s A ! l [ -IJ4- s s or s « M • O 1 * 11 • Z l U U I L _ » » * • • * X • • * • — M 3 S U • t u • ± o O * 3 • • CL a r a o • • O X ft. * • • « s at M X 09 «- • • • • • • t aa • *t x 4 * o I H W W W U I M I K U V I 3 -« • • H. I t I I I f «t I _ 19 • K « • * K l f X M U » 9 4 - W 3 3 Z X H U O W I u « > v i ( a«4 < A * V»A*t«* - - - - -- M i l _ » X X X • • 3 I 3 1 3 M M - • _ • * * m m « • • • » S X X X * * N * * * 0 3 M M M M • • • • * tfl tfl B X - f • 3 « • t 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 « - » o —* O O 4 O t a t • * X « •• — O K 3 a x o x x «- O fl O f f 3MM • e * * • * « + + + * + > =i mmmmm » SSSssSsSSissSsBSsSs „ ° c S o o c ^ o o o - o o o o o o ' o o o c o csl»i!lfilESilif!iil •* 11 * Aiiiiiii i! jj J •! -1=1111111111111111111 i i '! ! I! j ilillillllililllgll . 1 5 1 T.Y .hiHlHliiiliiiiiil!-s 1 A Il ' fllllllfflBHHII! i\ l\l S|i ] . i "SK»"«»" s s ' « ' s " n » RUN COLUMN COLUMN PELLET BED PELLET OIFFUSIVITV NO LENGTH OIAHETER OIANETER POROSITY POROSITY CMS CMS CMS 10 11I.8000 9.0000 0.2080 0.3660000 0. CARRIER MW TEMP KELVIN PRESS ATM VISCOSITY 29.00000 294.10000 1.26900 0.01815*6 0.162)81388 HVO D M DENSITY 0.0766935 0.0011219 I 2 1 • * 6 1 S 9 10 11 12 13 VELOCITY HETP MOLECULAR PELLET RE NTU WIDTH TOTAL 0 EOOY OIFF PECLET EMPTY RE 1/SCH «E MO CM/SEC CMS PECLET CN CN MLS/SEC I.3TT19 0.25976 1.76410 2.60161 6T4 2.1900 11.9000 12.7000 0.1T9 0.626 0.879 1.699 0. 1.32060 0.26547 1.70169 2.31632 637 0.6600 6.0000 12.2300 0.176 0.63B 0.8*8 1.678 3. 1.26873 0.28468 1.62319 2.21232 636 0.3000 6.2000 11.7000 O.180 0.684 0.810 1.313 3. 1.16031 0.34131 1.48629 2.02323 399 0.6000 6.6000 10.7000 0.198 0.821 0.761 1.661 3. 1.06645 0.33333 1.34066 1.82669 360 0.6700 3.0300 9.6900 0.185 0.869 0.668 1.390 9. 0.92174 0.38881 1.18069 1.60726 317 0.8100 9.8200 8.9000 0.179 0.935 0.388 1.302 3. 0.80266 0.44057 1.02790 1.39923 276 1.0000 6.7500 7.6000 0.177 1.059 3.512 1.682 3. 0.49003 0.66467 0.97646 0.78471 154 2.2200 12.2000 4.1500 0.190 1.998 0.287 1.254 3. 0.29279 1.16811 0.37504 0.31054 100 4.9000 20.4000 2.7000 0.170 2.786 0.187 1.421 . 3. -0.06964026 AA« 0.2799876 0.369)7887 CC—0.1302027 0.07176007 SIGMA 0.02973078 GAMMA. 1.06)680 LANDA. -0.167404  RIM COLUMN COLUMN PELLET BEO PELLET DIFFUSIVITY NO LENGTH OIAMETER DIAMETER PORO$ITV POROSITY CMS CMS CMS SIP 118.1000 2.6000 0.2080 0.1830000 0. 0.20921196* CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY HVO DIA OENSITV 24.00000 296.00000 1.02000 0.0182169 0.0T92282 0.00121T9 1 2 1 4 S 6 T a 9 10 11 12 13 VELOCITY HETP MOLECULAR PELLET RE NTU K10TM TOTAL 0 EDOV OIFF PECLET EMPTY RE 1/SCH IE Hf 3 CN/SEC CMS PECLET CN CM MLS/SEC 11.64811 0.69431 11.56801 18.97991 1851 0.3800 2.1000 28.5000 4.718 1.659 7.269 31.678 7.233 17.096*% 1.17195 16.99572 21.77486 4824 0.4000 1.7000 35.7000 10.035 2.822 9.106 67.092 9.355 22.02917 1.16291 21.89925 10.61427 6217 1.3700 5.8590 46.0000 12.899 2.796 11.731 85.619 11.66* 21.4658* 1.64875 23.32746 12.61216 6622 1.4500 5.2000 49.0000 19.145 3.963 12.498 129.133 12.433 S.8S9S6 0.47216 8.80711 12.12011 2500 0.5000 3.3500 18.5000 2.092 1.HS 4.719 11.990 4.693 9.26784 0.34159 5.23678 7.32559 I486 0.6600 5.2000 11.0000 0.900 0.821 2.806 6.315 2.713 1.91558 0.11749 1.90428 2.66185 340 1.4500 11.8500 4.0000 0.304 0. 763 1.329 2.311 1.315 10.91881 0.58901 10.85441 15.18194 1081 0.4500 2.7000 22.8000 1.216 1.416 5.815 21.499 5.784 A 8 C SIGMA N -0.26517320 0.86951987 0.07300184 0.11659287 8 AA>-0.0705450 CC" 0.0640317 GAMMA* 2.077865 IANOA* •0.637416 R U N C O L U M N C O L U M N P E L L E T B E O P E L L E T O I F F U S I V I T V N O L E N G T H D I A M E T E R D I A M E T E R P O R O S I T Y P O R O S I T Y C M S C M S C M S 9 2 1 1 1 . 6 0 0 0 5 . 0 0 0 0 0 . 2 0 8 0 0 . 3 7 2 0 0 0 0 0 . 0 . 2 0 9 2 3 1 9 6 * C A R R I E R M M T E M P K E L V I N P R E S S A T M V I S C O S I T Y H Y D D I A D E N S I T Y 2 9 . 0 0 0 0 0 2 9 6 . 0 0 0 0 0 1 . 0 2 0 0 0 0 . 0 1 8 2 1 6 9 O . 0 T B 6 6 6 1 O . O S 1 2 1 7 9 V E L O C I T Y C M / S E C 2 3 . 9 9 8 0 2 2 1 . 3 3 1 5 7 2 0 . 3 1 1 6 5 1 8 . 6 6 5 1 2 1 7 . 3 3 1 9 0 1 6 . 3 9 8 6 4 1 4 . 6 6 5 4 5 1 0 . 6 6 5 7 8 1 4 . 3 3 2 1 5 1 1 . 0 6 5 7 5 8 . 8 6 5 9 3 7 . 1 ) 6 6 0 2 7 . 8 6 6 0 2 7 . 7 9 9 3 9 7 . 2 6 6 0 7 7 . 2 6 6 0 7 6 . 5 1 2 7 9 6 . 1 1 2 8 1 9 . 8 2 6 1 8 9 . 0 1 2 9 2 9 . 0 1 2 9 2 4 . 2 3 2 9 8 1 . 6 2 6 1 7 2 . 7 1 3 1 1 1 . 6 6 6 5 3 1 . 6 7 9 8 6 1 . 4 3 9 8 8 1 . 1 9 9 9 0 0 . 8 7 9 9 3 0 . 6 1 9 9 5 2 H E T P C M S 1 . 0 4 8 7 3 0 . 8 8 5 9 4 0 . 7 6 5 7 3 0 . 8 4 0 7 1 0 . 7 5 7 3 8 0 . 6 3 8 7 1 0 . 5 4 6 8 9 0 . 4 7 0 5 7 0 . 6 5 1 2 7 0 . 5 0 5 2 6 0 . 4 3 2 7 3 0 . 1 3 1 3 1 0 . 1 7 9 7 7 0 . 3 6 0 7 9 0 . 3 5 4 1 7 0 . 1 4 9 4 6 0 . 1 1 5 7 9 0 . 3 2 2 1 6 0 . 2 9 1 3 1 0 . 2 6 2 1 8 0 . 2 6 2 1 8 0 . 2 6 7 5 4 0 . 2 4 6 4 5 0 . 2 6 0 8 2 0 . 2 9 2 7 1 0 . 2 8 8 4 5 0 . 3 0 3 4 5 0 . 3 6 0 9 2 0 . 9 0 3 2 3 0 . 5 8 4 1 4 M O L E C U L A R P E C L E T 2 3 . 8 5 6 4 9 2 1 . 2 0 5 7 7 2 0 . 2 1 1 7 5 1 8 . 5 5 5 0 4 1 7 . 2 2 9 6 8 1 6 . 3 0 1 9 3 1 4 . 5 7 8 9 6 1 0 . 6 0 2 8 8 1 4 . 2 4 7 6 2 1 1 . 0 0 0 4 9 8 . 8 1 3 6 5 7 . 8 1 9 6 3 7 . 8 1 9 6 3 7 . 7 5 3 3 6 7 . 2 2 3 2 1 7 . 2 2 3 2 1 6 . 4 9 4 2 7 6 . 0 9 6 6 6 5 . 7 9 1 8 2 4 . 9 8 3 3 5 4 . 9 8 3 3 5 4 . 2 0 8 0 2 3 . 6 0 4 9 8 2 . 7 1 6 9 9 1 . 6 5 6 7 0 1 . 6 6 9 9 5 1 . 4 3 1 3 9 1 . 1 9 2 8 2 0 . 8 7 4 7 4 0 . 6 1 6 2 9 4 9 6 T a 9 1 0 1 1 P E L L E T R E N T U W I D T H T O T A L 0 E D D Y O I F F P E C L E T E M P T Y R E C M C N M L S / S E C 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 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 2 8 . 2 7 3 6 7 5 4 1 1 1 . 2 5 0 0 6 . 4 3 1 * 0 1 5 2 . 5 0 0 0 7 . 7 8 4 1 . 8 4 1 1 0 . 5 1 8 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 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 2 2 . 8 0 4 3 3 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 2 0 . 3 9 4 1 2 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 1 4 . 8 3 2 0 9 2 8 4 9 1 . 6 0 0 0 1 0 . 4 5 O O 8 0 . 0 0 0 0 2 . 5 1 9 1 . 1 3 1 9 . 5 1 8 1 9 . 9 3 0 6 2 3 8 2 9 1 . 4 5 0 0 8 . 0 9 O O 1 0 7 . 5 0 0 0 4 . 6 6 7 1 . 5 5 6 7 . 4 1 4 1 5 . 3 8 8 2 9 2 9 5 6 1 . 6 9 0 0 1 0 . 4 0 0 0 8 1 . 0 0 0 0 2 . 7 9 6 1 . 2 1 5 5 . 7 2 4 1 2 . 3 2 9 1 7 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 4 . 9 8 6 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 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 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 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 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 9 6 8 0 . 8 0 0 0 7 . 2 2 0 8 2 7 . 2 0 0 0 0 . 4 4 7 0 . 5 9 2 1 . 8 7 6 3 . 8 0 0 7 2 7 3 0 1 . 1 0 0 0 9 . 6 5 0 8 2 0 . 9 0 0 0 0 . 3 9 6 0 . 6 2 7 1 . 4 1 4 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 2 . 0 0 2 3 3 3 8 4 2 . 2 5 0 0 1 8 . 3 0 0 8 1 0 . 8 0 0 0 0 . 2 1 8 0 . 7 2 9 0 . 7 4 5 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 0 . 2 1 7 0 . 8 6 8 0 . 5 2 1 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 S I G M A N 1 2 1 3 1 / S : H I E 1 Y 3 8 4 . 1 3 0 6 3 . 1 7 9 5 2 . 3 4 4 9 2 . 4 5 6 4 3 . 8 8 1 3 5 . 3 1 3 2 6 . 8 1 1 1 6 . 7 7 8 3 1 . 2 0 3 1 8 . 6 9 0 1 2 . 8 2 5 8 . 7 1 2 9 . 9 8 6 9 . 4 0 7 8 . 6 0 7 8 . 4 8 8 7 . 3 3 3 6 . 6 0 9 9 . 7 1 2 4 . 3 9 3 4 . 9 9 3 3 . 7 8 6 2 . 9 8 8 2 . 3 8 3 1 . 6 3 1 1 . 6 2 0 1 . 4 6 1 1 . 4 4 8 1 . 4 8 0 1 . 2 1 1 1 2 . 6 2 1 1 1 . 2 1 9 1 3 . 5 9 3 9 . B I T 9 . 1 1 5 8 . 6 2 9 7 . 7 1 3 9 . 6 1 3 7 . 5 3 8 9 . B 2 0 4 . 6 6 3 4 . 1 3 7 4 . 1 3 7 4 . 1 3 2 3 . 3 2 1 3 . 8 2 1 3 . 6 3 6 3 . 2 2 3 3 . 3 6 4 2 . 6 3 6 2 . 5 3 6 2 . 2 2 6 1 . 3 0 7 1 . 4 3 7 3 . 1 7 5 3 . 8 8 4 3 . 7 9 7 3 . 5 1 1 9 . 4 5 1 0 . 1 2 5 0 . 0 0 1 9 3 6 8 6 A A * 0 . 0 3 2 7 5 4 2 0 . 3 6 6 5 2 5 5 8 C C " 0 . 0 3 8 9 1 8 1 0 . 0 4 0 7 5 3 3 6 0 . 0 3 2 7 1 1 6 9 1 0 G A M M A " 0 . 8 7 5 8 7 9 L A M U A " 0 . 0 0 3 6 9 4 RUN COLUMN COLUMN PELLET BED PELLET OIFFUSIVITV NO LENGTH OIANETER DIAMETER POROSITY POROSITY CMS CMS CMS 11 186.1000 6.2700 1.0300 0.4010000 0. 0.209233164 CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY HVD OIA OENSITY 29.00000 296.00000 1.02000 0.0182169 0.3947390 0.0012179 I 2 3 4 1 * T S 9 10 11 12 IS VELOCITY HETP MOLECULAR PELLET RE NTU WIDTH T3TAL 0 EOOV DIFF PECLET EMPTY RE 1/SCH »fc HV3 CM/SEC CMS PECLET CN CN MLS/SEC 0.21928 1.24171 3.73771 1.2281B 338 2.2000 11.4000 9.7100 0.471 0.631 2.118 1.162 2.334 1.38227 0.92379 6.80414 9.11869 611 1.7100 14.6000 17.7100 0.638 0.448 3.811 4.269 1.648 2.06168 1.01114 10.13890 14.21101 918 1.8100 22.1300 26.1000 1.067 0.493 1.711 7.003 1.446 2.14610 1.01431 12.13170 17.13181 113) 3.1000 17.8000 12.7000 1.292 0.412 7.102 8.636 6. 723 1.01268 0.998)0 11.02710 21.02117 1319 2.6000 11.0300 39.2000 1.124 0.48> 8.114 10.187 9.116 1.14329 1.12271 17.44264 24.40001 1377 2.4000 11.1000 61.1000 1.989 0.16* 9.882 13.298 9.311 1.91320 1.21201 19.26117 26.94729 1742 2.2100 11.8200 10.2100 2.171 0.188 10.914 11.811 13.327 4.36098 1.03526 21.46786 30.03081 1941 1.9000 10.8000 16.0000 2.217 0.131 12.162 11.092 11.139 4.69194 1.08964 23.09712 32.30993 2088 1.8100 10.2600 60.2100 2.116 0.129 11.386 17.090 12.38) S.81181 1.00200 18.78438 26.27696 1698 2.2100 13.0000 49.0000 1.912 0.486 10.642 12.781 13.173 4.19460 1.02629 22.61792 31.63960 2043 1.8900 10.7900 19.0000 2.118 0.498 T 2 . S U 11.763 12.126 1.41122 1.10811 26.83482 37.13811 2426 1.7200 9.4100 70.0000 1.020 0.118 11.203 23.193 14.386 6.10784 1.18432 31.01172 43.43742 2808 1.1900 8.4100 81.0000 1.711 0.171 17.192 24.973 16.647 A B C SIGMA H 0.68256106 0.31607676 0.07115107 0.06606999 11 . AA» 0.7240111 CC- 0.0641448 GAMMA. 0.810906 LAHDA- 0.111162 RIM COLUMN COLUMN PELLET BED PELLET NO LENGTH OIAMtTER DIAMETER POROSITY POROSITY CMS CMS CMS I* IBS.4000 0.4150 0.2975 0.6300000 0. D I F F U S I V I T Y 0 . 2 0 9 2 3 3 9 6 4 CARRIER MW 2 9 . 0 0 0 0 0 TEMP K E L V I N PRESS ATM 2 9 6 . 0 0 0 0 0 1 . 0 2 0 0 0 VISCOSITY 0 . 0 1 8 2 1 6 9 HVD OIA 0 . 1 6 7 3 6 2 1 DENSITY 0 . 0 0 1 2 1 7 9 1 2 3 6 3 6 7 a 9 10 11 12 V E L O C I T Y H E T P MOLECULAR P E L L E T RE NTU WIDTH TOTAL 0 EDDY D I F F PE : L E T EMPTY RE I / S C H <t • CM/SEC CMS P E C L E T CN CM M L S / S E C 8 . 3 4 2 0 3 0 . 8 7 ) 7 3 1 1 . 8 6 1 1 3 1 6 . 5 9 2 2 6 3695 1 . 4 5 0 0 8 . 9 5 0 0 0 . 7 3 0 0 3.666 1.45B 1 0 . 4 3 3 2 4 . 3 6 6 1 3 . 8 2 7 2 0 1 .12457 1 9 . 6 6 0 2 6 2 7 . 5 0 2 2 0 6 1 2 6 1 . 0 5 0 0 5 . 7 0 0 0 1 . 2 1 0 0 7 . 8 0 9 1 .898 1 7 . 1 2 6 6 2 . 2 1 1 l i 1 7 . 5 9 8 2 6 1 .44479 2 5 . 0 2 2 1 4 3 5 . 0 0 2 8 0 7 7 9 6 1 . 0 0 0 0 6 . 8 0 0 0 1 . 5 4 0 0 1 2 . 7 1 3 t . 4 2 8 2 2 . 9 5 2 8 4 . 9 9 4 11 2 2 . 7 4 0 6 1 1.33151 3 2 . 3 3 3 8 1 4 5 . 2 ) 0 8 9 10075 0 . 4 2 0 0 4 . 6 0 0 0 1 . 9 9 0 0 1 6 . 1 6 0 ( . 2 3 8 2 8 . 4 9 5 1 0 1 . 2 1 9 21 2 5 . 1 4 0 3 7 1 .86726 3 5 . 7 4 5 9 2 5 0 . 0 0 4 0 0 11138 0 . 9 0 0 0 3 . 8 0 0 0 2 . 2 0 0 0 2 3 . 6 7 2 1.138 3 1 . 5 0 3 1 5 6 . 9 2 5 I< 2 9 . 8 2 5 6 2 2 . 1 9 ) 7 8 4 2 . 4 0 7 6 6 5 9 . 3 2 2 9 3 13214 0 . 8 6 0 0 3 . 3 5 0 0 2 . 6 1 0 0 3 2 . 7 1 5 1 .687 3 7 . 3 7 3 2 1 8 . 7 2 5 21 3 8 . 2 8 1 9 ) 2 . 8 2 9 0 4 5 4 . 4 3 1 2 9 7 6 . 1 4 2 4 6 16960 0 . 8 6 0 0 2 . 9 5 0 0 3 . 3 5 0 0 5 4 . 1 5 0 k.7S5 6 7 . 3 7 0 3 6 2 . 0 3 3 31 2 8 . 5 6 8 6 0 2 . 3 ) 8 7 3 4 0 . 6 2 0 3 6 5 6 . 8 2 2 7 3 12657 0 . 9 6 0 0 3 . 4 4 0 0 2 . 5 0 0 0 3 6 . 2 6 6 < k .257 3 5 . 7 9 8 2 4 2 . 4 4 9 21 2 1 . 0 2 6 4 9 1.69844 2 9 . 8 9 6 5 9 4 1 . 8 2 1 5 3 9 3 1 5 0 . 9 6 0 0 4 . 2 5 0 0 1 . 8 4 0 0 1 7 . 8 5 6 1 .855 2 6 . 3 6 B 1 1 9 . 3 8 3 23 1 2 . 7 9 8 7 4 1 .15076 1 8 . 1 9 7 9 2 2 5 . 4 5 6 3 8 5670 1 . 1 1 0 0 5 . 9 7 0 0 1 . 1 2 0 0 7 . 1 6 6 1 .934 1 6 . 9 3 8 4 9 . 2 3 6 11 3 YD - 0 . 2 2 8 6 3 9 2 6 AA' 0 . 1 8 6 7 1 0 8 3 . 8 8 1 3 * 1 9 9 C C 0 . 0 6 8 8 3 7 1 0 . 0 7 8 8 2 7 2 1 SIGMA 0 . 1 8 2 2 1 1 2 1 N 10 GAMMA' 9 . 2 7 1 1 2 6 LANDA" -0.38*100 RUN COLUMN COLUMN PELLET BED PELLET NO LENGTH OlAHETER OIAMETER POROSITY POROSITY CMS CMS CMS 55 121.0000 1.1500 1.0050 0.4540000 0. OlffUSIVlTY 0.19&6992U CARRIER MW 29.00000 TEMP KELVIN PRESS ATM VISCOSITY HYD OIA DENSITY 296.00000 1.0S500 0.0132169 0.2695177 0.0012955 I 2 3 4 5 6 7 8 9 10 VELOCITY HETP MOLECULAR PELLET RE NTU WIDTH TOTAL Q EDDY OIFF PECLET CM/SEC CMS PECLET CM CM ' MLS/SEC 6.67826 1.10204 34.12138 47.73147 2054 1.2500 5.5500 3.4400 3.680 0.548 4.69808 0.81204 24.00399 33.57854 1445 1.4500 7.5000 2.4200 1.908 0.404 4.6980S 0.92790 24.00399 33.57854 1445 1.5500 7.5000 2.4200 2.180 0.452 3.57209 0.86016 18.25097 25.53079 1098 1.9500 9.8000 1.8400 1.536 0.428 2.64024 0.79122 13.48985 18.87058 812 2.5000 13.1000 1.3600 1.045 0.394 2.09666 0.75765 10.71253 14.98546 644 3.1000 16.6000 1.0800 0.794 0.377 1.70839 0.79877 8.72872 12.21038 525 3.9500 20.6000 0.8800 0.682 0.397 1.04833 0.88864 5.35626 7.49273 322 7.2000 35.6000 0.5400 0.466 0.442 0.75713 0.91842 3.86841 5.41142 232 2.2000 10.7000 0.3900 0.348 0.457 0.43680 0.93106 2.23178 3.12197 134 5.9000 28.5000 0.2250 0.233 0.453 A 8 C SIGMA N 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 0.60157888 0.15542669 0.06020300 0.05700413 10 AA* 0.4058721 CC* 0.0954678 GAMMA* 0.395087 LAMDA* 0.299293 RUN COLUMN COLUMN PELLET BED PELLET DlfFUSIVITV NO LENGTH DIAMETER DIAMETER POROSITY POROSITY CMS CMS CMS 6 ) 4 2 1 . 0 0 0 0 0 . 6 6 0 0 0 . 5 6 8 0 0 . 4 4 7 0 0 0 0 0 . 0 . 7 6 1 4 9 6 2 ) 8 CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY HVD OIA DENSITY 2 9 . 0 0 0 0 0 2 9 6 . 5 0 0 0 0 0 . 9 8 ) 0 0 0 . 0 1 8 2 4 0 9 0 . 1 7 4 7 8 4 8 0 . 0 0 1 1 7 1 8 t 2 3 4 S 6 T B 9 1 0 11 1 2 1 3 V E L O C I T Y HETP MOLECULAR P E L L E T RE NTU WIDTH TOTAL 0 EDOV 01 FF P E C L E T EMPTY RE 1 /SCH « E NVO C M / S E C CMS P E C L E T CN CN M L S / S E C 9 . 4 6 4 9 5 0 . 4 9 9 6 3 7 . 0 5 9 9 0 3 4 . 5 3 5 2 8 2 6 1 6 1 . 0 0 0 0 1 2 . 3 3 0 0 1 . 5 9 0 0 2 . 3 6 6 0 .462 1 1 7 . 1 6 4 1 1 . 1 8 9 1 0 . 6 2 7 8 . 6 ) 1 5 6 0 . 4 9 8 1 6 6 . 4 3 8 2 8 3 1 . 4 4 4 4 3 2 3 8 6 1 . 1 0 0 0 1 3 . 5 6 0 0 1 . 4 5 0 0 2 . 1 3 0 0 . 4 1 « t 1 5 . 6 5 3 1 3 . 8 1 1 9 . 6 9 1 7 . 9 1 7 2 2 0 . 4 9 3 5 9 5 . 9 0 5 4 6 2 8 . 8 8 8 0 0 2 1 8 8 1 . 2 0 0 0 1 4 . 8 3 0 0 1 . 3 3 0 0 1 . 9 5 6 0 . 6 ) 4 > 1 4 . 3 S 7 1 2 . 5 5 2 B . B 8 9 7 . 2 0 7 8 9 0 . 4 9 2 8 3 5 . 3 7 2 6 3 2 6 . 2 8 1 5 6 1491 1 . 3 0 0 0 1 6 . 1 0 0 0 1 . 2 1 0 0 1 . 7 7 3 0 . 4 3 « > 1 3 . 0 6 2 1 1 . 4 0 2 6 . 3 6 7 6 . 7 2 6 6 6 ' 0 . 4 9 3 6 6 5 . 0 1 7 4 2 2 4 . 5 4 3 4 4 1859 1 . 3 9 0 0 1 7 . 2 9 0 0 1 . 1 3 0 0 1 . 6 6 0 0 . 4 3 ! 1 2 . 1 9 8 1 3 . 6 6 6 7 . 1 1 1 5 . 9 5 2 8 0 0 . 5 3 2 5 7 4 . 4 4 0 1 9 2 1 . 7 2 0 3 0 1 6 4 5 1 . 6 2 0 0 1 9 . 3 0 0 0 1 .0000 1 .585 0 . 4 6 4 1 1 0 . 7 9 5 1 3 . 1 8 1 6 . 6 8 6 6 . 2 9 7 9 9 0 . 3 4 3 6 6 3 . 9 3 1 7 7 1 9 . 3 3 1 0 7 1464 1 . 8 7 0 0 2 2 . 0 6 0 0 0 . 8 9 0 0 1 .440 0 .471 » 9 . 6 9 8 9 . 2 1 1 1 . 9 4 9 3 . 9 2 8 8 6 0 . 6 4 6 2 8 2 . 9 3 0 5 3 1 4 . 3 3 5 4 0 1086 2 . 7 0 0 0 2 9 . 2 0 0 0 0 . 6 6 0 0 1 . 2 7 0 0 . 5 6 9 7 . 1 2 5 8 . 1 1 6 4 . 4 1 1 2 . 6 7 8 7 6 0 . 9 1 6 6 1 1 . 9 9 8 0 9 9 . 7 7 4 1 3 740 , 0 . 9 3 0 0 8 . 4 5 0 0 0 . 4 5 0 0 1 . 2 2 6 o.eje 4 . 8 6 8 7 . 8 7 8 1 . 0 0 8 1 . 6 6 6 7 8 1 . 2 4 3 4 4 1 .24325 6 . 0 8 1 6 8 4 6 0 9 . 3 3 0 0 7 2 . 9 0 0 0 0 . 2 8 0 0 1 . 0 3 6 1.093 1 . 3 2 3 6 . 6 1 7 1 .871 • B C SIGMA N 0 . 1 0 6 1 1 3 2 6 1 . 8 7 8 7 2 8 0 1 0 . 0 1 8 8 9 4 2 1 0 . 0 2 2 4 9 6 0 0 1 0 A A . 0 . 6 0 7 0 6 0 1 C C — 0 . 0 2 0 9 6 S 4 GAMMA. 1.21137T RUN COLUMN COLUMN PELLET BEO PELLET OlFFUSIVITV NO LENGTH DIAMETER OIAMETER POROSITY POROSITY CMS CMS CMS 6* 421.0000 0.6600 0.5680 0.4970000 0. 0.200754301 CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY HVO OIA OENSITV 29.00000 297.00000 1.05000 0.0182648 0.1747848 0.001249S VELOCITY CM/SEC 8.57)53 7.81529 7.14541 6.36388 5.58235 4.63335 3.62853 2.45623 1.50723 HETP CMS 0.71677 0.67842 0.64067 0.59639 0.52179 0.46804 0.47013 0.47348 0.58603 MOLECULAR PECLET -23.69146 22.11203 20.21671 18.00651 15.79431 13.10928 10.26630 6.94950 4.26446 PELLET RE 32.53779 30.36861 27.76558 24.72872 21.69186 18.00424 14.09971 9.54442 5.85680 -0.05827764 AA> 0.2393774 0.79323625 CC« 0.0486036 NTU 8780 8194 7492 6672 5853 4858 3804 2575 1580 6 MIOTM CN 1.3000 1.3500 1.4500 1.5500 1.6700 1.9200 2.5000 3.7000 1.4000 0.0813202* 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 N 0.01178047 9 12 i/ s : H IE 20.530 18.131 15.65' 12.98; 9.96' 7.41 5.83! 3.37 3.021 GAMMA. 1.975639 LAMOA. -0.051301 RUN COLUMN COLUMN PELLET BED PELLET DIFFUSIVITY NO LENGTH OIAMETER DIAMETER POROSITY POROSITY CMS CMS CMS 65 119.5000 2.1750 0.5680 0.4630000-0. 0.1967*9071 CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY MVD OIA OEMSITV 29.00000 291.50000 1.05000 0.0180970 0.2465515 0.0012666 1 2 3 4 9 6 T 8 9 10 11 VELOCITY HETP MOLECULAR PELLET RE MTU MIOTH TOTAL 0 EOOY OIFF PECLET EMPTY RE CM/SEC CMS . PECLET CN CN MLS/SEC 7.08854 0.60142 20.46409 28.11152 2152 2.7000 16.1000 13.0000 2.119 0.911 11.825 6. 78864 0.55490 19.59830 26.94114 2061 2.7500 17.1000 12.4500 1.864 0.488 12.474 6.70685 0.54847 19.16218 26.61675 2016 2.7500 17.2000 12.3000 1.839 0.481 12.324 6.18864 0.51881 17.86671 24.56098 1879 2.9000 18.1000 11.3500 1.66T 0.474 11.372 5.61611 0.51662 16.21386 22.28882 1705 1.1500 20.1000 10.3000 1.451 0.455 10.320 5.34367 0.48800 15.42678 21.20684 1622 1.2500 21.9500 9.8000 1.304 0.410 9.619 4.77111 0.47337 13.773V1 18.93468 144B 1.5500 21.9000 8.7500 1.129 0.417 8. 767 4.49850 0.47137 12.98663 17.85270 1166 3.7500 25.3000 8.2500 1.060 0.419 8.266 1.57151 0.42841 10.31075 14.17396 1084 4.5500 12.2000 6.5500 0.765 0.177 6.563 2.90085 0.41559 6.37454 11.51228 880 9.4000 38.8000 5.3200 0.691 0.366 5.330 2.13747 0.40664 6.17071 6.48274 649 7.2000 52.3000 3.9200 0.439 0.158 3.928 2.26288 0.41787 6.53277 8.98045 687 1.6000 11.2000 4.1500 0.499 0.185 4.158 1.55401 0.44896 4.48616 6.16729 471 2.1000 19.9000 2.8500 0.149 0.193 2.899 0.98149 0.55904 2.83349 1.89511 298 4.1000 25.4000 1.8000 0.274 0.492 1.801 A B C SIGMA N 0.12397121 AA* 0.1811939 0.17161307 CC- 0.0491846 0.09709327 0.01086196 16 12 13 l/SCH « HYD 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 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 GAMMA* 0.944181 LAHOA* 0.109131 RUN NO COLUMN LENGTH CMS 119.5000 COLUMN OlAMETER CMS 2.1750 PELLET DIAMETER CMS 0.5680 0 SEO POROSITY PELLET POROSITY DIFFUSIVITY CARRIER MM 29.00000 TEMP KELVIN PRESS ATM 293.50000 1.00500 *630000-0. VISCOSITY 0.0180970 0.732060581 HYO DIA 0.2*65515 DENSITY 0.0012102 1 2 3 4 5 6 VELOCITY HETP MOLECULAR PELLET RE NTU WIDTH CM/SEC CMS PECLET CM 33.0*188 1.08620 25.63693 125.50986 2696 0.9000 26.*90*7 0.9562* 20.55375 100.62*29 2162 0.9500 21.79055 0.72807 16.90711 82.77159 1778 1.0500 16.09367 0.59599 12.*869S 61.13196 1313 1.2000 8.260*7 0.37585 6.*0923 31.377*7 67* 1.8000 7.*059* 0.37903 5.7*621 28.13152 60* 2.1000 7.1*958 0.*0062 5.5*730 27.1577* 583 2.2000 6.57989 0.37821 5.10529 2*.99377 537 2.3500 5.78233 0.39897 4.*86*6 21.96*23 * 7 l 2.7000 5.32658 0.39289 4.13285 20.23306 *3* 2.9500 5.01325 0.1001*. 3.8897* 19.0*288 *09 3.1000 3.81691 0.*2907 2.96151 l*.*9855 311 *.2000 3.30*19 0.*51S2 2.56369 12.55099 269 5.0500 2.53511 0.52718 1.96697 9.6296* 206 6.8500 1.70906 0.71956 1.32605 6.*9189 139 12.0500 2.07936 0.5*927 1.61336 7.898*7 169 2.0000 1.39573 0.82132 1.0829* 5.30171 113 3.6000 0.82605 1.36681 0.6*092 3.13775 67 7.9000 0.*8*23 2.11930 0.37571 1.83937 39 3.3000 0.17091 5.09239 0.13260 0.6*919 13 13.3000 A 0 C 7 TOTAL CM *.0000 *.5000 S.7000 7.2000 13.6000 15.8000 16.1000 17.7000 19.3000 21.8000 22.7000 29. 7000 3*.8000 43.7000 65.8000 12.5000 18.*000 31.3000 10.5000 27.3000 0.12011538 AA—0.129712* 0.86630061 CC- 0.0396874 0.027*5608 SIGMA 0.07*73090 N 20 17.9*5 12.666 7.933 *.796 1.552 8 9 0 EDDY OIFF MLS/SEC 58.0000 *6.5000 38.2500 28.2500 1*.5000 13.0000 12.5500 11.5500 10.1500 9.3500 8.8000 6.7000 5.8000 *.*500 3.0000 3.6500 2.4500 l.*500 0.8500 0.3000 10 11 PECLET EMPTY RE 12 13 I/S:H « *»o i . * o * l.*32 1.2** 1.153 1.0*6 1.003 0.819 0.7*6 0.668 0.615 0.571 0.573 0.565 0.513 0.*35 0.956 0.8*2 0.6*1 0.525 0.331 0.33* 0.353 0.333 0.351 0.3*6 0.352 0.378 0.398 0.*6* 0.633 0.*8* 0.723 1.233 1.866 *.*83 58.111 *6.589 38.323 28.30* 1*.628 13.325 12.57* 11.572 10.167 9.363 8.817 6.713 5.811 *.*S9 3.006 3.6S7 2.*5S l.*53 0.852 0.301 120.008 8*.701 53.3*9 32.072 10.381 9.386 3.578 8.321 7.71* 6.998 6.708 5.*76 *.992 *.*69 *.112 3.819 3.833 3.775 3.431 2.910 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 GAMMA- 0.591686 LAMOA* 0.105735 RUN COLUMN COLUMN PELLET BED PELLET DIFFUSIVITY NO LENGTH OIAMETER DIAMETER POROSITY POROSITY CMS CMS CMS 69 186.1000 6.2700 1.0100 0.4040000 0. 0.209211964 CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY HYD OIA OENSITV 29.00000 296.00000 1.02000 0.0182169 0.1947190 0.0012179 1 2 1 4 5 6 VELOCITY HETP MOLECULAR PELLET RE NTU MIOTH CM/SEC CMS PECLET CM 26.08798 2.35665 128.42380 179.64859 11614 2.1500 21.90749 2.09059 117.68987 164.61318 10641 2.2500 21.41551 2.08198 105.4225? 147.47272 9514 2.4200 18.61202 I.91579 91.62175 128.16720 8285 2.5600 15.57491 1.81694 76.6709? 107.25289 6913 2.6900 15.57491 1.65182 76.6709? 107.25289 6931 2.8000 12.45991 1.66300 61.33674 85.80211 5547 3.3000 8.72195 1.27502 42.93572 60.06162 1882 4.1000 1.58221 1.00956 17.63431 24.66816 1594 2.0500 4.61154 1.03061 22.80960 11.90771 2062 1.6500 1.95211 1.02574 19.45525 27.21542 1759 1.9000 1.12914 0.92915 16.38841 22.92510 1462 2.2000 2.60880 0.92915 12.84218 17.96486 1161 2.8000 2.12208 0.95168 10.44641 14.61321 944 1.5000 1.46015 1.11456 7.18790 10.05496 650 1.1500 0.79821 1.14091 1.92918 5.49671 155 2.0500 A B C 0.70091049 0.11824411 0.06312746 AA* 0.7012348 CC* 0.0610176 7 8 9 10 11 12 13 TOTAL 0 EDDY OIFF PECLET EMPTY RE 1/SCM »E «V» CM MLS/SEC 8.1000 115.0000 30.740 1.144 72. 7! 18 205.514 68.849 9.0000 107.0000 24.990 1.015 66. 61 r6 167.378 63.994 9.7000 275.0000 22.293 1.011 59. 7, 16 149.046 56.918 10.6000 219.0000 18.014 0.940 51. 908 120.439 49.119 12.4000 200.0000 14.149 0.862 41, 43 If 94.598 41.104 12.6000 200.0000 12.864 3.832 41. 43 1? 86.301 41.194 14.8000 160.0000 10.160 0.607 34. 7! 10 69.26? 32.889 21.0000 112.0000 5.560 0.619 24. 3i E5 17.175 23.918 11.8300 46.0000 1.808 0.493 9, 91 11 12.089 9.494 9.4000 59.5000 2.188 0.500 12. 91 t l 15.964 12.228 10.6500 50.7500 2.027 0.498 11. Si •2 19.551 10.499 11.2000 42.7500 1.547 0.461 9. 21 15 10.340 8.789 16.8000 13.5000 1.212 0.451 7. 21 r6 8.109 6.889 70.7500 27.2500 1.010 0.462 S. 9 18 6.751 9.609 6.1000 18.7500 0.814 0.341 4. 3 r2 5.440 9.B99 11.1000 10.2500 0.455 0.554 2. .21 E6 9.944 2.10? SIGMA N 0.07076120 16 GAMMA* 0.760496 LANOA* 0.34024S RUM COLUMN COLUMN PELLET BED PELLET DIFFUSIVIfV NO LENGTH DIAMETER DIAMETER POROSITV POROSITY CMS CMS CMS TO 186.3000 6.2700 1.0300 0.6030000 0. 0.768330810 CARRIER MK TEMP KELVIN PRESS ATM VISCOSITY • HVD DIA DENSITY 29.00000 296.30000 1.00000 0.0182409 0.3967)90 0.0011920 1 2 1 4 1 * VELOCITY HETP MOLECULAR PELLET RE NTU KlDTH CM/SEC CMS - PECLET CN 4.61473 0.92911 6.37736 31.19642 176 1.6000 3.36167 1.00033 4.62164 22.62741 418 2.3000 2.88428 1.00200 3.96874 19.41408 318 2.TO0O 2.40688 1.10177 3.31181 16.20072 299 3.4000 1.69078 1.19846 2.32650 11.38067 210 1.1000 0.99418 1.18211 1.16811 6.69411 121 10.2000 1.34178 1.01170 4.19827 22.49316 411 2.4000 10.98014 0.98300 11.10819 73.90740 1166 1.0000 14.00366 1.11381 19.26892 94.21871 1742 2.1000 17.82284 1.20311 24.32408 119.96563 2217 2.2000 21.37371 I.12841 29.34111 143.51031 2613 1.8000 24.62171 1.16736 33.88484 165.71609 1064 1.7000 19.41416 A 1.33798 26.71373 B 130.67681 2411 C 2.1000 0.67149119 0.81181611 0.0232)081 AA* 0.1)90088 CC- 0.0297571 T B 9 10 11 If 11 TOTAL 0 EOOV DIFF PECLET EMPTY RE 1SSCM *E HYO CN MLS/SEC 9.6000 18.2100 2.111 0.411 12.611 14.371 11. * 11.3000 42.2100 1.6SI 0.486 9.164 10.988 8. 6 11.6000 16.2100 1.441 0.486 7.461 9.441 T. 4 18.TO0O 10.2100 1.111 0.117 6.161 8.696 6. 2 28.0300 21.2100 1.013 3.182 4.60* 6.621 4. 1 46.9000 12.1000 0.787 0.768 2.711 1.142 2. 1 11.8000 42.0000 1.690 0.491 4.110 11.042 a. 6 17.1000 118.0000 3.1*7 0.477 29.9)2 11.268 20. 1 11.7000 176.0000 7.799 0.141 18.171 13.966 16. 1 11.6000 224.0000 10.722 0.184 48.186 TO.066 41. 9 9.8000 268.0000 12.0)1 0.148 58.1)6 78.621 11. 0 9.1300 309.3000 14.374 0.167 67.111 91.910 61. .1 10.1000 244.0000 12.988 0.610 12.924 84.871 13. .3 SIGMA N 0.06791941 IS GAMMA. 0.171649 LAMDA* 0.127910 RUN COLUMN COLUMN PELLET BEO PELLET DIFFUSIVITY NO LENGTH DIAMETER DIAMETER POROSITV POROSITY CMS CMS CMS TI 122.0000 I.1100 1.0050 0.4540000 0. 0.725219170 CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY HVO DIA DENSITY 29.00000 290.00000 0.99400 0.0179282 0.2695177 0.0012114 1 2 3 4 5 6 VELOCITY HETP MOLECULAR PELLET RE NTU WIDTH CM/SEC CMS PECLET CN 26.98967 1.10044 37.40195 183.28539 22TO 1.3000 26.47064 1.05343 36.68268 1T9.7606T 2226 1.2500 24.39451 1.04817 13.80560 165.66179 2051 1.4000 20.76129 1.01516 28.77073 140.9B8T6 1746 1.5500 18.16613 0.88690 25.17439 123.36516 1527 1.6500 14.13290 O.B0681 20.13951 98.69213 1222 1.9000 11.93774 0.74504 16.54317 81.06854 1006 2.2500 B.82155 0.73362 12.22756 59.92022 742 2.8000 S.70935 0.69510 7.91195 38.77191 480 4.4000 1.55710 1.01222 2.15780 10.57416 130 15.8000 5.50174 0.62926 7.62424 37.36202 462 1.0000 1.J2I81 0.77561 4.60332 22.55820 279 I.T500 1.45329 1.11689 2.01395 9.86921 122 4.9000 0.5T094 2.34615 0.79119 3.87719 48 3.6000 a B c 0.30983961 AA- 0.3436004 1.13807026 CC" 0.0267854 0.02831543 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 0 EOOV OIFF MLS/SEC 13.0000 12.7500 11.7500 10.0000 8.7500 7.0000 5.7500 4.2500 2.7500 0.7500 2.6500 1.6000 0.7000 0.2750 10 11 PECLET EMPTY RE 12 IS l /SCH «E HYO SIGMA 0.03608801 N 14 16.850 13.942 12.785 10.538 8.056 5.863 4.467 3.2)7 1.9B4 0.788 1.731 1.288 0.812 0.670 0.547 0.524 0.521 0.535 0.441 0.401 0.371 0.365 0.346 0.536 0.313 0.386 0.556 1.167 83.212 81.611 75.210 64.009 56.308 44.806 36.835 27.204 17.602 4.801 16.362 10.241 4.481 1.760 100.346 94.212 86.389 71.207 54.4)4 39.615 33.350 21.870 13.408 5.325 11.697 8.705 5.484 4.526 49.15) 48.208 44.427 37.113 S3.384 26.467 21.741 16.369 13.398 2.8)6 10.323 6. 353 2.647 1.040 GAMMA* 0.784638 LAHOA* 0.154149 RUN COLUMN COLUMN P H I E T BED P E l l E T DIFFUSIVITY NO LENGTH OIAMETER OIAMETER POROSITY POROSITY CMS CMS CMS 72 122.0000 1.1500 1.0050 0.4540000 0. 0.210971311 CARRIER MW TEMP KELVIN PRESS ATN VISCOSITY HYD 01* DENSITY 29.00000 294.00000 1.00000 0.0181210 0.2695177 0.0012022 VELOCITY CM/SEC 27.19777 26.67474 25.62867 24.05957 21.96743 19.87530 17.78116 15.69102 12.02978 8.36855 5.23034 1.56910 2 3 4 S 6 7 HETP MOLECULAR PELLET RE NTU U10TH TOTAL CMS PECLET CM CN 2.35064 129.56047 181.33542 7863 1.9000 9.8000 2.13912 127.06892 177.84820 7712 2.0000 6.4000 2.17880 122.08583 1T0.87376 7410 2.090O 6.5000 2.21762 114.61119 160.41210 6956 2.1000 6.6000 1.93606 104.64500 146.46322 6351 2.2000 7.4000 1.87034 94.67881 132.51434 5746 2.2500 7.7000 1.73712 84.71262 118.56546 5141 2.4900 8.7000 1.57176 74.74641 104.61659 4936 2.6000 9.7000 1.39214 57.30559 80.2060$ 3478 3.0000 11.9000 1.15284 39.86476 55.79551 2419 3.9000 17.0000 0.99599 24.91548 34.87220 1912 S.8000 27.2000 0.84553 7.47464 10.46166 493 16.7000 83.0000 • C SIGH A 8 9 0 EOOV OIFF MLS/SEC 13.0000 12.7500 12.2900 11.9000 10.5000 9.9000 8.9000 10 11 12 19 PECLET EMPIY RE 1/SCH « HYJ 7.3000 5.7900 4.0000 2.5000 0.7900 31.966 28.930 27.920 26.67? 21.269 18.987 19.446 12.347 8.374 4.824 2.609 0.663 1.16 1.06' 1.08 1.13 0.96i 0.99 0.86 0.79 0.69 0.57 0.491 0.42 82.326 80.743 TT .5T7 72.827 66.696 60.162 93.829 4T.496 16.414 25.331 15.832 4 . 7 9 ) 212.967 189.279 189.224 176.981 141.375 123.307 102.469 81.911 99.951 32.002 17.280 6.401 48.633 47 .699 45.826 49.319 39.2TB 39.937 91.79? 28.099 2 1 . 9 0 * 16.969 9.392 2 .9 )9 0.64111141 0.17188688 0.06044230 0.09391870 N 12 AA- 0.9882264 CC« 0.0629386 GAMMA* 0.407367 TINE 16HRS 41MIN I2.6SEC LAMOA* 0.318961 • OAVIS FORTRAN SOURCE LIST GAAA 01/27/69 PAGE 1 IM SOURCE STATEMENT C PROGRAM FOR COMPUTATION OF RESULTS FROM POROUS PELLET RUNS I 61 REACH,NO,CL,CO,OP ,EB jEP*D08S_ 1*1 FORMAT lllr)Fi0.4,2FiO.T(FIZ.9 I * IF <NOI 64,61.84 * 1 64 CALL SKIP TO III A PRINT 16 T 16 OFORKATIRH COLUMN. 2X,6HC0LU*N,)X.6HC0LURN,4K,6HPELLET.1S,1MBED,68« liHPeilET.̂ X.llHDIfFUSIVITT I _ 10 PRINT IT 11 IT 0F0RMATt4X,2KN0 .4It6HLENGTH,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 -17J- COLUMN COLUMN COLUMN P E L L E T BEO P E L L E T O I F F U S I V I T Y NO LENGTH OIAMETER OIAMETER POROSITY POROSITY CMS CMS CMS 9 6 129.6000 0.6600 0.59T0 0.•.710000 0.3100000 0.207605682 CARRIER MW TEMP K E L V I N PRESS ATM VISCOSITY HYD OIA DENSITY 29.00000 2 9 6 . 0 0 0 0 0 1.02800 0.0182169 0.16559*5 0 . 0 0 1 2 2 7 5 VELOCITY CM/SEC 0.95941 1.31919 2.27860 2.87823 3.05812 4.79705 5.75646 6.17620 7.37546 7.97509 8.63469 8.75461 9.59410 10.31365 11.99262 HETP CMS 0.58797 1.00155 2.14265 2.85963 3.46638 5.35024 7.57958 7.30867 9.87107 10.41804 12.80894 11.75458 12.69080 12.82180 14.12322 R E C I P VEL SEC/CM 1.04231 0.75804 0.43887 0.34744 0.32700 0.20846 0.17372 0.16191 0.13558 0.12539 0.11581 0.11423 0.10423 0.09696 0.08338 RE 3.85938 5.3066S 9.16602 11.57814 12.30177 19.29689 2 3 . 1 5 6 2 7 2 4 . 8 4 4 7 5 2 9 . 6 6 8 9 8 3 2 . 0 8 1 0 9 34.73441 3 5 . 2 1 6 8 3 3 8 . 5 9 3 7 9 4 1 . 4 8 8 3 2 4 8 . 2 4 2 2 4 NTU 299 411 711 898 954 1497 1796 1927 2302 2489 2 6 9 5 2732 2994 3219 3 743 2. 2. 1. 1. 7. 5. 1. 5. 4. 4. 4. 4. 4. 3. 3. WIDTH CM 7500 17. 3900 7600 5600 7000 8500 1700 1000 9500 5500 6 0 0 0 3000 1000 8600 3500 1 1 . 5. 4. 1 9 1 2 . 2 9. 7. 6. 6. 6. 5. 5. 4. TOTAL CM 3000 5 2 0 0 8000 4 5 0 0 9 5 0 0 2000 0 5 0 0 1000 6000 8000 2 0 0 0 0500 5500 2 0 0 0 3000 Q CC/SEC 1600 2 2 0 0 3 8 0 0 4 8 0 0 5100 8 0 0 0 9 6 0 0 0 3 0 0 2300 3300 4 4 0 0 4 6 0 0 6000 7200 0 0 0 0 - 0 . 2 6 1 0 1 7 4 8 B - 0 . 6 3 9 1 4 2 6 5 1.32167980 SIOMA 0.6 3 2 8 3 8 8 6 H 15 EFFECTIVE O I F F U S I V I T Y * 0 . 0 0 0 8 5 0 2 0 CORRECTION OF SLOPE FOR O I F F U S I V I T Y TERM « 0.0605318 AND NEW SLOPE C - 1.26115 COLUMN COLUMN COLUMN PELLET BEO PELLET OIFFUSIVITY NO LENGTH OIAMETER OIAMETER POROSITY POROSITY CMS CMS CMS 57 421.0000 0.6600 0.5970 0.4.710000 0.3100000 0.210264673 CARRIER Mil TEMP KELVIN PRESS ATM VISCOSITY HYD OIA DENSITY 29.00000 296.00000 1.01500 0.0182169 0.1655945 0.0012120 VELOCITY CN/SEC 10.62794 9.53478 8.32016 7.28773 6.07311 5.22287 4.12971 2.97582 2.00413 1.21462 0.48585 HETP CMS 16.53753 15.40599 13.49911 11.65843 9.47155 8.50542 6.47520 4.53345 3.37384 2.20735 1.75714 RECIP VEL SEC/CM 0.09409 0.10488 0.12019 0.13722 0.16466 0.19147 0.24215 0.33604 0.49897 0.82330 2.05825 B RE 42.21196 37.87016 33.04593 28.94534 24.12112 20.74416 16.40236 11.81935 7.95997 4.82422 1.92969 NTU 10639 9545 8329 7295 6079 5228 4134 2979 2006 1215 486 8. 9 10. 10 12 2 3, 3 4, I 3, WIDTH CN 7000 18. ,3000 1000 8000 ,0000 ,6500 0000 6000 ,5000 ,3500 ,4000 20. 23. 27, 33. 7, 10. 14, 21, 7. 22, TOTAL CM 6000 ,6000 ,9000 ,5000 ,9000 ,9000 ,2500 ,7000 ,3000 ,9000 ,3000 0 CC/SEC ,7500 ,5700 ,3700 ,2000 ,0000 ,8600 ,6800 ,4900 ,3300 ,2000 ,0800 -0.22380747 0.57967149 1.60895641 SIGMA 0.19640320 N 11 EFFECTIVE OIFFUSIVITY » 0.00069313 CORRECTION OF SLOPE FOR OIFFUSIVITY TERM 0.0620292 ANO NEW SLOPE C > 1.54693 COLUMN COLUMN COLUMN PELLET BED PELLET OIFFUSIVITY NO LENGTH DIAMETER OIAMETER POROSITY POROSITY CMS CMS CMS SB 421.0000 0.6600 0.9970 0.4710000 0.5000000 0.204915337 CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY HYO DIA DENSITY 29.00000 296.00000 1.00790 0.0182169 0.1655945 0.0012035 VELOCITY CM/SEC 0.73391 2.20172 3.18026 S.32082 6.11589 6.72748 7.33907 8.68456 9.17384 10.76397 HETP CMS 5.41801 3.35020 4.786*6 8.01792 8.98325 10.17581 11.14604 12.20968 13.47536 17.10708 RECIP VEL SEC/CM 1.36257 0.45419 0.31444 0.18794 0.16351 0.14864 0.13626 0.11515 0.10901 0.09290 RE NTU 2.89453 753 21c 8.68360 2261 4. 12.54298 3266 3, 20.98537 5465 2, 24.12112 6282 2, 26.53323 6910 2, 28.94534 7539 2, 34.25199 8921 2, 36.18168 9423 2. 42.4531711057 9, WIDTH CM .9000 81. 8000 8500 8400 6200 5500 4000 1100 0900 8000 22. 15. 8. 7. 6. 6. 5. 4. 20. TOTAL CN 8000 8000 3000 7200 6000 9500 2500 2500 9500 6000 Q CC/SEC 1200 3600 5200 8700 0000 1000 2000 4200 5000 7600 -2.20553330 B 4.65074056 1.69947962 SIGMA 0.40248025 N 10 EFFECTIVE OIFFUSIVITY « 0.00126920 CORRECTION OF SLOPE FOR OIFFUSIVITY TERM - 0.0613776 AND NEW SLOPE C * 1.63810 OIFFUSIVITY WITH K»l/(bP»ADS> - 0.00450397 TORTUOSITY* 22.74828 AOSORPTION - 1.37000MLS GAS/ML OF PELLET COLUMN COLUMN COLUMN PELLET BED PELLET OIFFUSIVITY NO LENGTH OIAMETER DIAMETER POROSITY POROSITY CMS CMS CMS 39 421.0000 0.6600 0.5970 0.4710000 0.5000000 0.211745851 CARRIER HM TEMP KfcLVIN PRESS ATM VISCOSITY HYD DIA DENSITY 29.00000 296.00000 " 1.00790 0.0182169 0.1655945 0.0012095 VELOCITY CM/SEC 11.98715 17.43029 22.93459 27.88846 32.1084J 39.44749 47.70395 54.73722 63.91106 72.16751 77.67181 80.729 76 HETP CMS 17.32317 25.13199 30.70657 36.22900 40.22707 42.23292 42.51876 42.09674 35.33998 35.09355 31.52819 35.13947 RECIP VEL SEC/CM 0.08342 0.05737 0.04360 0.03586 0.03114 0.02535 0.02096 0.01827 0.01565 0.01386 0.01287 0.01239 RE NTU 47.2773911916 9, 68.7451917327 7 90.4541922799 5, 109.9923027724 4, 126.6358731919 4 155.5812139215 3 188.1447247423 3 215.8840154415 2, 252.0656963535 8, 284.6291971742 6 306.3382077214 6, 318.3987680254 6, MIDTH CM ,0000 18. 1500 8000 9500 4500 7000 0000 5000 0000 .9500 2000 0000 12. 9. 7. 6. 4. 4. 3. 11. 10. 9. 8. TOTAL CM 8000 4000 1000 1500 1000 9500 0000 3500 7000 2000 6000 8000 0 CC/SEC 9600 ,8500 ,7500 ,5600 ,2500 ,4500 ,8000 ,9500 ,4500 ,8000 ,7000 ,2000 68.01311398 B -592.80078888 -0.33316236 SIGMA 2.48643523 N 12 INVALIO OUTPUT FORMAT. -0.59035632E-02 EFFECTIVE OIFFUSIVITY » XXXXXXXXXX CORRECTION OF SLOPE FOR OIFFUSIVITY TERM = 0.0190111 AND NEM SLOPE C - -0.35217 COLUMN COLUMN COLUMN PELLET BED PELLET DIFFUSIVITY NO LENGTH DIAMETER DIAMETER POROSITY POROSITY CHS CMS CMS 60 420.0000 1.6000 1.3000 0.5220000 0.3800000 0.205018722 CARRIER MU TEMP KELVIN PRESS ATM VISCOSITY HYO OIA OENSITV 29.00000 295.00000 1.03500 0.0181690 0.4436744 0.0012400 VELOCITY CM/SEC 0.58324 0.95687 1.16647 1.52189 1.913TS 2.39674 2.73393 3.23515 3.71814 3.71814 HETP CMS 1.43971 1.17827 1.20172 1.25354 1.45079 1.59910 1.72312 1.88001 2.01567 2.04571 RECIP VEL SEC/CM 1.71457 1.04507 0.85728 0.65708 0.52253 0.41723 0.36577 0.30911 0.26895 0.26895 RE 5.17474 8.48981 10.34948 13.50284 16.97962 21.26496 24.25660 28.70365 32.98898 32.98898 NTU 597 980 1194 1558 1960 2454 2800 3313 3808 3808 UIOTH CN 5.9000 3.1500 2.5500 2.0500 8.3500 7.1500 6.5000 5.7000 5.1500 5.1800 42, 25, 20. 15. 60. 49. 43. 36. 31. 31. TOTAL CN ,7000 ,2000 ,2000 ,9000 ,2000 ,1000 ,0000 ,1000 ,5000 ,4500 Q CC/SEC ,6400 .0500 ,2800 ,6700 ,1000 ,6300 .0000 ,5500 ,0800 ,0800 0.29405186 0.49/73393 0.43967650 SIGMA 0.04052734 N 10 EFFECTIVE DIFFUSIVITY * 0.01938486 CORRECTION OF SLOPE FOR DIFFUSIVITY TERM - 0.1182664 ANO NEW SLOPE C - 0.32141 COLUMN COLUMN COLUMN PELLET BEO PELLET OIFFUSIVITY NO LENGTH OIAMETER OIAMETER POROSITY POROSITY CMS CMS CMS 61 621.0000 0.6600 0.S970 0.6710000 0.5000000 0.773230260 CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY HYO OIA DENSITY 29.00000 299.00000 0.98200 0.0183602 0.1635965 0.0011608 VELOCITY CM/SEC 0.95112 1.71202 2.28269 3.42604 3.07265 6.15059 7.16512 T.79920 8.94055 10.52575 11.28665 15.78863 25.23645 HETP CMS 1.80907 1.48969 1.55270 1.58861 1.87352 2.07753 2.33134 2.57822 2.79545 3.05998 3.42417 4.24708 6.23979 REC1P VEL SEC/CM 1.05139 0.58411 0.43808 0.29205 0.19714 0.16259 0.139S6 0.12822 0.11185 0.09501 0.08860 0.06334 0.03963 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 NTU 258 466 621 932 1380 1674 1950 2123 2433 2865 3072 4298 6870 WIDTH CN .2500 40. 9200 2000 3500 4000 7000 2500 1000 7500 3500 1500 5600 7000 20. 15. SO. 34. 28. 24. 22. 19. 16. 14. 10. 26. TOTAL CM 4000 8000 3500 7000 3000 3500 2000 2000 5000 6500 8000 8000 8000 Q CC/SEC ,1500 ,2700 ,3600 ,5400 ,8000 ,9700 ,1300 ,2300 ,4100 ,6600 ,7800 ,4900 ,9800 0.64075078 0.87210897 0.22362800 SIGMA N 0.074S0189 13 EFFECTIVE OIFFUSIVITY » 0.01016580 CORRECTION OF SLOPE FOR OIFFUSIVITY TERM » 0.0191109 AND NEW SLOPE C > 0.20452 COLUMN COLUMN COLUMN PELLET BEO PELLET DIFFUSIVITY NO LENGTH OIAMETER DIAMETER POROSITY POROSITY CMS CHS CMS 62 421.0000 0.6600 0.5970 0.4710000 0.3400000 0.767428882 CARRIER MM TEMP KELVIN PRESS ATM VISCOSITY HYO OIA DENSITY 29.00000 297.50000 0.98100 0.0182887 0.1655945 0.0011655 VELOCITY CN/SEC 1.89463 3.41034 4.42081 5.36812 6.31544 7.01014 7.83114 7.83114 8.58899 9.34685 10.23101 22.79873 HETP CMS 1.61910 1.64960 1.76095 1.89269 2.13187 2.208 75 2.39186 2.49881 2.56615 2.73856 3.06171 5.85278 RECIP VEL SEC/CM 0.52781 0.29323 0.22620 0.18628 0.15834 0.142A5 0.12770 0.12770 0.11643 0.10699 0.09774 0.04386 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 NTU 519 935 1212 1472 1732 1922 2148 2148 2355 2563 2806 6253 13, 7, 5, 4, 4, 4, 3, 3, 3, 3, 3, 1, MIOTH CM ,1500 89. 1500 8000 8500 4000 ,0000 7000 8000 5100 3500 2000 9200 48, 38. 30. 26. 23, 20. 20. 1». 17. 15. 6. TUTAL CM 8500 4000 0000 6500 2000 4000 8000 9000 0500 6000 9000 9000 0 CC/SEC ,3000 ,5400 ,7000 ,8500 ,0000 ,1100 ,2400 ,2400 ,3600 ,4800 ,6200 ,6100 0.37897281 1.49184255 0.23788515 SIGMA 0.04411453 N 12 EFFECTIVE DIFFUSIVITY » 0.00561640 CORRECTION OF SLOPE FOR DIFFUSIVITY TERM « 0.0192997 ANO NEH SLOPE C - 0.21859 COLUMN COLUMN COLUMN PELLET BED PELLET DIFFUSIVITY NO LENGTH DIAMETER OIAMETER POROSITY POROSITY CMS CMS CMS 73 119.4000 2.1700 0.3200 0.3900000 0.3100000 0.742124423 CARRIER MW TEMP KELVIN PRESS ATM VISCOSITY HYO OIA OENSITV 29.00000 295.00000 1.00000 0.0181690 0.1174626 0.0011981 VELOCITY CM/SEC 9.09386 8.92227 8.40753 7.72120 T.03487 6.00538 *.14747 4.11797 2.91690 1.88740 0.68633 HETP CMS 1.01778 0.98153 0.96925 0.89770 0.88500 0.81740 0.72322 0.72916 0.76273 0.92221 2.14016 RECIP VEL SEC/CM 0.10996 0.11208 0.11894 0.12951 0.14215 0.16652 0.19427 0.24284 0.34283 0.52983 1.45703 RE 19.18923 18.82717 17.74099 16.29275 14.84450 12.67214 10.86183 8.68946 6.15504 3.98267 1.44824 NTU -0.01542345 1.43413471 WIDTH CN 4.7500 4.9000 5.0500 5.3000 5.7500 1.6500 1.8000 2.2500 3.1500 -.6000 4.8500 0.09974282 21. 22. 23. 25. 28. 8. 9. 12 16. 27. 15. TOTAL CN 8000 9000 7500 9000 3000 4500 8000 2000 7000 0000 3500 Q CC/SEC ,2500 ,0000 ,2500 ,2500 ,2500 ,7500 .5000 ,0000 ,2500 ,7500 ,0000 SIGNA 0.01539098 N 11 EFFECTIVE DIFFUSIVITY - 0.00447061 CORRECTION OF SLOPE FOR DIFFUSIVITY TERM « 0.0166358 AND NEW SLOPE C » 0.07911 TIME 17HRS 21MIN 55.4SEC - 181 - TABLE A II.1 Low Flow (LF) Manometers Inches O i l •1.8 3.65 7.9 10.7 17.5 23.7 26.1 Flow Rate cm 3/sec. at 298°K 760 mm Kg O.338 0.674 1.416 1.87 2.82 3.65 3.96 High Flow (HF) Very High Flow (VHF) .91 1.5 2.8 4 . 1 5 . 1 7 . 7 7 . 0 1 0 . 2 15.1 8 . 1 1 4 . 4 18.1 2 1 . 1 2k.9 2 4 . 0 I.65 2.35 5.3 8 . 7 1 4 . 4 20 . 0 5 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 II.1 (Continued) Manometers Inches O i l 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 Flow Rate cm 3/sec. a t 298°F 760 mm Hg 287.5 271 246 233 204 171 139 106 6 4 . 4 4 8 E x t r a High Flow (EHF) 26.75 23.3 17.6 II.85 8.25 6.35 5.77 1.88 .95 3 4 8 323 278 221 180 152 112.5 73.6 5̂.3 J - 183 - 1 0 0 0 . CL* 5 t o o O LU 10 UJ cr o 0-1 V H F J i i i I m i I 1 0 M A N O M E T E R R E A D I N G I N C H E S O I L Too Figure A II.1 Flow Meter Calibration - 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 diffusion of a gas from a spherical pel le 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 (MM ) i s given by, OO M t = 1 = 6_ 1 exp / - D E n 2 T T 2 t \ n { a* J where a is the pel let radius. For the second term i n the series to be less than 1$ of the f i rs . t , { - D E TT 2 t \ = 100 exp / - k DE TT2t \ ^ P ( ) — ( — £ 2 — ) Ln ( 25)= ̂ TT g D E t - 7T 2 D E t = 5TT 2 DEt a 2 a 2 a 2 I f D£ = 0.01 cm2/sec and a = 0.5 cms t = .25 Ln(25) '=- 2.82 seconds 3 TT 2 x .01 Thus after ten seconds M+ = 1 - 6_ e (- .01 2 x ION = .9881 Moo T T 2 \ .25 J = 93.81$ Thus i n ten seconds 98.8$ of the gas w i l l diffuse out of a 1 cm diameter pel le t i f the effective gas d i f f u s i v i t y is 0.01 cm2/sec 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 Private Communication Apparent Porosity 0.4l .36 - AO Water absorption 20$ 15 - 19$ Bulk density 2.1 grams/ml 2.1 - 2.3 grams/ml Apparent Specific Gravity 3.4 - 3.6 Packing Density 75 - 73 l b / f t 3 Surface Area less than 1 meter2/gram Pore Diameter Range 9°$ i n 2-40 microns Using the bulk density value of 2.1 and the s p e c i f i c gravity of 3-5> a porosity of O.382 can be calculated. As a further check on the consistency of the data, i f the water adsorbed i s assumed to ex i s t as l i q u i d water occupying the pores, then 17$ water indicates a porosity of 0.37$. In an experimental check, a p e l l e t was dried by heating to 500°F, and then put i n a vacuum which was released by water so that the p e l l e t absorbed as much water as possible. Weighing the p e l l e t between each operation showed that the i n i t i a l water content was n e g l i g i b l e , but the evacuation and saturation procedure yielded a water content of 16.8$ which again i s an i n d i c a t i o n of a 37$ porosity i f the p e l l e t s p e c i f i c gravity of 3.5 i s accepted. In conclusion, a value of 38$ porosity has been taken for the pulse apparatus experiments and the information available suggests an error l i m i t of ± 1$. * Activated Alumina Pe l l e t s Alcoa H 151 l/4" and l/8" diameter Two sources of information were again used to determine the properties of these p e l l e t s , but some of these data were contradictory. Because one of these sources was private communication i n the form of a l e t t e r from the supplied, which d i f f e r e d from the manufacturer's data i t was concluded that the supplier had not furnished the correct data. ' This conclusion i s j u s t i f i e d i n the following paragraphs. - 186 - Ivianufacturer' s Data Private Communication Packing Density 52-55 l b / f t 3 51-53 Ib /Tts Specific Gravity 5.1-3.3 Pore .Volume 0.3 mls/gm O.5-G.55 ials/gm Pore Diameter 50 A° 40 A" Surface Area (BET) 350 meter2/gm 390 meter2/gm Pore Volume greater than 3OA0 0.28 mls/gm Average Pore Diameter i n 90 to 30A° region k2A° Static Adsorption at 60% RH 22-25$ A pore volume of 0.3 mis per gram represents a porosi'cy of 50$ i f the specific gravity of the pellets i s taken as 3.2, while a pore volume of 0.5 indicates a porosity of 63$. The problem amounts to deciding which set of data above are consistent. Placing the pellets i n a vacuum and releasing with water to measure the water absorbed showed a porosity of around 50 "to 55$ which is somewhat indeterminate, lying as i t does between the data from the two sources. However, for the l/8 inch activated alumina pellets the test described below was applied. The test pellets were placed i n the sample loop of a gas chromatograph and a hydrogen purge put on the loop. A i r carrier gas was put on the column, which consisted of a 20 f t . length of l/2" plast ic hose to cause dispersion and create a Gaussian pulse d is t r ibut ion . 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 pellets i n the sample loop, and without, the volume of the solids i n the pellets could be determined. The peak heights were calibrated i n terms of gas volume by injecting known volumes of 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 pellets occupied the tube was 4.40 mis. From the mean diameter the overall volume of the pellets was calculated to he 0.8l mis and so the porosity is given by (0.8l-0.40)/0.8l = 50.5% Similarly for 47 wet (12$) pe l le ts , porosities of 28.4 and 33$ were obtained. If the water i s assumed to exist as l i q u i d water (3l)> then the porosity of the wet pellets can be calculated from the moisture content i f the dry pel le t porosity is known. If a 50$ dry pel le t porosity is assumed, then for a 12$ wet pel le t the porosity comes out to be 30.8$. Thus, the manufacturer's data appears to give the best agreement with the observations. F i n a l l y , two individual pellets were weighed and the dimensions of three diameters measured on c a l l i p e r s . 'This allowed the apparent density of the pellets to be calculated, and i f the water content i s taken into account, the porosity of the dry pellets can be obtained from a knowledge of the true specific gravity. 0.699 cm pel le t weighed 0.3140 grams so density = 1.76" gms/ml 0.617 cm pel le t weighed O.23OO grams so density = 1.87 gms/ml I f 12$ water i n the pel le t i s assumed then the densities become 1.57 and I.67 respectively. In conjunction with the specific gravity the porosities of these two pellets 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$ static adsorption of moisture i n a i r of 60$ RH. Even with soaking i n water only i4$ water was adsorbable. However, i n an Alcoa product data - 188 - b u l l e t i n , ("Activated and Catalytic Aluminas", Feb. 1, 1963, Section GB2A, Figure 2, page 8), i t is shown that after about six months operation the adsorptive capacity of this material drops to 13 or lhc/a. The samples used i n this work were stored for six months before work was started, and so i t i s possible that the low moisture contents are to be expected. The dry pellets were assumed to have a 50$ porosity i n the pulse apparatus determinations and the porosity of the wet pellets was taken as 31$ corresponding to a 12$ wet p e l l e t . The l/k" and l/8" pellets were assumed to have the same properties. POROSITY OF PACKED BEDS Two general methods were used to obtain the porosity of the non porous pellet beds A.) If the density of the pel let packing was known, the bed was weighed before and after f i l l i n g and the pel let density used to convert the packing weight to a packing volume. The overal l volume of the vessel was calculated from the internal 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 = 1051 grams + packing = 35.5 l h . = 161028 grams Weight of packing = 15,°51 grams Volume of packing 15,051/10.808 = 1392 ml. Volume of bed T(5.0) 2 111.8 = 2195 ml. 4 Bed Porosity = 2195 - 1592 = 36.6$ 2195 J - 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. If 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 = 4J.0 grams " " " + packing =72.0 grams 11 " " " + water = 89.0 grams " " " + water = 70.0 grams Porosity of bed = (89.0 - 72.0)/(70.0 - H3.0) = 63$ The porosities of the beds of porous pellets are treated individual ly 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 pellets was found to be negligible and so the manufacturer's pel let density was accepted as a value of 2.05 grams/ ml. With the weight of pellets i n the bed measured, the porosity of the bed (not including pel let pores) was calculated by method A) above. Activated Alumina Pellets Alcoa iii51 l/4" diameter, RUNS 56,57,58,59,61,62 The bed used for these runs was a single pel le t diameter and the porosity was measured as follows. The average diameter was used to calculate the mean pel le t volume and then the number of pellets i n a measured length of tube was measured. The volume of the pellets i s thus known and the volume of the bed over the measured length can be calculated from the internal diameter of the vessel . - 1 9 0 - Example Forty-four pellets i n l ine occupy 2 5 . 0 cm. making a mean pel let diameter of O . 5 6 8 cm. (Note the pellets were graded so that a small pel let was not adjacent to a large pe l le t , which would introduce an error into the result . ) In the paeH©d "baa 18 pellets occupied 10 cm. The voluma of the pellets is thus 1 8 T T ( 0 . 5 6 8 ) 3 / 6 = 1 . 7 2 7 mis. and the volume of the vessel i s given "by 1 0 1 T ( 0 . 6 6 ) 2 / 4 = 3.I1-2 mis. The bed porosity i s ( 3 . 4 2 - 1 . 7 2 7 ) / 3 . 4 2 = 4 9 . 7 $ Activated Alumina Pel lets , Alcoa H 1 5 1 1 - / 8 " Diameter, RUN 7 3 The above method could not be applied to a bed of several part ic le diameters thick. The moisture content of the pellets was determined to be 1 2 $ , and weighing the bed before and after f i l l i n g showed that 4 6 7 grams of the wet pellets packed the bed. Thus the weight of dry pellets was 4 l l grams and since the density of the dry pellets was given as 3 . 1 "to 3 . 3 gm/cm. by the manufacturer, the volume of sol id can be calculated to be 1 2 9 ml. 'The 1 1 9 . 4 cm. long by 2 . 1 7 5 cm. diameter bed contains a volume of 4 4 l ml. and so the porosity of the bed can be evaluated by method A i f the dry pellets 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 corollary to the above calculation, the porosity of the dry activated alumina pellets i s unlikely to be 6 5 $ as claimed i n the supplier 's l i te ra ture . The assumption of a 6 5 $ pel let porosity yields impossible bed porosit ies , for example, a value of 5 ^ $ i s obtained and a 5 4 $ bed porosity i s extremely unlikely i n a random packed bed of uniform spheres. - 191 - E s t i r a a t i o n o f the Molecular D i f f u s i v i t y o f the Methane A i r System Ref vj>2) A i r Mol Wt. 28.97 or 3.617 A K 97.0 Tc 132 Pc 36.4 86.6 Methane 16.04 3.822 137.0 190.7 45.8 99-7 °AB - 3.7195 A € A B = K J 91 x 137 = 115.4°K At 298°K KT_ = 6AB 298 115.4 2.54 From Table B.2 J l O.990 = 0.212 cm2/sec - 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 pel le t s , adsorption i s known to influence the effective d i f f u s i v i t y i n a porous pel le t (2). The following two assumptions were made, the adsorption isotherm i s l inear , i . e . moles adsorbed/gm. so l id = W(partial press.) and the presence of other gases does not affect the adsorption isotherm. The apparatus i s shown i n Figure A IV.1 . The test chamber BC could be evacuated while the burette zone AB was purged with the test 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 of gas used i n the test could be adjusted by regulating the burette zone vent at B and adjusting the manometer l e v e l . The vacuum i n the test chamber could be shut off at C, and the "burette and test chamber connected at B. Thus by knowing the volume of the test chamber and the tube connecting to the burette zero, a series of measurements of the volume and pressure of trapped gas could be made by al ter ing the manometer posi t ion. A series of burette readings (volume) and manometer readings (pressure) were taken at corresponding points as well as the atmospheric pressure. The volume of sample solids was also obtained. Total gas i n the system = P QV Q Moles RT A material balance of the trapped gas y i e l d s , P Q V Q = P_V + W P p p Q R T R T - 195 - T E S T C H A M B E R V E N T B U R E T T E T E S T G A S v V A C U U M C O N N E C T I O N M E T E R R U L E >! M E R C U R Y 1 R E S E R V O I R y Figure A VI.1 Adsorption Measurement Apparatus - 19k - or V - FQVQ - WRT (QLQ P Where p i s the pel let density, and Q the volume of p e l l e t s . 1? Hence a plot 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 of gas adsorbed. If the overall volume of the pellets (including pores) i s known then the volume of adsorbed gas per unit volume of pellets i s easi ly obtained. RESULTS Table A IV. I shows the manometer and burette readings along with data i n terms of volume and inverse pressure for the methane, hydrogen and nitrogen. Two sets of data are recorded for methane, one assumed atmospheric pressure and the other at about l/2 atmospheres i n order to t ry and approach the concentration i n the pulse apparatus. These points are plotted i n Figure A IV.2. Following are the characteristics of the apparatus which had to be known to prepare Table A IV. 1, Volume from zero of burette to stop cofek B = 6.7 ml. Volume of empty test chamber = 25.29 ml . Pellets ALCOA H 151 activated alumina spheres 1 / V ' diameter. Weight of dry pellets i n sample = 7.68 grams Volume of so l id excluding pores = 7.65/3.2 = 2.k ml . Overall volume of pel le t (50$ porosity) = 4.8 ml. The overall volume of the pellets was also computed from the dimensions of the pellets and the same result was obtained. Hence volume to be added to the burette reading = 25.29 .+ 6.7 - 2.4 = 29.59 ml. Volume of gas = 29.59 + Burette reading Pressure i n chamber = Atmospheric ± manometer pressure dif ference RESULTS AND CONCLUSIONS - 195 - 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 least squares technique and methane showed 5.22 ml . adsorbed at a half 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 result giving 5«22 + .31 = 5.53 ml. i n k.8 ml. of pel let at a half atmosphere and 6.60 ml,, i n k.Q ml. pel le t at one atmosphere. Hence the methane adsorbed per unit volume of pel let material i s 1.15 ml. at a half atmosphere and 1.375 m l . / m l . pel le t at one atmosphere. The results for nitrogen are not part icular ly of interest 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. A least squares computation for the nitrogen data was not carried out. Figure A IV.2 Gas Volume v s . Inverse Pressure For Adsorption Measurement TABLE A IV.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 Burette Man. Press. V o l . Burette Man. Press. V o l . mis. cm. mm Hg a t m . " 1 mis. mis. cm. mm Hg atm mis. 0 - 1.5 74.4 .1.02 29.59 41.3 -14.0 62.5 1.215 70.89 2.5 - 6.2 69.7 1.09 51.89 43.0 -15.4 61.1 1.242 72.59 4.2 - 8.2 67.7 1.12 35.79 36.7 - 9.7 66.8 1.138 66.29 6.5 -12.8 63.I 1.20 36.09 30.7 - 3.2 73.3 I .O38 60.29 11.9 -19.7 56.2 1.35 41.49 25.3 + 4.3 80.8 .940 59.89 19.1 -26.8 49.1 1.55 48.69 23.9 + 6.4 82.9 .917 55.99 25.4 -31.8 44.1 1.72 5^.99 21.6 +10.7 87.2 .872 51.19 29.4 -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 H V O -4 <— p 25.4517 15.33 477.71 799.7635 8.86418 8.356 487.62 Intercept= -5.2208 mis. = 517.23384 Intercepts O.3167 mis, TABLE A IV. 1 (Continued) NITROGEN Atm. Pressure Room Temp. Man. Burette cm. mis. 4 2 . 3 - 1 7 . 0 3 9 . 0 -14.0 3 5 - 2 - 1 0 . 5 3 1 . 0 - 6 . 1 2 5 - 9 5 0 2 3 . 4 + 3 . 3 2 0 . 6 + 7 . 7 7 6 5 . 2 mm Hg 2 2 °C Press. mm Hg atm. 1 V o l . mis. 5 9 - 5 1 . 2 7 9 7 1 . 8 9 6 2 . 5 1 . 2 1 5 6 8 . 5 9 6 6 . 0 1 . 1 5 2 64 . 7 9 7 0 . 4 1 . 0 8 6 0 . 5 9 7 6 . 5 . 9 9 ^ 55.54 7 9 . 8 . 9 5 3 5 2 . 9 9 84 . 2 . 9 0 1 5 0 . 1 9 METHANE High Pressure 7 ^ 9 . 5 2 2 . 5 ° C Man. Press. V o l . Burette cm. mm Hg. atm. mis. mis. 31.0 -10 . 6 64 . 3 I.I85 60 . 6 3 3 . 3 -12 .7 62.2 1.225 62 . 9 37 . 5 5 -16.4 5 8 . 5 1.30 67.2 40 . 8 -18 . 9 56.O 1.354 70. ii 26 . 8 - 6 . 5 68.4 1.11 56.4 23 . 3 - 2 . 3 72 . 6 1.045 52 . 9 21.2 0 7 ^ . 9 1.018 50 . 8 19.0 + 3 . 3 78.2 O.965 48.6 7 1 -— p 2 = 10.71984 9.202 = 548.0431 = 1 : 6 9.8 n = 8 In"l ex c cpt = -6. 387 moles PROGRAM APPENDIX V B DAVIS FORTRAN SUURCE LIST GAAA 04/29/65 IStt SOURCE STATEMENT 1 PI*3.14159 2 6 READ l.DB.EL.ELE.E 3 1 FORMAT (4F10.5I <V IFIDB>70.70,5 S 5 READ3,S,q,P,T,TB,00.J 7 2 FaRNATIFlO.a.SFlO.5 .12) 10 1F1S16.6.9 11 9 0«.0*TB/T 12 CALL SKIP TO 111 13 1-0 14 IFIJ-621101,102,103 15 101 PRINT61 16 61 F0RMATI20H HYDROGEN-NITROGEN / I 17 G0T0105 20 102 PRINT62 21 62 FORMATI 20H NITROGEN-ETHANE / ) 22 G0T0105 23 103 PRINT63 24 63 FORMAT! 20H NITROGEN-BUTANE /) 25 105 PRINT3.DB.0.EL,T,ELEtP 26 3 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 27 PRINT31.E.TB 30 J l FORMAT(9H POROSITY .F10.5.20X,18HBED TFMPERATURE K> .F10.4//1 31 CALLABCOIS) 32 A.1.S6/EL 33 20 D-S/A..2 34 H*.4.*0/(D<>E*PI*0B**2> 35 tK-ELE/E 36 0EL-H-EK*A*.2-A*SINIA.ELI/COS!4*EL1 37 1*I»1 40 IFUBSIDELI-.00001) 31.50.50 41 50 DDEL«-2.«fcK«A-A«EL/lC0SIA*El»>**2-SlN(A«EL)/C0S(A»Elt 42 A.A-OEL/DOEL 43 IFII-20) 71.71,51 44 71 G0TO20 46 51 0-<D»760./P 46 AMDA*D0/D 47 OEF'OoE 50 PRINTS2.0.AMDA,DEF,00.A.I 51 52 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) 52 GOT06 53 70 STOP 54 ENO NO MESSAGES FOR ABOVE ASSEMBLY TIME 17HRS OOMIN 06.0SEC 6 OAVIS ISM SOURCE S T A T E M E N T H I R A M SOURCE L I S T J A A A 04/i">/t» 1 S U B R O U T I N E A B C O ( B ) 2 D I M E N S I O N V ( 1 0 0 1 i O l M C N S I O N T 1 1 0 0 ) * 2 S Y - 0 . 0 5 U'l 6 PR I NT 50 7 SO FORMAT I ? O X i 3 0 H TI ME S t C . PEAK H E I G H T S ) 10 2 6 C O N T I N U E 11 REAI)25,T ( N>,Y ( N I 12 2 S F C R H A T I F 1 0 . 1 . F 1 0 . 2 1 13 I F I T I N M -.0,40,41 14 4 1 P R I N T 2 7 . T I N ) , V ( N ) 15 2 7 F O R M A T I 2 0 X . F 1 0 . 1 . F 1 0 . 2 ) 16 S Y - S Y » Y ( N I 17 N-N»l 2 0 C 0 T 0 2 6 21 4 0 NU»N-1 22 0 0 7 1 1 - 1 , 1 0 2 3 tl S E B T - 0 . 0 2 4 S T t B T - 0 . 0 25 S Y E 6 T - H . 0 2 6 S Y T E B T - 0 . 0 2 7 S T E 2 B T - 0 . 0 30 S E 2 d T » 0 . 0 31 ST2E8T*0.0 32 S T 2 E 2 B - 0 . 0 3 3 S Y T 2 E B - 0 . 0 34 0 0 1 2 N - I . N U 34 E B T " £ X P I - 8 « T ( N > ) 3 6 E 2 B T » E X P t - B » T < N > « 2 . ) J 7 S E 8 T « S C B T » E B T 4 0 S Y T E B T « S » T E 8 T » Y ( N I « T I N I « E B r 4 1 S Y E B T " S Y t B T » Y ( N ) « E B T 4 2 S T E H T " S T f c B T » T ( N > » £ B T 4 3 S Y T 2 E B • S Y T 2 E 8 • Y ( N ) • ! I N I • • 2 « E 8 T 4 4 S T 2 E 2 B - S T 2 E 2 B » T I N ) • • ? « E 2 B T 4 5 S T 2 E B T .ST<:£BT • T ( N ) « « 2 « E B T 4 6 S T E 2 B T • S T £ 2 B T « T C N ) « E 2 B T 4 7 S E 2 3 T * S E 2 B T » E 2 B T 5 0 12 C O N T I N U E 52 EN-NU 5 3 EROR- - S Y E B T » S T E B T « S E 8 T / E N » S Y e B T « S r E 2 B r » i r t a T » S Y » i E 2 l J T / E % - 5 Y T F ( i r « 1 S E 2 B T - S Y « S E B T « S T E 2 B T / E N • S Y T E B T « S E B T » « 2 / E N 5 4 16 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 • S T £ S T « S F r l I » S Y T E l ! T / t " « l - 2 . » S Y > : B T « S T 2 t 2 D - 5 T E 2 B T » S Y T t B T - ( 2 . » S Y « S T t B T » S T E 2 b I » S Y « S E 2 H T « S I 2 F 2 B T I / E N • 2 . « S Y T E 6 T « S T E 2 o T • S E 2 ( i T » S Y T 2 t 8 « 2 . ' S Y < i S F B T « S T 2 E 2 B / L N • 3 S Y » S T f c 2 B T » S T E B T / E N - 2 . « S V T E 8 T « S E » T » i T E d T / E N - S E B T « » < « S Y T 2 f . 6 / f c N 5 5 7 B-B-ERUR/IJEROR 6 7 15 A " ( S T E B T . S Y / t N - S Y T E B T I / ( S T £ B T « S E B T / E N - S T E 2 B l l 6 0 C - ( S Y - A » S b B T I / E N 61 P R I N T 3 7 , A , B . C 6 2 i l F O R M A T ( / / 6 0 H C O N S T A N T S FOR LEAST S A U A R E S F I T OF OATA I N Y - C " A » t X P K-BTI / 2 0 X . 3 H A < i H t . t / 2 0 X . 3 H B • I F 1 6 . S / 2 0 X . 3 H C • , F | 6 . B / / / 2 5 0 H SUMMATIONS FROM L E A S T SQUARES C A L C U L A T I O N I 6 3 P R I N T D t S E B T i S Y E B T . S V » S T k B T f S T E 2 B T i S f c 2 a T f S T 2 E B T » S T 2 E ? b . 1 -200- 8 DAVIS ISN SOURCE STATEMENT FORTRAN SOURCfc LIM JAAA 6*, 13 65 66 ISYT2EB .SYTEBT F0«MATU6X,7HSEBT » , £ 1 2 . 5 / 1 6 X , 7 H S Y E B T » ,F1 2 . b / 1 6 X . 7 H S Y U l 2 . 5 / l 6 X , 7 H S r E 8 T » ,E 1 2 . 5 / 16X , 7HSTE26T" ,fc12.5/16X,7HSfc?8T » 2E12 . 5 , /16X,7HST2EBT» ,E12.5/16X,7HST2E2.* , £ 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 P A R A L L E L T U B E B E D H Y D K O G E N - N l T R O G t U -201 h fcO D A T A B fcO D I A M E T E R C M S 5 . 0 3 0 0 0 H E O L E N G T H C M S 1 0 . 0 1 0 0 0 E N O Z O N f c M f c l & H T 0 . 2 7 0 0 0 P O R U S I T Y 0 . 5 2 0 0 0 K U ' 4 O A T A FLOW SATE C C / S E C « O . S I O ? ' ) MUtlH T f M P t R A T U R C * 2 " > 5 . 5 C 0 0 A T M . P R c S S U K E MM HI} 7 5 7 . h b E O T E M P E R A T U R E K» 1 0 4 . 0 0 0 U U M f c S E C . P E A K H E I G H T S 2 0 0 . 0 2 9 2 5 . 0 0 3 0 0 . 0 ^ 0 5 0 . 0 0 8 E J B 0 T B ) K U B T S 4 C 0 . 0 1 4 0 0 . 0 0 MO *°«> 5 0 0 . 0 1 0 0 0 . O C 6 0 0 . 0 7 0 0 . 0 0 7 0 0 . 0 4 H 0 . 0 O C O N S T A N T S F O R L E A S T S A U A R E S H I OF D A T A I N V - C « A » E X P | - B T > A * 6 0 3 5 . 5 1 4 8 3 11 » 0 . 0 0 3 8 4 1 8 2 C » 1 3 . 9 6 3 7 0 3 6 1 S U M M A T I O N S F R O M L t A S T S Q U A R E S C A L C U L A T I O N S E b T » 0 . 1 4 0 3 5 E 0 1 S V L U T » 0 . 2 7 0 3 7 E 0 4 S V » 0 . 8 5 5 5 0 E 0 4 S T E B T • ft.4<l346t 0 3 S T E 2 B T ' O . 1 2 7 0 7 E 0 3 S t 2 B T « ( I . 4 4 4 7 1 E - U 0 ST^£OT> 0 . 2 0 6 0 2 t 0 6 S T 2 f 2 H « . I . 4 2 2 2 5 E 0 5 S V T ^ E B . ( > . 2 5 » 7 3 F 0 9 S Y T E S T " « . 7 7 3 5 3 C 0 6 U I F F U S I V | T V » 0 . 5 5 1 2 1 7 1 6 L A M O A • 1 . 4 8 7 6 1 6 9 6 E F F E C T I V C O I F F U S I V I T Y . 0 . ? H 6 6 3 . " ) 3 P U B L I S P E O 0 | F F l ' S I V I T Y « O . H 7 0 0 0 0 0 0 A L P H A * U . I » 1 2 ? " » T 7 N u M u f c R I T F R A T I U N S * 2 1 - 2 0 2 - H Y Q R U G t N - M T R Q G E N B E D D A T A HEO D I A M E T E R C C S b . J J O O O 0 6 0 L E N G T H C M S l U . u S O O O b N O IQHt H E I G H T 0 . 2 7 0 0 0 P O R O S I T Y 0 . 5 2 0 0 0 R U i DAT 4 F L O W K A T E C C / S c C * 0 . 5 4 i 7 b ROOM T E M P E R A T U R E * 2 9 S . 5 O 0 0 A T M . P R C S S U R t M " HG 7 6 7 . 6 •tED T f c M P f c R A T U R e K * 1 0 9 . 0 0 0 0 l i f t S f c C . P E A K M H O M T S 3 0 ' ) . 0 3 6 0 . OQ 4 0 0 . 0 2 4 5 . C O 5 0 0 . 0 1 6 7 . 6 0 6 0 0 . 0 1 1 5 . 0 0 7 0 0 . 0 R 2 . 0 0 B C O . O 6 S . 0 0 1 0 1 0 . 0 3 6 . 5 0 1 1 0 0 . 0 3 0 . 0 0 1 2 0 0 . 0 2 4 . i O 1 3 U 0 . 0 2 0 . 7 0 1 4 0 0 . 0 IB .20 1 3 U 0 . 0 1 6 . 5 0 £21X0233) 800. 490. C O N S T A N T S F O R L E A S T S A U A 4 E S F I T OF D A T A I N Y - C - A * E * P ( - B T I A • 1 1 6 4 . 0 3 9 0 3 B • 0 . 0 0 4 0 6 2 1 8 C • I S . 6 3 1 0 0 4 0 3 S U M M A T I O N S F R O M L E A S T S U U A R E S C A L C U L A T I O N S E U T > C . 8 S 4 3 6 E 0 0 S V E B T • 0 . 1 9 S 3 0 E 0 3 S Y « O . U B O t - 0 4 S T C B T • C . 4 1 0 4 0 L 0 3 S T E 2 B T * 0 . 5 ' ( 0 2 5 t 0 2 S E 2 B I • 0 . l b 6 » H E - 0 0 S T 2 E B I * 0 . 2 3 7 7 1 F 0 6 S T 2 E 2 H - G . 2 4 3 3 8 E O b S Y T 2 E U * 0 . 3 2 0 4 0 t 0 8 S Y I E B I * 0 . 7 i O « F O S D I F F U S I V I T Y * 0 . 7 1 4 6 1 0 1 0 L A M O A • 1 . 1 4 7 4 7 8 8 8 E F F L - C T I V E O I F F U S I V I T Y - 0 . 3 ? l i 9 7 2 S P U B L I S H E D O l F F U i l V I T Y * 0 . B 2 Q O 0 0 0 0 A L P H A * 0 . 0 7 5 2 2 8 2 0 N U M B E R I T E R A T I O N S - 2 1 - 2 0 3 - H Y D R U G C N - N I T R U G E N B E O O A T A B E O O l A M E t E H C M S 5 . 0 3 0 0 0 060 L E N G T H C M S 1 0 . 0 5 0 0 0 E N O Z O N E H E I G H T 0 . 2 7 0 C O POHUSm 0.52000 R U N DA I 4 F L O W R A T E C C / S E C ' 0 . 5 6 2 4 8 ROOM TfcMHC*ATU*E» 7 9 5 . 0 0 0 0 A T M . P R E S S U R E MM HG 7 4 9 . 8 b E O T E M P E R A T U R E K» 3 0 9 . 0 0 0 0 T I M E S E C . P E A K H E I G H T S HUSCTEU P0JHX3 150.0 655.00 2 0 0 . 0 530.00 250.0 430.00 300.0 350.00 400.0 227.50 700.0 HO. 00 «M> 173. 800. 0 " "58.1)0 TOO. ISO. 900.0 43.00 £00. lot. IOOO.O 3 3.00 i l o o . o 26.00 12U0.O 21.50 1300.1 18.50 C O N S T A N T S F U R L t A S T S V J A R E S F I T fJF O A T A I N Y - C « A » E » ( M - H T I A * 1 2 V 2 . H 0 4 7 5 b » 0 . 0 0 4 3 4 2 1 3 C • 1 5 . 9 1 9 S 9 U 9 S U M M A T I O N S F R O M L c A S T S U U A R b S C A L C U L A T I O N S E 8 T • 0 . 1 8 5 . S 0 E 0 1 S Y t b T • C1 . 85162E 0 1 S Y • P . 2 4 7 2 5 E 0 4 S T E f l T • C . 5 0 8 3 3 C 0 3 S I E 2 B T " 0 . 1 4 2 1 0 E 0 3 S E 2 6 T • 0 . 6 7 0 / 6 6 0 0 S T 2 F B T - C 1 9 8 8 6 F . 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 O I F F U S I V I T Y * 0 . 8 7 2 6 3 7 3 9 L A M O A • 0 . 9 3 9 6 8 0 1 1 E F F E C T I V E O I F r U S I V I T Y . 0 . 4 5 3 7 7 1 4 5 P U B L I S H E D D I F F L ' S I V H Y - 0 . 8 2 0 0 0 0 0 0 A L P H A ' 0 . 0 7 0 6 0 9 5 7 N U M 8 E R I T E R A T I U N S * 2 1 - 2 0 4 - H Y D R U G E N - N I T R O G k N B E D O A T A t t E O D I A M E T f c R C M S 5 . 0 3 0 0 0 H 8 D L E N G T H C M S 1 0 . 0 5 0 0 0 kHO Z C N f c H E I G H T 0 . 7 7 0 0 0 P O R O S I T Y 0 . 5 2 U 0 0 R U N D A T A Kov. R A T E cc/sfc« z.nan-t R U U M TtNPt«ATURfc= 2 9 5 . 0 G U O A T M . f M t . S S U R E MM H G 7 5 5 . 3 i iCO T F V P F R A T U R C K* 3 0 9 . 0 0 0 0 T I K E S b C . P E A K W I G H T S E J E C T S ' rvJHTS 1 C U . 0 3 3 . S O S O . i»5. 1 5 0 . 0 I " . 5 0 2 0 0 . 0 l < r . O U 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 C O N S T A N T S F O R L E A S T S A U A R E S H I T b f D A T A I i 4 Y - C " A ' E X P I - b T ) A • 1 0 0 . 1 6 8 7 8 R » 0 . 0 1 1 5 4 3 6 4 C • 1 . 8 9 3 5 1 0 1 3 S U M M A T I O N S F R O M L E A S T S C U A R t S C A L C U L A T I O N S E B T • C . 7 0 6 2 6 E 0 0 S Y E r t T • 0 . 1 5 8 7 7 E 0 2 S V » 0 . 6 4 0 " 0 C 0 2 S T f c B l « 0 . l l l 4 l t 0 3 S T E 2 B T ' 0 . 1 7 8 3 5 E 0 7 S E 2 H T • O . i 4 M 0 t - O ( > S T 2 E B T » G . 2 U u 4 t 0 ' . 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 . 7 7 6 0 2 > 9 l L A M O A « 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 ' 0 . 4 0 3 5 3 5 5 6 P U B L I S H E D O I F F U S I V f T Y . O . f l Z O U O O O O A L P H A ' 0 . 1 2 7 . 3 4 J u l D U M B E R t T F R A T I O N S » 1 8 N H H O G E N - E T H A N i B E D D A T A BfcU O I A M E T E R C H S S . 0 3 C O O B E O L E N G T H C M S 1 0 . 0 5 0 0 0 E N O Z O N E H E I G H T 0 . 2 7 0 0 0 P O R O S I T Y O . H 2 0 0 0 R U N D A T A F L O W «ATE C C / S E C * 0 . 4 8 4 9 7 * 0 0 « r£.*PtrRATURE = 2 9 S . 0 n . 1 0 A T H . P R E S S U R E MM H G 7 * 9 . 8 HEO T E M P t K A T U R E K * 3 0 4 . 0 0 0 0 T l * f c S E C • too.o 5 0 0 . 0 7 0 0 . 0 1 0 0 . 1 I 0 5 0 . i l 1 2 0 0 . 0 1 1 5 0 . a I S O O . O 1 6 3 0 . 0 1 8 0 0 . 0 1 9 3 0 . 0 2 1 0 0 . 0 2 3 0 0 . 0 P E A K H E I G H T S 4 7 0 . 0 0 1 1 8 . 0 0 2 1 7 . 0 0 1 5 , ? . 0 0 1 1 6 . J O 9 0 . 0 0 7 2 . SO 5 7 . 0 0 4 5 . 5 0 3 7 . 0 0 3 1 . 0 0 2 6 . 0 0 2 1 . 5 0 C O N S T A N T S E Q * L E A S T S U I A 4 E S F I T t i t D A T A IH Y - C * A ' E X P I - B T ) A * R27.H814') B • 0 . 0 0 l ' ) 9 9 4 0 C • 14.535 )2410 S U M M A T I O N S F R O M L F A S l S Q U A R E S C A L C U 1 1 T I 0 N s t a r * o . i 7 t ' * o t - o i S Y E B T - 0 . 4 8 / 7 U 0 1 S Y • O . l G S l S t 0 4 S T t B T • 0 . 1 2 / 7 7 t 0 4 S T E 2 B T * 0 . 2 6 S 9 U 0 3 S E . H r • 0 . 5 5 P 0 5 E 0 0 S T 2 E B T - 0 . 1 2 « 2 4 t 0 7 S T 2 C 2 B * 0 . 1 6 4 4 3 < " 0 6 S Y T 2 b b ' 0 . 1 6 4 8 9 t 0 9 S Y T E H T - C . 2 3 8 7 2 t 0 6 D I F F U S I V I T Y * 0 . 1 3 5 6 4 1 7 7 L A M O A • 1 . 1 1 3 / 2 6 4 1 E F F t C T I V E O U F u S l V I l Y - 0 . 0 7 0 5 1 3 7 2 P U B L I S H E D D I F F U S I V I T Y ' 0 . 1 6 1 0 0 0 0 0 A L P H A * 0 . 1 2 2 2 3 2 S 2 NOMHFR I T E R A T I O N S * I d -206- N r - T R U G E N - f c T H A N E B E D D A T A B E D C1AHETEH C H S » . 0 3 0 0 0 H B O L E N G T H C M S l O . O i O O O E N D Z O N E H E I G H T 0 . 2 7 0 0 0 P O R U S I T Y 0 . 5 2 0 0 0 F L C w K A T f c C C / S E C * l . 4 h l ? 0 «(J0« IE>P£«ATUf<E» 2 9 5 . 0 0 i / i i A T M . P R E S S I M E MM tr» 7 5 5 . 1 tttO rt«°e«ATL;i«C K * t O . 0 0 0 0 T I K E S E C . 2 0 0 . 0 4 0 0 . 0 6 C 0 . 0 ' 0 0 . 0 1 ) 0 0 . 0 1 0 0 0 . > 1 2 0 0 . 0 P E A K H E I G H T S 2 3 0 . 0 0 1 3 0 . 0 0 7 6 . 0 0 5 9 . 0 0 4 6 . f n ! 2 B . 0 I ) I ' ) . 0 0 C O N S T A N T S FOR. L c A S T S A U A R E S 'IT Of OA TA I N Y - C * A » F X P ( - 8 T » A * 4 0 0 . > > 7 ' > 2 l B * ' ) . 0 0 ^ 9 4 5 9 4 C - / . 4 7 5 8 0 4 9 8 S U P I N A T I O N S F R O M L l - A S T S O U A K L S C A L C U L A t l O N S L O T * 0 . 1 1 3 5 ' J F 0 1 S Y E B T * 0 . 1 m Ic 0 3 S Y * C . 5 i l « ) 0 £ 0 3 S T E B T * O . S B B r l h u 0 3 S T E 2 B T * 0 . i i 9 ? 2 E 0 3 S E 2 0 T * 0 . 4 6 0 4 2 F - 0 0 S T 2 E B T * 0 . 3 5 0 3 K E 0 6 S T 2 t 2 B * O . V > 6 l / t 0 5 S Y T 2 E B " 0 . 2 4 9 1 2 L " 0 8 S Y T t B T * 0 . 6 0 1 H 5 F 0 5 O I F F U S I V I T Y * P . 1 4 7 S 2 4 0 * L A H O A • 1 . 0 2 1 4 B 4 6 7 E F F E C r i V E O I F F U S I V I T Y * 0 . 0 7 6 8 6 8 5 0 P U B L I S H E D D I F F U S I V I T Y * 0 . 1 5 1 0 0 0 0 0 A L P H A * 0 . 1 4 1 6 0 7 6 2 N U M n f c K I T E R A T I O N S * 1 2 -207- N U R O G E N - E T H A N E B E O O A TA B E O D I A M E T E R C M S 5 . 0 3 0 0 0 H 6 0 L E N G T H C M S 1 0 . 0 5 0 0 0 E N O Z O N E H E I G H T 0 . 2 7 0 0 0 P O R O S I T Y 0 . S 2 C 0 0 R U N D A T A E L O w R A T E C C / S E C « 2 . 2 6 6 0 9 R U O " T E M P E R A T U R E - ' 2 9 6 . 5 0 0 0 A T M . P R E S S U R E MM H G 7 5 5 . 3 K E O T C M P E R A T U R E K * 3 0 9 . 0 0 0 0 T I K E S E C . 2 0 0 . 0 3 0 0 . 0 4 0 0 . 0 5 0 0 . 0 6 0 0 . 0 7 0 0 . 0 P E A K H E K . H T S 1 5 0 . 0 0 1 1 1 . 0 0 8 4 . 0 0 6 2 . 5 0 4 6 . 5 0 3 5 . 5 0 C O N S T A N I S F U R L E A S T S A U A R E S F I T O F D A T A I N Y - C * A * E X P ( - B T ) A * 2 6 8 . 3 7 3 0 3 b « 0 . 0 0 7 9 9 7 1 4 C • 2 . 4 4 4 4 0 4 6 0 SUMMATIONS FROM LEAST SQUARES CALCULATION S E f i T * 0 . 1 7 6 9 3 E 0 1 S Y E B T « 0 . 1 7 6 8 9 E 0 3 S Y * C . 4 B 9 * 0 E 0 3 S T E r t T * C . 6 4 0 4 8 F 0 3 S T £ 2 e i » 0 . 1 9 8 3 O L 0 3 5 E 2 B I « C . 6 5 0 4 4 E 0 0 5 T 2 E 8 T * 0 . 2 8 2 4 3 E 0 6 S T 2 E 2 8 * 0 . 7 1 2 4 1 E 0 5 S Y T 2 t b " C . 1 9 8 1 0 E 0 C S Y T E B T * 0 . 5 4 8 0 7 E 0 5 D I F F U S I V I T Y * 0 . 1 3 9 5 3 9 4 6 L A M O A « 1 . 0 8 2 1 3 1 1 5 E F F E C T I V E U I F F U S I V I T Y * 0 . 0 7 2 5 6 0 5 2 P U b L I S h k U O I F F U S I V I T Y * 0 . 1 5 1 0 3 0 0 0 A L P H A * 0 . 1 4 7 0 1 1 6 4 ' 1 U M B E K I T E R A T I O N S " I I • 2 0 8 ' N I T R O G E N - E T H A N E B E D D A T A B E D C F A M E T E H C M S 5 . 0 3 0 0 0 rtEO L E N G T H C M S 1 0 . 0 5 0 0 0 E N O Z O N E H E I J H T G . 2 7 0 0 0 P O R O S I T Y 0 . 5 2 0 0 0 R U N O A T A FLO« H A T E C C / i E C ' 7 . 9 4 3 3 6 R O O M T E M P E R A T U R E * 2 9 5 . 0 0 0 0 A T M . P R E S S U R E MM H G 7 5 7 . 6 B E D T f M P M A T U R t K * » 0 9 . 0 0 0 0 I l e t S E C . 2 0 0 . 0 3 0 0 . 0 4 C 0 . O 5 0 0 . 0 o O O . O 7 0 0 . 0 8 C 0 . 0 9 0 0 . 0 1 too.o P E A K H F I G H T S 1 2 4 . 0 0 8 8 . 0 0 6 5 . 0 0 4 7 . 5 0 3 5 . 5 0 2 7 . 0 0 2 1 . 5 0 1 7 . 5 0 1 2 . 0 0 C O N S T A N T S F O R L E A S T S A U A R f c S F I T OF O A T A I N Y - C » A * E X P ( - a T I A « 2 3 6 . 8 4 6 9 4 tt • 0 . 0 0 3 5 6 9 5 0 C » 7 . 6 8 5 2 6 9 6 5 S U M M A T I O N S F R O M L E A S T S O U A R t S C A L C U L A T I O N S E d T « 0 . 1 5 5 7 3 E 0 1 S V E H T « 0 . 1 2 3 0 1 L 0 3 S Y * 0 . 4 3 8 0 0 E 0 3 S T E 8 T • 0 . 6 1 2 5 6 1 0 3 S T E 2 8 T * 0 . 1 3 7 8 4 E 0 3 S E 7 8 T • 0 . 4 6 8 S 5 E - 0 0 S T 2 E B T ' 0 . 3 0 6 6 0 E 0 6 S T 2 C 7 U ' 0 . 4 8 5 1 7 E 0 5 SYT2ctt» 0 . 1 3 8 5 8 E 0 8 S Y T E 8 T • 0 . 3 7 3 5 4 E 0 5 O I F F U S I V I T Y ' 0 . 1 6 4 9 4 4 6 1 L A M U A • 0 . 9 1 5 4 5 8 8 4 E F F E C T I V E O I F T U S I V I T Y ' 0 . 0 8 5 7 7 1 2 0 P U I J L I S H E U D I F F U S I V I T Y ' 0 . 1 5 1 0 0 0 0 0 A L P H A * 0 . 1 4 7 8 2 8 8 0 N U M B E R I T E R A T I O N S ' l u 209- N H R O G E N - E T H A N b B E D O A T A B E D O I A M E T S K C M S 5 . 0 3 0 0 0 atC L E N G T H C M S 1 0 , 0 5 0 0 0 E N D ZO'it H E I G H T J . 2 7 0 0 0 P O K O S I T Y 0 . 5 2 0 0 0 R U N D A T A F L O W R A T E C C / S E C - 3.0 79 J ) ROHM T I M P E R A T U R t * 2 9 5 . 0 0 0 0 A T H . P R t S S U R b M.M HG 75c- .O OEO T t M P E R A T U R E K» 3 0 9 . 0 0 0 0 T 1 M b ScC. 2 0 0 . 0 3 0 0 . 0 4 C 0 . O 5 0 0 . 0 6 0 0 . ' ) 7 0 0 . 0 8 0 0 . 0 9 C 0 . 0 1 0 0 0 . I 1 1 0 0 . J PtA« H E I G H T S 1 1 5 . 0 0 d 9 . 0 u 6 7 . 0 0 5 2 . 0 0 4 2 . 0 0 3 4 . 0 0 2 9 . 0 0 2 5 . 0 0 2 2 . 5 0 2 0 . 5 0 C O N S T A N T S FOR L E A S T S A U A R b S F I T OF D A T A I t Y - C - A*EXt><-fcT» A « 1 9 4 . 3 7 1 8 1 b - 0 . 0 0 3 2 8 7 5 4 C « 1 4 . 9 9 7 0 6 3 * 0 S U M M A T I O N S F R C M L E A S T S Q U A R E S C A L C U L A T I O N S E B T - 0 . l 7 B ' ) 2 t 0 1 S Y k B T = 0 . 1 3 * ' < * E 0 3 S Y • 0 . 4 4 6 0 0 E 0 3 S T E B T • 0 . 7 4 4 3 ' > E 0 3 S T E 2 8 T - 0 . 1 7 0 3 2 E 0 3 S E 2 B T » 0 . 5 5 6 3 7 E 0 0 S T 2 E B T - 0 . 4 0 2 7 3 E 0 6 S T 2 b 2 b - 0 . 6 3 7 7 9 E 0 5 S Y T 2 E B - 0 . 1 8 * 2 8 1 0 8 S r T c t t T - 0 . 4 * 2 6 9 1 - 0 5 D I F F U S I V I T Y * 0 . 1 4 8 6 7 5 5 5 L A M O A - 1 . 0 1 5 6 3 * 4 7 t F F t C T l V E D I F F U S I V I T Y - 0 . 0 7 7 3 1 1 2 9 P U B L I S H E O D I F F U S I V I T Y - 0 . 1 6 1 0 0 0 0 0 A L P H A * 0 . 1 * 8 8 ^ 7 7 6 N U M B E R I T E R A T I U N S * 1 0 •210- M'TRUGEN-ETHANE ato DATA' d E U DIAMETER CMS S.OJGOO UCD LENGTH CMS l O . C O O O G E N D Z U N t H t 1 G H T 0 . 2 / 0 0 0 POROSITY 0 . 5 2 0 0 0 R U * DATA F L O W R A T E C C / S E C = 4 . 5 5 o 4 4 R O O M T F M P l . R A T U R E = 2 0 5 . 0 0 0 0 A T M . P R E S S U R E MM HG 7 5 5 . 6 B E O T E M P E R A T U R E K * 3 0 9 . 0 O 0 C T I M E S E C . P E A K H E I G H T S 2 0 0 . 0 7 8 . 0 0 l0C> , ' i ' 1 8 1 0 0 . 0 5 2 . 0 0 4 0 0 . 0 3 9 . 0 0 5 0 0 . 0 2 6 . 0 0 6 0 0 . 0 2 0 . 5 C 7 0 0 . 0 1 5 . 5 0 8 0 0 . 0 1 2 . C C C O N S T A N T S F O R L E A S T S A U A R E S F I T uf D A T A I N Y - C ' A ' E X P I - U T ) A = 1 5 8 . 9 9 5 5 7 H » 0 . 0 0 3 9 9 9 4 8 C • 5 . 9 2 0 1 3 3 6 3 S U M M A T I O N S F R U M L E A S T S C U A R t S C A L C U L A T I O N S E C T * O . I 7 8 0 3 E 0 1 S Y E b T « C . 6 5 6 7 4 6 0 2 i Y • C . 2 4 5 0 0 F 0 3 S T c b T * P . 4 5 8 3 6 E 0 3 S T C 2 i i T » 0 . 1 O 1 9 5 L - 0 3 S h 2 t l T = 0 . 1 6 5 3 9 E - 0 0 S T 2 E 8 T " 0 . 1 9 9 R U 0 6 S T 2 E 2 H * C . J H 9 3 F 0 5 S Y T 2 S 8 * 0 . 6 4 6 4 4 E 0 7 S Y T E 8 T * 0 . 1 8 9 ? I t 0 5 D I F F U S I V I T Y " D . 1 7 8 1 4 4 7 1 L A M D A * 0 . 0 4 7 6 ? 5 4 9 E F F E C T I V E O I F F u S I V I T Y = 0 . 0 9 2 6 1 5 2 5 PUaLISHED O I F F U S I V I T Y " 0 . 1 5 1 0 0 0 0 0 A L P H A * 0 . 1 5 0 2 7 1 4 1 N t U t o B E R I T E R A T I O N S * 9 -211- N 1 T R 0 G E N - B U T A N E B E D O A T A R U N C A T A B E D D I A M E T E R C M S BEO L E N G T H C M S ENO ZONE H E I G H T P O R O S I T V 0 . S 2 0 0 0 5 . 0 3 0 0 0 1 0 . 0 5 0 0 0 0 . 2 7 0 0 0 B E D T E M P E R A T U R E K - 3 0 9 . 0 0 0 0 FLOW R A T E C C / S E C - RUOM T E M P E R A T U R E - A T M . P R E S S U R E MM HG 0 . 4 6 1 M 2 4 6 . 0 0 0 0 7 6 1 . 0 T I M E S E C . P E A K H E I G H T S 3 0 0 . 0 4 0 0 . 0 5 0 0 . 0 5 1 0 . 0 0 4 4 0 . 0 0 3 8 0 . 0 0 6 0 0 . 0 3 2 5 . 0 0 7 0 0 . 0 2 8 5 . 0 0 8 0 0 . 0 2 5 0 . 0 0 9 0 0 . 0 2 1 5 . 0 0 1 0 0 0 . 0 1 9 0 . 0 0 1 2 0 0 . 0 1 4 2 . 0 0 1 4 0 0 . 0 1 0 8 . 0 0 1 6 0 0 . 0 8 2 . 0 0 1 8 0 0 . 0 6 1 . 0 0 2 0 0 0 . 0 4 5 . 0 0 2 2 0 0 . 0 3 4 . 0 0 2 4 0 0 . 0 2 6 . 0 0 2 6 0 0 . 0 1 9 . 0 0 C O N S T A N T S F O R L E A S T S A O A R E S F I T OF D A T A I N Y - C - A - E X P 1 - B T ) S U M M A T I O N S F R O M L E A S T S Q U A R E S C A L C U L A T I O N S E B T » 0 . 3 9 7 5 7 E 0 1 S Y E B T ° 0 . 1 2 S 0 B E 0 4 S Y - O . 3 1 1 2 0 E 0 4 S T E B T - 0 . 2 9 7 0 2 E 0 4 S T E 2 B T - 0 . 8 8 8 0 1 E 0 3 S E 2 B T • 0 . 1 6 0 2 7 6 01 S T 2 E B T - 0 . 3 0 8 3 1 E 0 7 S T 2 E 2 B " 0 . 6 2 1 8 8 F . 0 6 S Y T 2 E B - 0 . 4 8 8 0 2 b 0 9 S Y T E 8 T - 0 . 6 9 4 1 9 E 0 6 D I F F U S I V I T Y - 0 . 0 8 1 7 2 5 2 6 LAMOA • 1 . 1 6 2 4 3 1 2 3 E F F E C T I V E D I F F U S I V I T Y - 0 . 0 4 2 4 9 7 1 4 P U B L I S H E D D I F F U S I V I T Y - 0 . 0 9 5 0 0 0 0 0 A L P H A - 0 . 1 3 2 2 5 3 3 3 NUMBER I T E R A T I O N S - 15 A B c 7 7 6 . 6 9 5 2 1 0 . 0 0 1 4 3 1 3 3 1 . 5 0 5 6 7 8 1 8 - 2 1 2 - N I T R O G E N — B U T A N E B E D O A T A B E D O I A N E T E R C N S 5 . 0 3 0 0 0 B E D L E N G T H C H S 1 0 . 0 5 0 0 0 E N D Z O N E H E I G H T 0 . 2 7 0 0 0 P O R O S I T Y 0 . 5 2 0 0 0 R U N O A T A F L O W R A T E C C / S E C « 0 . 9 0 2 9 4 ROOM T E M P E R A T U R E * 2 9 6 . 0 0 0 0 A T M . P R E S S U R E MM HG 7 6 1 . 0 B E D T E M P E R A T U R E K * 3 0 9 . 0 0 0 0 T I M E S E C . 3 0 0 . 0 4 0 0 . 0 5 0 0 . 0 6 0 0 . 0 7 0 0 . 0 8 0 0 . 0 9 0 0 . 0 1 0 0 0 . 0 1 1 0 0 . 0 1 2 0 0 . 0 1 4 0 0 . 0 1 6 0 0 . 0 1 8 0 0 . 0 P E A K H E I G H T S 2 8 5 . 0 0 2 4 0 . 0 0 2 0 5 . 0 0 1 7 4 . 0 0 1 4 8 . 0 0 1 2 6 . 0 0 1 0 4 . 0 0 9 0 . 0 0 7 6 . 0 0 6 3 . 0 0 4 6 . 0 0 3 3 . 0 0 2 3 . 0 0 C O N S T A N T S F O R L E A S T S A U A R E S F I T O F D A T A I N Y - C * A . E X P I - B T ) A • 4 6 6 . 9 9 1 1 2 B • 0 . 0 0 1 6 3 1 9 1 C » - 1 . 7 4 8 5 7 6 2 2 S U M M A T I O N S F R O M L E A S T S Q U A R E S C A L C U L A T I O N S E B T « 0 . 3 5 0 2 7 E 0 1 S Y E B T * 0 . 6 0 8 3 9 E 0 3 S Y « 0 . 1 6 1 J O E 0 4 S T E B T = 0 . 2 3 8 9 0 E 0 4 S T E 2 B T * 0 . 7 0 1 9 6 E 0 3 S E 2 B T • 0 . 1 3 1 5 9 E 0 1 S T 2 E B T * O . 2 0 5 9 6 E 0 7 S T 2 E 2 B " 0 . 4 6 0 2 2 E 0 6 S Y T 2 E B " 0 . 2 1 1 3 0 E 0 9 S Y T E B T * 0 . 3 2 3 6 3 t 0 6 D I F F U S I V I T Y " 0 . 0 7 9 3 9 0 6 3 L A M O A * 1 . 1 9 6 6 1 4 6 B E F F E C T I V E D I F F U S I V I T Y * 0 . 0 4 1 2 8 3 1 3 P U B L I S H E D D I F F U S I V I T Y * 0 . 0 9 5 0 0 0 0 0 A L P H A * 0 . 1 4 3 2 7 7 3 1 N U M B E R I T E R A T I O N S * 1 2 -213- N 1 T R 0 G E N - B U T A N E B E O O A T A B E D D I A M E T E R C M S 5 . 0 3 0 0 0 B E D L E N G T H CMS 1 0 . 0 5 0 0 0 t N O ZONE H E I G H T 0 . 2 7 0 0 0 P O R O S I T Y 0 . 5 2 0 0 0 RUN OATA FLOW R A T E C C / S t C ' 2 . 0 * 6 0 ' ) ROOM T E M P E R A T U R E - 2 9 6 . 0 0 U 0 A T M . P R E S S U R E MM HG 7 6 1 . 0 BEO T E M P E R A T U R E K " 3 0 9 . 0 0 0 0 T I M E S E C . 3 0 0 . 0 4 0 0 . 0 5 0 0 . 0 6 0 0 . 0 7 0 0 . 0 8 0 0 . 0 9 0 0 . 0 1 0 0 0 . 0 1 1 5 0 . 0 P E A K H E I G H T S 1 3 8 . 0 0 1 1 0 . 0 0 9 2 . 0 0 7 6 . 0 0 6 1 . 0 0 5 1 . 0 0 4 2 . 0 0 3 4 . 0 0 2 5 . 0 0 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 D A T A I N Y - C ' A ' E X P I - B T I A • 2 4 9 . 2 1 8 2 6 B • 0 . 0 0 2 0 2 6 0 6 C - 1 . 2 3 3 9 6 2 1 6 S U M M A T I O N S FROM L E A S T S Q U A R E S C A L C U L A T I O N S E 6 T • 0 . 7 4 7 9 3 E 0 1 S Y E B T • 0 . 2 1 8 5 5 E 0 3 SY - 0 . 6 2 9 0 0 E 0 3 S T E B T « 0 . 1 4 1 7 4 E 0 4 S T E 2 B T ' 0 . 4 1 0 7 9 E 0 3 S E 2 B T • 0 . B 6 4 6 8 E 0 0 S T 2 E B T ' 0 . 9 5 4 2 0 E 0 6 S T 2 E 2 B ' 0 . 2 2 7 7 1 E 0 6 S Y T 2 E 6 ' 0 . S 7 9 3 B E 0 8 S V T E B T ' 0 . 1 0 4 1 2 E 0 6 D I F F U S I V I T Y ' 0 . 0 9 0 5 7 9 1 3 ' LAMOA ' 1 . 0 1 5 6 8 6 4 7 E F F E C T I V E O I F F U S I V I T Y ' 0 . 0 4 7 1 0 1 1 5 P U B L I S H E O O I F F U S I V I T Y ' 0 . 0 9 2 0 0 0 0 0 A L P H A ' 0 . 1 4 9 4 6 0 6 8 NUMBER I T E R A T I O N S ' 10 E N O - O F - O A T A E N C O U N T E R E D ON S Y S T E M I N P U T F I L E . T I M E 1 7 H R S 0 1 M I N 2 S . 3 S E C P O R O U S S O L I D B E D H Y O R O G C N - N l T R O C t f J -214- B E O O A T A 6 b 0 0 S A M E T 6 R C M S 2 . 6 1 0 C 0 B O O L E N G T H C M S 7 . 0 0 0 0 0 E N D Z O N E H E I G H T 0 . 2 7 0 0 0 P O R O S I T Y O . 5 9 C 0 0 R U N O A T A F L O n , U T E C C / S E C * 0 . 5 6 2 3 8 R O O M T E M P E R A T U R E * 2 9 6 . 0 0 0 0 A T M . P R E S S U R E MM H G 7 6 5 . 1 B E O T E M P E R A T U R E . K * 3 0 6 . 0 0 0 0 T I M E S E C . 1 5 0 . 0 2 0 0 . 0 2 5 0 . 0 3 0 0 . 0 3 5 0 . 0 4 0 0 . 0 4 5 0 . 0 " 5 0 0 . 0 P E A K H H G H T S 1 2 3 . 0 0 6 1 . 0 0 3 4 . 0 0 1 8 . 0 0 1 0 . 0 0 _t>j-JO 4 . 5 0 3 . 2 0 C O N S T A N T S F O R L E A S T S A U A R E S F I T O F O A T A I N V - C = A o E X P ( - B T ) A * 9 6 1 . 0 6 7 3 4 B " » 0 . 0 1 3 8 7 8 3 2 C * 2 . 6 J 4 5 8 3 9 5 S U P M A T I Q N S F R O M L E A S T S Q U A R E S C A L C U L A T I O N S E B T * 0 . 2 4 8 2 7 E - 0 0 S Y E B T * 0 . 2 0 5 9 2 E " " 0 " 2 S Y « 0 . 2 5 9 9 0 6 0 3 S T E r t T * 0 . 4 9 2 4 7 C 0 2 S T C 2 B T * 0 . 3 4 5 3 6 C O l S E 2 B T * 0 . 2 0 7 2 7 E - 0 1 S T 2 E B T * 0 . 1 0 B 5 2 C 0 5 S T 2 E 2 B * 0 . 5 9 8 3 8 b 0 * S Y T 2 E B * 0 . 6 0 4 4 3 E 0 6 S Y T E B T * 0 . 3 4 5 2 0 t 0 4 D I F F U S I V I T Y * 0 . 5 6 4 7 6 9 5 9 L A M O A « 1 . 4 5 1 9 1 9 5 3 E F F E C T I V E D I F F U S I V I T Y * 0 . 3 1 3 2 1 4 0 6 P U B L I S H E D D I F F U S I V I T Y * 0 . 8 2 0 0 0 0 0 0 A L P H A * 0 . 1 5 6 2 3 2 5 1 D U M B E R I T E R A T I O N S * 2 1 £ 1 > H Y D R O G E N — N I T R O G t N B E O D A T A O E D D I A M E T E R C M S 2 . 6 1 0 0 0 B E D L E N G T H C M S 7 . C 0 O 0 0 E N D Z O N E H E I G H T 0 . 2 7 0 0 0 P O R O S I T Y O . 5 9 C 0 0 RUN O A T A F L O W R A T E C C / S t C - 0 . 7 9 2 9 1 R O O M T E M P E R A T U R E * 2 9 6 . 0 0 0 0 A T M . P R E S S U R E M M H G 7 6 5 . 1 B E D T E M P E R A T U R E K» 3 0 6 . 0 0 0 0 T I M E S E C . 1 5 0 . 0 2 0 0 . 0 2 5 0 . 0 3 C 0 . 0 3 5 0 . 0 4 0 0 . 0 P E A K H E I G H T S 6 6 . 5 0 2 9 . 0 0 1 3 . 5 0 6 . 5 0 3 . 5 0 2 . 2 0 C O N S T A N T S F O R L E A S T S A U A R E S F I T O F D A T A I N Y - C - A » E X P I - b T ) A * 8 3 2 . 0 2 7 1 6 b * 0 . 0 1 6 9 8 6 1 3 C * 1 . 3 5 6 6 9 6 6 1 S U M M A T I O N S F R O M L E A S T S C - U A R F S C A L C U L A T I O N S E B T ' 0 . 1 3 5 8 8 F - 0 0 S Y E 8 T * 0 . 6 4 1 B 4 E 0 ! S Y * 0 . 1 2 1 2 0 F C 3 S T E O T = 0 . 2 5 2 0 9 E 0 2 S T E 2 B T * O . 1 2 0 7 7 E 0 1 S E 2 B T * C . 7 4 9 2 6 E - 0 2 ST2Et tT» 0 . 5 0 4 4 7 E 0 4 S T 2 E 2 B * 0 . 1 9 9 7 7 E 0 3 S Y T 2 E H * 0 . 1 7 3 0 7 E 0 6 S Y T E B T * U . 1 0 J 9 0 E 0 4 O I F F U S I V I T Y " 0 . 5 9 6 2 0 0 1 5 L A M U A « 1 . 3 7 5 i 7 7 0 4 E F F E C T I V E D I F F U S I V H Y » 0 . 3 5 1 7 5 8 0 9 P U B L I S H E D O I F F U S I V I T Y - 0 . 8 2 0 0 0 0 0 0 A L P H A * 0 . 1 6 8 2 2 8 2 2 N U M B E R I T E R A T I O N S * 2 0 -216- H^DRUGEN-MTROGfcN «tO DATA BED OIAMETER CHS 2 . 6 1 0 C 0 H8U LENGTH CMS 7 .00000 END ZONE HEIGHT 0 . 2 7 0 0 0 P U R O S H V 0 . 5 9 C 0 0 RUN DATA FLOW RATE C C / S E C * 0 . 9 2 B 3 4 RllOM TEMPERATURE* 2 4 6 . 0 0 0 0 ATM.PRESSURE MM HG 7 6 b . I BED TFNPFHATURE K« 106 .0000 TIME S E C . PEAK HEIGHTS 1 0 0 . 0 1 2 4 . 0 0 1 5 0 . 0 5 4 . 0 0 2 0 0 . 0 2 3 . 0 0 2 5 0 . C 10 .00 3 0 0 . 0 5 . 0 0 3 5 0 . 0 i . 0 0 4 0 0 . 0 2 . 0 0 " ~ CONSTANTS FOR LEAST SAUARES F IT uF OATA IN V - C " A * E X P ( - B T > A * 6 7 3 . 3 1 6 0 1 B » 0 . 0 1 7 0 6 4 1 6 C * I .O01777O9 SUMMATIONS FROM L t A S T i0UAf .CS CALCULATION SFBT * C . 3 1 5 4 5 F - 0 0 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 ST2EBT" 0 .67745E 04 ST2E2H* 0 .5239HC 03 SYT2EB* 0 . 3 6 2 1 0 C Ob S Y T E I T " 0 .30764b 04 O I F F U S I V I I Y - 0 . 5 J 3 0 9 6 B 5 LAMOA * 1.53811)204 E F F E C T I V E O I F F U S I V I T Y * 0 .31452714 PUBLISHED D I F F U S I V I T Y * 0 .82000000 ALt»HA« 0 . 1 7 3 3 1 4 6 6 NUMBER ITERATIONS* 18 217- HYUKUGbN-NITKOGEN BED DATA BbU DIAMETER CMS 2 . 6 1 0 C 0 i)60 LEN6TH CMS 7 . 0 0 0 0 0 ENO ZONE HEIGHT 0 . 2 7 0 0 0 POROSITY 0 . 5 9 0 0 0 RUN OATA FLOw RATE C C / S E C " 1 .24571 ROOM TEMPtftATURfc* 2 9 6 . 0 0 0 0 ATM.t»ReSSUKE MM HG 765 .1 BCD TEMPERATURE K» 3 0 6 . 0 0 0 0 I I M t S E C . 1 0 0 . 0 1 5 0 . 0 2 0 0 . 0 2 5 0 . 0 1 C 0 . 0 3 5 0 . 0 PEAK M U G H T S 7 1 . 0 0 2 b . 00 1 0 . 5 0 ' . . 5 0 2 . 2 0 1 .10 CONSTANTS FUR LEAST SAUARES F IT OF OAIA li Y-C» A»EXP|-HI) A » 4 7 4 . 7 3 4 2 1 B « 0 . 0 1 9 0 S U U 6 C • 0 . 4 6 2 4 3 6 8 ) SUMMATIONS FROM LEAST SQUARES CALCULATION StBT » 0 . 2 4 1 2 4 6 - 0 0 SVEBT « '>.1244 3E 02 SY « ' I . U T 3 0 F 03 STCJT • 0 . J 1 4 S 6 E 02 STE2BT* U . 2 8 2 4 3 E 01 SE2BT • O . 2 5 9 7 5 C - 0 1 ST2EBT" ' , . 4 6 4 5 9 t 04 ST2E2B* 0 . 3 2 0 3 8 E O i SYT2EB* U . 1 5 4 2 1 E 06 SYTEBT* 0 . 1 3 5 5 3 f 04 D I F F U S I V I T Y " 0 . 5 3 7 7 I 2 G 2 LAMOA « 1 . 5 2 4 9 7 7 5 8 E F F E C T I V E O l f F U S I V H Y * 0 . 3 1 7 2 5 0 5 6 PUBLISHED D I F F U S I V I T Y * 0 . 8 2 0 0 0 0 0 0 A L P H A " 0 . 1 8 7 6 3 4 1 6 .4UMBER I T E R A T I O N S " lo 218- HYDROGEN-NlTROGfcN BEO OAT A BEO OIAMETER CMS 2 . 6 1 0 0 0 BEO L6NSTH CMS 7 .00000 ENO ZONE HEIGHT 0 . 2 7 0 0 0 POROSITY 0 . 5 9 C 0 0 MUN DATA FLCW RATE C C / S E C * 1 . 8 3 4 9 7 ROOM TEMPERATURE" 2 9 6 . 0 0 0 0 A T K . P R E S S U R t MM HG 7 6 5 . 1 8ED TEMPERATURE K« 3 0 6 . 0 0 0 0 TIME S L C . PEAK HEIGHTS 5 0 . 0 " 1 2 4 . 0 0 1 0 0 . 0 3 7 . 0 0 1 5 0 . 0 1 4 . 0 0 2 0 0 . 0 5 . 0 0 2 5 0 . 0 2 . 0 0 CONSTANTS FOR LEAST SAUARES F I I UF DATA IN Y - C " A»£KP(-BTt A « 4 1 4 . 3 5 1 8 2 B • 0 . 0 2 4 4 5 5 9 8 C • 1 .89934120 SUMMATIONS FROM LEAST SQUARtS CALCULATION SE8T » 0 . 4 1 L 3 2 C - 0 0 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 STE2BT* O.Sr?51f.~CT SE2UT • 0 . 9 4 8 9 9 E - 0 1 ST2EBT* 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 D I F F U S I V I T Y ' 0 . 6 5 0 9 3 1 1 7 LAMDA • 1 . 2 5 9 7 3 3 8 0 E F F E C T I V E O I F F U S I V I T Y * 0 . 3 8 4 0 4 9 3 9 PUBLISHED D I F F U S I V I T Y * O.B2OU0O0O A L P H A * 0 . 1 9 3 1 C 4 7 0 NUMBER I T E R A T I O N S * 15 -21* MiTRUGEN-ETHANfc BEO OATA BEO DIAMETER CMS 2.61000 BEO LENGTH CMS 7.00000 ENO ZONE HEIGHT 0.27000 POROSITV 0.59COO RUN OATA FLOW RATE CC/SEC* 0.39180 ROOM TEMPERATURE* 296.0000 ATM.PRESSURE MM H i 766.9 BEO TEMPERATURE K> 306.0000 T IMF. S t C . 250.0 " 400.0 550.0 700.0 aso.o 10C0.0 PtAK HEIGHTS 212.50" 117.00 68.00 43.50 30.50 24.50 0 5 0 7 0 ?0T5fT CONSTANTS FOR LEAST SAUARES FIT OF OATA IN Y-C- A'EXP(-BT) A » 598.64044 B * 0.00448394 C » " "IT. 36491060 SUMMATIONS FROM LEAST SUUARFS CALCULATION SEBT * 0.65974C 00 SVEBT » 0 . 974 5 9 E 02 SY - C5T6"50E~0~3 STEBT « 0.26179f O i STE2BT* 0.4J494E 0? SC2BT * 0.14366E-00 ST2EBT* C.12BB0L 06 ST2E2B» 0.1469*C 05 SYT2EB* 0 . 1 l 0 3 4 t 08 SYTEBT* 0.30583E 05 DIFFUSIVITY- 0.11308980 LAMOA • 1.33522204 EFFECTIVE DIFFUSIVITY* 0.06672298 PUBLISHED OIFFUSIVITY* 0.15100000 ALPHA* 0.19822170 NUMBER ITERATIONS- U -220' N H R U G t N - E THANE BEO OATA RUN DATA BEO OtAMETER CMS BEO LENGTH CMS END ZONE HEIGHT PURQSITY 0 . 5 9 0 0 0 2 . 6 1 0 0 0 7 .00000 0 . 2 7 0 0 0 BEO TfcMPtRATURC K« 3 0 6 . 0 0 0 0 FLOW RATE C C / S E C - ROOM I f M P F R A T U R t - ATM.PRESSURE MM HG 0 . 8 2 0 8 2 2 9 6 . 0 0 0 0 7 6 6 . 9 TIME S E C . PEAK HEIGHTS 2 C 0 . 0 3 0 0 . 0 4 0 0 . 0 5 0 0 . 0 6 0 0 . 0 7 0 0 . 0 120O.') " 1 6 . O f f 1 1 6 . 0 0 7 7 . 5 0 5 3 . 0 0 3 8 . 5 0 2 9 . 5 0 2 4 . 0 0 CONSTANTS FOR LEAST SAUARES F IT OF OATA IN Y - C - A»EXK(-tJT( SY - 0 ; i ? 4 5 0 t 03 - - - - STE8T » C . 3 0 4 0 6 E 03 STE2BT- 0 . 5 8 9 1 0 c 02 SE2i3T » O . 2 2 7 A 0 E - 0 O ST2EBT- 0 . 1 2 0 0 7 L 06 ST2E2B* 0 .17230F. 05 S Y T 2 E B - 0 . 6 4 * 3 3 F 07 - - - SYTEBT- 0 . 2 0 3 8 3 t OS D I F F U S I V I T Y - 0 . 1 0 7 * 6 8 0 7 LAMOA • 1 . 3 9 9 S 5 8 1 6 E F F E C T I V E D I F F U S I V I T Y ' 0 . 0 6 3 6 4 2 1 6 PUBLISHED D I F F U S I V I T Y - 0 . 1 5 1 0 0 0 0 0 A L P H A - 0.21I6&63'~> NUMBER I T E R A T I O N S - 11 A « B - C - 2 h 7 . 69002 0 . C 0 4 8 7 6 6 5 1 5 . 1 7 3 8 2 1 2 1 SUMMATIONS FROM LEAST SOUARtS CALCULATION SEOT » 0 . 9 2 7 5 0 r 00 SYEBT = 0 . 7 4 9 9 9 F 02 22b N H R U G E N - E T H A M E bEO DAT« BED DIAMETER CMS BCD LENGTH CMS END ZONE HEIGHT POROSITY 0 . 5 9 C O U 2 . 6 1 0 0 0 ' . o o o o o 0 . 2 7 0 0 0 RUN DATA F L O * HATE C C / S E C " 1 .32324 ROOM TEMPERATURE" 2 9 6 . 0 0 0 0 ATM.PRESSURE MM HG 7 6 6 . 9 BEO TEMPERATURE K» 3 0 6 . 0 0 0 0 TIME S E C . 100 .0 2 C 0 . 0 3 0 0 . 0 4 0 0 . 0 5 0 0 . 0 uOO.O PEAK HEIGHTS 1 2 2 . 0 0 7 8 . 5 0 5 3 . 0 0 3 7 . 5 0 2 8 . 5 0 2 3 . 5 0 CONSTANTS FOR LEAST SAUARES FIT OF DATA IN Y-C« A»fcXP(-BT) A » 1 7 9 . 8 9 0 9 7 8 « 0 . 0 0 5 2 5 7 5 5 C » 1 5 . 6 7 2 7 4 1 5 3 SUPMATIONS FRUM LEAST SQUARES CALCULATION SCBT » 0 . 1 3 8 4 0 E 01 SYEBT * 0 . 1 l 8 1 3 t 03 SY • 0 . 3 4 3 0 0 E 03 STEBT * 0 . 3 0 « 7 E ~ 0 3 STE2BT" 0 . 8 1 8 1 4 E 02 SC26T 0 . 5 3 6 0 9 E 00 S T 2 E K T " 0 . 9 1 4 0 9 k 05 S T 2 E 2 B " 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 O I F F U S I V I T Y " 0 . 1 1 1 6 0 2 7 0 LAMOA « 1 .35301381 E F F E C T I V E O I F F U S I V I T Y " 0 . 0 6 1 8 4 5 5 9 PUBLISHED O I F F U S I V I T Y " 0 . 1 5 1 0 0 0 0 0 A L P H A " 0 . 2 1 6 0 6 8 7 3 NUMBER I T E R A T I O N S " 10 -222- N H R L i G E N - E T H A N f c B E D C A I A B E D U I A M E T E K C M S 2 . 6 1 0 0 0 B 6 0 L E N G T H C M S / . C O O O O E N D Z O N E H E I G H T 0 . 2 7 0 0 0 P O R O S I T Y 0 . 5 9 C O O RUN O A I A FlCk» R A T E C C / S E C * 1 . 9 0 2 1 6 R O H M T t M P f c R A T U R f c " 2 9 6 . 0 0 0 0 A T M . C R f S S U R E MM H G 7 6 6 . 9 H E O T E M P t R A T U R t K • 3 0 6 . 0 0 0 0 T I M E S F t . 1 0 0 . 0 - 2 0 0 . 0 3 0 0 . 0 4 0 0 . 0 5 0 0 . J 6 0 0 . J 70ovor~~ P E A K H E I G H T S 8 7 . 5 0 5 8 . 0 0 4 0 . 5 0 3 0 . 0 0 2 4 . 2 0 2 0 . 5 0 1 8 . 0 0 C O N S T A N T S F O R L E A S T S A U A H E S F I T O F O A T A I N Y - C » A » f X P ( - B T ) A * 1 2 2 . 1 8 3 7 9 8 • 0 . 0 0 5 2 2 6 9 1 C * 1 5 . 0 4 5 6 7 6 3 5 S U M M A T I O N S F R O M L E A S T S Q U A R E S C A L C U L A T I O N sear S Y E B T = S Y S T t B T 0 . 1 4 1 9 0 E 0 1 0 . 8 7 S 4 9 E 0 2 0 . 2 7 3 7 0 C O 3 ~ 0 . 3 2 2 J 2 6 0 3 S T F 2 8 T * 0 . 8 3 3 0 2 E 0 2 S C 2 b T * 0 . 5 4 1 8 0 F 0 0 S T 2 E 8 1 " 0 . 1 0 5 1 1 E 0 6 S T 2 E 2 8 - 0 . 1 7 1 6 1 E 0 5 S Y T 2 £ B = 0 . 3 O 7 8 7 E 0 7 S Y T f c B T " 0 . 1 5 0 2 8 E 0 5 D I F F U S I V I T Y " 0 . 1 0 8 2 8 4 3 3 L A M C A • 1 . 3 9 4 4 7 6 9 1 E F F E C T I V E D I F r U S l V I T Y * 0 . 0 6 3 8 8 7 7 6 P U B L I S H E C D I F F U S I V I T Y " 0 . 1 5 1 0 0 0 0 0 A L P H A " 0 . 2 1 6 7 1 4 3 2 " NUMBER ITERATIONS" 8 - 2 2 3 - 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 C M S H E D L E N G T H C M S L N L ) Z O N E H E I G H T P U R O S I T Y O . 5 9 C 0 O 2 . 6 1 0 0 0 7 . 0 C O O O O . 2 7 0 C O R U N D A T A F L O W R A T E C C / S E C » O . S 9 4 4 3 R O O M T E M P E R A T U R E * 2 9 6 . 0 0 0 0 A T M . P R E S S U R E M M H G 7 6 1 . 4 B E O T F K P E R A T U R E K * 3 0 6 . 0 0 0 0 T I M E S E C . 6 0 0 . 0 7 0 0 . 0 B C O . O 9 C 0 . 0 1 0 0 0 . 0 1 1 0 0 . 0 1 2 0 0 . 0 P E A K H E I G H T S 5 8 . 2 0 4 2 . 0 0 3 0 . 5 0 2 2 . 0 0 1 6 . 0 0 1 2 . 0 0 9 . 0 0 C U N S I A N T S F O R L E A S T S A U A R E S F I T ( I F D A T A I N Y - C > A » E X P ( - B T ) A * 4 2 9 . 9 2 2 0 7 B ' 0 . 0 0 3 3 7 3 2 b C ' 1 . 4 3 0 0 4 9 b ! S U M M A T I O N S F R O M L E A S T S Q U A R E S C A L C U L A T I O N S E B T « 0 . 4 1 7 O 6 E - 0 0 S Y E 8 T » 0 . 1 5 7 T 9 E 0 2 S Y » G . 1 8 9 / 0 E 0 3 S T E r t T » 0 . 3 2 4 - . 9 E 0 3 S T E 2 B T » 0 . 2 4 S 9 8 F 0 2 S E 2 8 T « 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 * 0 . 0 7 4 7 2 3 7 5 L A M U A * 1 . 3 2 4 8 7 9 9 9 E F F E C T I V E D l F r u S I V I ' T Y * 0 . 0 4 4 0 8 7 0 1 P U U L I S H E O O I F f U S I V i T Y * 0 . 0 9 9 0 0 0 0 0 A L P H A * 0 . 2 1 2 2 7 3 6 6 N U M B E R I T E R A T I O N S * 11 - 2 2 4 - N H R O G L N - B U T A N f c B E O D d T A H U N O A T A B E C D I A M E T E R C M S 1360 L E N G T H C M S E N D 2 . C N E H E I G H T P O R O S I T Y 0 . 5 9 C 0 0 2 . 1 1 0 0 0 7 . C O 0 0 O 0 . 2 / 0 0 0 B E O T L M P L R A T U R E K « 3 0 6 . 0 0 0 0 F L O W K A T E C C / S E C * R O O M 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 1 . 1 4 7 5 0 2 9 6 . 0 0 0 0 7 6 1 . 4 T I M E S E C . P E A K H E I G H T S 5 0 0 . 0 6 0 0 . 0 7 0 0 . 0 8 0 0 . 0 9 0 0 . 0 1 0 0 0 . ) 30.50 2 1 . 0 0 1 5 . 0 0 1 0 . 5 0 7 . 5 0 5 . 5 0 C O N S T A N T S F O R 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 » E X P C - H T I S E B T • U .424 . '»9C -U0 S Y E B T • U . H 6 2 1 B F 0 1 S Y * O . V O O J O t U 2 S T t B T • 0 . 2 7 4 9 0 6 0 ) S T t 2 B T » 0 . 2 4 1 J b E 0 2 S E 2 h T *, 0 . 4 I 5 0 7 E - 0 1 S T 2 E 1 T * 0 . 1 8 7 ? 6 t 0 6 S T 2 E 2 B * C . l 4 5 - > 9 F 0 5 S Y T 2 E 0 * 0 . 3 0 6 1 0 F 0 / S Y T C B l * 0 . 5 0 4 5 0 F 0 4 D I F F U S I V I T Y * 0 . 0 8 0 2 1 0 6 1 L A M C A • 1 . 2 3 4 2 S 0 7 1 E F F E C T I V E O l F F l l S I V l T V * 0 . 0 4 7 3 2 4 2 6 P U B L I S H E D D I F F U S I V I I Y * 0 . 0 9 9 0 0 0 0 0 A L P H A * 0 . 2 1 7 5 0 3 2 1 N U M B E R I T E R A T I O N S * 9 A • » » C • 1 9 6 . 2 1 1 2 a 0 . 0 0 3 8 0 1 5 7 1 . 1 2 4 9 6 3 2 0 S U M M A I I U N S F K L - M L E A S T S Q U A R E S C A L C U L A T I O N • 2 2 5 - N H R C G E N - B U T A N k B E D O A T A etC D I A M E T E R C M S 2 .61CC0 B60 L E N G T H C M S 7.CO0CO t N D Z O N E H E I G H T 0 .27000 P O R O S I T Y 0 .59000 R O N O A T A F L O U R A T t C C / S E C * 2 . 0 6 7 5 7 R O O M T£MPtRATURc» 2 9 6 . 0 0 0 0 A T M . P R t S S U R E MM HG 7 6 1 . 4 B E O T E M P E R A T U R E K* 3 0 6 . 0 0 0 0 T I M E S C C . 1 0 0 . 0 i 5 0 . 0 4 0 0 . 0 4 5 U . 0 5 0 0 . 0 P E A K HE1GHTS 1 3 . 0 0 2 7.CO 2 3 . 0 0 1 9 . 5 0 1 6 . 0 0 C O NSTANTS FUR L E A S T S A U A R E S F I T OF DATA I N Y-C* A - E X P I - B T I A * 1 0 0 . 5 3 7 S 7 B * 0 . 0 0 4 0 O 6 4 J C * 2 . 6 3 2 2 2 9 4 ? SUMMATIONS FROM L E A S T SQUARE S C A L C U L A T I O N SEBT « 0 . 1 0 4 / H E 0 1 SYERT * 0 . 2 6 5 6 8 E 0 2 SY « 0 . U 8 5 0 E 0 3 STEBT = 0 . 3 9 8 4 7 E 0 3 S T E 2 8 T « 0 . 8 5 8 4 4 E 0 ? S E 2 8 T » 0 . 2 3 6 8 2 E - O 0 S T 2 E B T * 0 . 1 5 6 5 7 E 0 6 S T 2 E 2 8 « 0 . 3 2 0 1 8 E 0 5 • S Y T 2 E B * U . 3 6 3 8 1 E 0 7 S Y T E B T * 9 . 9 6 7 9 4 E 0 4 D I F F U S I V I T Y * 0 . 0 8 2 3 0 6 8 9 LAMOA • 1 . 2 0 2 8 1 5 4 7 E F F E C T I V E O I F F U S I V I T Y * 0 . 0 4 8 5 6 1 0 6 P U B L I S H E D O I F F U S I V I T Y * 1 1 . 0 9 9 0 0 0 0 0 A L P H A * 0 . 2 2 0 4 2 5 1 2 NUMBER I T E R A T I O N S * 8 S P H E R I C A L P A C K I N G B E O - H Y D R O G E N - N I T R O G E N B E O O A T A B E O O I A M E T E R C M S 2 . 6 1 C 0 O B E O L E N G T H C M S 7 . 0 0 0 0 0 E N C Z O N E H E I G H T 0 . 2 7 0 0 0 P O R O S I T Y 0 . 3 9 3 0 0 R U N O A T A F L O W S A T E C C / S C C " 0 . 4 5 5 6 4 R u O M T E M P E R A T U R E * 2 9 5 . 0 0 0 0 A T M . P R E S S U R E MM H G 7 4 2 . 5 B E O T E M P E R A T U R E K« 3 0 9 . 0 U 0 0 T IN£ S t C . 5 0 . 0 1 0 0 . 0 1 5 0 . 0 2 0 0 . 0 2 5 0 . 0 3 0 0 . 0 J 5 0 T O - 4 0 0 . 0 4 5 0 . 0 5 0 0 . 0 P E A K H E I G H T S 4 6 0 . 0 0 2 0 7 . 5 0 9 5 . 0 0 5 0 . 5 0 2 8 . 0 0 1 8 . O C ~ T J V O ~ 0 1 1 . 0 0 1 0 . O C 9 . O 0 CONSTANTS FOR L E A S T " STACl«R£S F I T o r O A T A " I N V - C » A*E"5SPT-BT> A « 1 0 2 3 . 4 1 0 8 5 8 « 0 . 0 1 6 4 3 9 5 2 C > 9 . 9 5 3 5 1 4 1 0 SUMMATIONS FROM LEAST"SOUAKES C A L C U L A T I O N S E 6 T • 0 . 7 8 4 1 I E 0 0 S Y E B T • O . 2 5 2 9 0 t 0 3 S Y • 0 . 9 0 2 0 0 t 0 3 S T t B T " 0 . 6 9 B 4 9 E 0 2 S T E 2 B T " 0 . 1 4 3 4 2 6 0 2 S E 2 B T • 0 . 2 3 9 4 9 E - 0 0 - - - - - S T 2 E B T " 0 . B 9 1 2 B E 0 4 S T 2 E 2 B " 0 . 1 0 9 7 5 E 0 4 S Y T 2 E B * 0 . 1 2 1 3 1 E 0 7 S Y T 6 B T " G . 1 5 8 8 5 6 0 5 D I F F U S I V I T Y " 0 . 6 8 1 8 8 8 1 2 L A M O A • l . 2 0 2 5 « Y 3 o ~ E F F E C T I V E D I F F U S I V I T Y . 0 . 2 6 7 9 B 2 0 3 P U B L I S H E D D I F F U S I V I T Y " 0 . 8 2 0 0 0 0 0 0 A L P H A " 0 . 1 5 7 0 8 6 1 8 N U M B E R I T E R A T I O N S " 21 •22T H Y 0 R U 6 E N - N I T R U G l N BEO OATA BED DIAMETER CMS 2 . 6 1 0 0 0 ' HBO LENGTH CHS 7 .00000 fcNO /ONE HEIGHT 0 . 2 7 0 0 0 POROSITY 0.39300 RUN DATA F L O * R A T E C C / S E C * 0 . 8 3 1 6 8 R O O M TEMPERATURE* 2 9 5 . 0 0 0 0 ATM.PRESSURE MM HG 7 * 2 . 5 BED TEMPERATURE M 3 0 9 . 0 0 0 0 TIME S E C . PEAK HEIGHTS 5 0 . 0 2 2 9 . 0 0 1 0 0 . 0 8 0 . 0 0 1 5 0 . 0 3 1 . 4 0 ' 2 0 0 . 0 1 6 . 0 0 2 5 0 . 0 9 . 5 0 1 0 0 . 3 6 . 8 0 3 5 0 . 0 5 . 8 0 4 0 0 . 9 S . 0 0 CUN3TANTS FOR LEAST SAUARES F I T OF OATA I N Y - C " A«EXPI-BTI A • 6 6 4 . 1 2 6 1 7 B • 0 . 0 2 1 8 4 3 7 7 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 . 3 3 3 5 0 E 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 .37986E 04 S T 2 E 2 8 - 0 . 4 4 7 3 8 E 03 S Y T 2 E B ' O .32075E 06 SYTEBT* 0 .49743E 04 O I F F U S I V I T Y * 0 . 6 8 7 1 1 5 8 * LAMDA « 1 .193 )9411 EFFECTIVE O I F F U S I V I T Y * 0 . 2 7 0 0 3 6 5 2 PUBLISHED O I F F U S I V I T Y * 0 . 8 2 0 0 0 0 0 0 A L P H A * 0 . I 8 0 3 H 7 9 I NUMBER I T E R A T I O N S * 17 228' H¥DROG£N-NITRUGEN BED OATA BEO DIAMETER CMS 2.61000 REO LENGTH CMS 7.00000 tNO ZONE HEIGHT 0.27000 POROSITV 0.39300 RUN DATA 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 PEAK HEIGHTS 122.25 38.00 15.00 6.80 *.10 3.20 CONSTANTS FOR LEAST SAUARES FIT OF DATA IN Y-C* A » E X P I - B T I * * 399.9892.* . B • 0.02*29557 C • 3.4*35383* SUPMAIIONS FROM LEAST SQUARES CALCULATION SEOT « 0 .*2173O00 SYEBT • O.*00d*b 02 SY • 0.18935E 03 STEHT « 0.?9899t 0?" STE2BT* 0.52955fc 01 St2BT * 0.96582F-01 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 2.61000 ilBO LENtTH CMS T.00000 END ZONE HEIGHT 0 .27000 POROSITY 0 .39400 RUN OATA FLOW RATE C C / S E C * 0 . 6 0 4 ) 8 ROOM TEMPERATURE" 295.0D00 ATM.PRFSSURf MM HG 754 .5 BED TEMPERATURE K» 309.0000 TIME S E C . 150 .0 300 .0 4 5 0 . 0 600 .0 700.0 800 .0 9 0 0 . 0 1050 .0 PEAK HEIGHTS 131.00 6 8 . 0 0 4 0 . 5 0 2 7 . 0 0 22 .50 20 .00 18 .50 17 .00 CONSTANTS TOR LEAST SAUARES FIT 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 .854I6E 00 SYEBT * 0 .798S2E 0? SY « 0 .34450E 03 STEBT * 0 . 2 4 I I 6 E 03 STE2BT* 0.49.->98E 07 SE2BT • 0 .26731C-00 ST2EBT* 0.97431E 05 ST2E2B" 0.11567E US SVT2EB" 0 .44904L 07 SYTEBT" 0.164S8E 05 D I F F U S I V I T Y " 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 ITERAIIUNS* 11 -230- NITROOEN-ETHANE BEO OATA BEO OIAMETER CMS 2.61000 BEO L6.U6TH CMS T.00000 END ZONE HEIGHT 0.27000 POROSITY 0.39300 RUN DATA F L O W RATE CC/SEC* 0.91862 ROOM TEMPERATURE* 295.0000 ATM.PRESSURE MM HG '55.5 BEO TEMPERATURE «.* 309.0000 TIME SEC. 200.0 300.0 400.0 5C0.0 600.0 700.0 B O O . U 900.0 l o o o . r 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* 0.122HF 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 - 2 3 1 NHROQEN-E THANE BED DATA OEC DIAMETER CMS 2.610C0 BEO LENSTH CMS 7.00000 ENO ZONE HEIGHT 0.27000 POROSITY 0.39300 R U N D M A FLO* HATE CC/SEC* 1.36169 ROOM TEMPERATURE* 295.0000 ATM.PRFSSURE MM HG 755.5 BED TEMPERATURE K> 309.0000 TIME SEC. 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 P E A K H E I G H T S 4 7 . O C 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 -232- NMRUGEN-BUJANE BED OATA BEO DIAMETER CMS 2.6100U BSD LkNGTH CMS f.OOOCO ENO ZONE HEIGHT 0.2/000 POROSITY 0.39300 R U N O A T A F L O W R A T E C C / S E C * 0.59619 R O O M T E M P E R A T U R E * 297.5000 A T M . P R E S S U R E MM H(, 7*6.B D E O T E M P f c R A T U R C K* 309.0000 T I M E S E C . 4 0 0 . 0 5 0 0 . 0 6 0 0 . 0 7 0 0 . 0 8 0 0 . . 3 7 0 0 . 0 1 0 0 0 . 0 1 1 0 0 . 0 P E A K H E I G H T S 6 3 . 0 0 4 5 . 0 0 3 2 . 0 0 2 3 . 5 0 1 7 . 0 0 1 3 . 0 0 1 0 . 0 0 8 . 0 0 C O N S T A N T S F O R L E A S T S A U A R E S F I T U F O A T A I N Y - C * A o F X P I - b T I A * 2 5 6 . 2 9 7 3 6 B • 0 . 0 0 1 6 3 8 2 2 C • 3 . 2 6 1 5 9 5 0 1 S U M M A T I O N S F R U M L E A S T S Q U A R E S C A L C U L A T I O N Sfc-BT * 0 . 7 2 3 4 1 E 0 0 S Y E 8 T - 0 . 2 9 2 7 2 E 0 2 S Y * 0 . 2 1 1 5 0 E 0 3 S I E 6 T =• 0.420HJE 0 3 S T E 2 B T * C . 5 l 5 o 4 E O ? S E 2 8 1 * U . 1 0 S n O L - 0 0 S T 2 E B T * 0 . 2 7 0 7 4 E 0 6 S T 2 E 2 B * 0 . 2 ' O I O E 0 5 S Y T 2 E B * C . 7 8 0 7 6 E 0 7 S Y T E B T * 0 . 1 4 5 8 8 E 0 5 O I F F U S I V I T Y * 0 . 0 7 9 4 7 4 5 ) L A M ' I A * 1 . 2 4 5 6 8 2 U 6 E F F E C T I V E D I F F U S I V I T Y * 0 . 0 3 1 2 3 3 4 9 P U 8 L I S H E O O I F F U S I V I T Y * 0 . 0 9 9 0 0 0 0 0 A L P H A * ( I . 2 1 5 8 4 U 0 N U M B E R I T E R A T I O N S * 1 0 - 2 3 3 - N l T R C G F N - B U T A N f c U E O O A T A b E O D I A M E T E R C M S 2 . 6 1 0 0 0 B E O L E N G T H C M S 7 . 0 0 0 0 0 t - N C Z O N E H E I G H T 0 . 2 7 0 0 0 P O R O S I T Y 0 . 3 9 3 0 0 R U N O A T A F L O W 4 A T E C C / S E C * 0 . - 7 7 9 4 5 R O O M T f - M P P K A T U R E " 2 9 7 . 5 0 0 0 A T M . P R t S S U R t MM H I . 7 4 6 . 8 BET) T E M P E R A T U R E K« 3 0 9 . 0 0 0 0 T I M E S E C . 3 0 0 . 0 4 0 0 . 0 5 0 0 . 0 6 0 0 . 0 7 0 0 . 0 8 0 0 . 0 9 C 0 . 0 1 0 0 0 . 0 PEAK HEIGHTS 5 5 . 0 0 3 8 . 0 0 2 7 . 5 0 2 0 . 0 0 1 4 . 2 0 1 0 . 7 0 8 . 0 0 6 . 2 0 C O N S T A N T S F O R 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 . F X « < - B T I A « 1 5 9 . 1 5 2 7 9 B • 0 . 0 0 3 7 0 3 5 4 C • 2 . 3 8 3 0 4 0 9 0 S U M M A T I O N S F R O M L E A S T S Q U A R E S C A L C U L A T I O N S E B T « 0 . 1 0 0 8 f t 0 1 S Y E b T » 0 . 3 5 2 3 2 E 0 2 S Y « 0 . 1 7 9 6 0 6 0 3 S T t B T > 0 . 4 H 3 6 7 E 0 3 S 1 E 2 B T " 0 . B 0 3 5 6 E 0 2 S E 2 B T • 0 . 2 0 6 5 B E - 0 0 S T 2 E B T " 0 . 2 6 7 5 3 E 0 6 S T 2 C 2 B ' 0 . 3 4 5 0 1 E 0 5 S Y T 2 E B " 0 . 6 1 2 9 6 t 0 7 SYTCBT« 0 . 1 3 9 4 2 E 0 5 D I F F U S I V I T Y ' 0 . 0 7 8 4 4 7 ) 4 L A M O A « 1 . 2 6 1 9 9 3 0 1 E F F E C T I V t O i r r u S I V K Y - 0 . 0 3 0 8 2 9 8 1 P U B L I S H E D D I F F U S I V I T Y " 0 . 0 9 9 0 0 0 0 0 ALPHA" 0 . 2 1 9 1 9 1 6 0 N U M B E R I T E R A T I O N S " 8 •lib- Nil TRUGEN-BUTANE BEO DATA BED DIAMETER CMS BEO LENGTH CMS END ZONE HEIGHT POROSITY 0.39300 2.O1000 T.00000 0.2T0OO RUN DATA F L O N RATE CC/SEC* 1.23600 RuOM TEMPERATURE' 297.5000 ATM.PRfcSSUKt MM HO 746.8 BED TEMPERATURE K» 309.0000 TIME SEC. 350.0 450.0 550.0 650.0 750.0 850.0 950.0 1050.0 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*EXP(-BT) A > 108.61678 B • 0.00369706 C • 1.28090966 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 NUMBER ITERATIONS* 6 TIME 20HRS 98MIN 39.5SEC

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