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UBC Theses and Dissertations

Diffusion of gases Cox, Kenneth Edward 1959

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DIFFUSION OF GASES by I KENNETH EDWARD COX B . S c , (Eng.) U n i v e r s i t y of London, 1956 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of CHEMICAL ENGINEERING We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1959 In 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 the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and s t u d y . I f u r t h e r agree 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 copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by t h e Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of 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 not be 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 . Department of ^/^ycetf*£lJs ^ ^ » ^ v - * » - i - ^ y The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date f~^£y/>Y^c<-6i-'i''^<i J? , /9S 9 ( i ) BSTR ACT The e f f e c t i v e d i f f u s i o n c o e f f i c i e n t of the b i n a r y gas p a i r , hydrogen and nitrogen,has been measured f o r d i f f u s i o n through s e v e r a l types of porous s o l i d s . E l e c t r i c a l c o n d u c t i v i t i e s through the pore spaces of the same s o l i d s were a l s o measured w i t h a view towards t e s t i n g the analogy between ordi n a r y d i f f u s i o n and e l e c t r i c a l c o n d u c t i v i t y . The r e s u l t s obtained show some d i s c r e p a n c i e s (up to 2B%) from exact equivalence, e s p e c i a l l y i n porous s o l i d s w i t h a mean h y d r a u l i c diameter of l e s s than 1 micron. The d i f f u s i o n apparatus was a l s o used to determine the temperature dependence of the o f d i n a t y a d i f f u s i o n c o e f f i c i e n t i n the range 20 - 300°C. The r e s u l t s obtained show clo s e agreement w i t h the H i r s c h f e l d e r , B i r d and Spotz t h e o r e t i c a l equation f o r non-polar gas p a i r s . The data were a l s o compared w i t h other values reported f o r t h i s system and good agreement was found. I t i s the r e f o r e concluded that the flow apparatus used i s s a t i s f a c t o r y f o r the i n v e s t i g a t i o n of the temperature dependence of the bi n a r y d i f f u s i o n c o e f f i c i e n t . C a l i b r a t i o n of the apparatus at one temperature w i l l y i e l d s a t i s f a c t o r y absolute values f o r b i n a r y d i f f u s i o n c o e f f i c i e n t s at other temperatures. ACKNOWLEDGMENT The author wishes to express h i s g r a t i t u d e t o Imperia l O i l L t d . f o r f i n a n c i a l a s s i s t a n c e during the period i n which the research was conducted, and a l s o to Mr. R. Muelchen f o r the e x c e l l e n t workmanship on the apparatus constructed and to Mr. A. Hawkins f o r the glassblowing. In p a r t i c u l a r , the author expresses h i s thanks to Dr. D. S. Scott under whose s u p e r v i s i o n the research was conducted. NOMENCLATURE a Constant A 1 C r o s s - s e c t i o n a l area of sample A, B, Molecular species Bo V i s c o u s - f l o w P e r m e a b i l i t y c, C, Constant C A, Cg, Concentrations d Diameter DAB' DBA> D i f f u s i o n c o e f f i c i e n t . De 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 Do D i f f u s i o n c o e f f i c i e n t , hydrogen n i t r o g e n . Do / De D i f f u s i o n r a t i o I Current d e n s i t y k Boltzmann constant k*, K, Constant K* Kozeny Constant Ko S p e c i f i c e l e c t r i c a l conductance L Length of sample m H y d r a u l i c radius M i, M A, Mg Molecular weights N i ? N A, N B Number of moles p, pA, pB P a r t i a l pressures P T o t a l pressure Pc C a p i l l a r y pressure P' P e r m e a b i l i t y r» rA' rB Molecular diameter ( i v ) Nomenclature (Cont.»> r^g C o l l i s i o n diameter r 0 R e c i p r o c a l s p e c i f i c c o n d u c t i v i t y R Resistance of saturated sample Rc C a p i l l a r y Radius R 0 Resistance of sample volume of e l e c t r o l y t e R/R0 E l e c t r i c a l r e s i s t i v i t y r a t i o R f Gas constant 5 Area per u n i t bulk volume So Sp e c i f i c s Surface Area T Absolute temperature u V e l o c i t y of gas f l o w V P o t e n t i a l V I Volume of gas d i f f u s e d /sec. Rate of flow V^, V B Molar volumes (1) W ( i ) C o l l i s i o n i n t e g r a l x Index of temperature (Andrussow Equation) z Length of d i f f u s i o n path B Maximum energy of a t t r a c t i o n 6 P o r o s i t y "7 V i s c o s i t y cr I n t e r f i a c i a l Surface t e n s i o n jb Angle of Contact (v) TABLE OF CONTENTS Page I n t r o d u c t i o n and Theory 1 A. D i f f u s i o n i n Porous S o l i d s 1 B. P r o p e r t i e s of Porous S o l i d s . . . . . 6 1. P o r o s i t y 7 2. Viscous Flow P e r m e a b i l i t y 7 3 . Pore S i z e D i s t r i b u t i o n . 8 4. S p e c i f i c Surface Area 8 5. E l e c t r i c a l R e s i s t i v i t y R a t i o 9 C. D i f f u s i o n of Gases 10 1. B i n a r y D i f f u s i o n C o e f f i c i e n t . . . . . . . 10 2. Use of Porous S o l i d s to Measure the Binary C o e f f i c i e n t of Gases 13 3 . The Temperature Dependence of the Binary D i f f u s i o n C o e f f i c i e n t . 13 L i t e r a t u r e Survey 17 A. The Binary D i f f u s i o n C o e f f i c i e n t of Gases . . . 17 B. D i f f u s i o n i n Porous S o l i d s 18 C. The E l e c t r i c a l R e s i s t i v i t y R a t i o 21 D. The Analogy Between D i f f u s i o n and E l e c t r i c a l C o n d u c t i v i t y i n Porous S o l i d s 23 E. The Temperature Dependence of the Binary D i f f u s i o n C o e f f i c i e n t 24 SECTION A. DIFFUSION IN POROUS SOLIDS 29 Apparatus 29 A. D i f f u s i o n Apparatus 29 ( v i ) Table of Contents (Cont.) Page B. D e s c r i p t i o n of Samples 31 C. D i f f u s i o n C e l l 32 D. E l e c t r i c a l R e s i s t i v i t y 32 Experimental Procedures 34 A. Thermal C o n d u c t i v i t y C e l l C a l i b r a t i o n . . . . 34 B. Measurement of E f f e c t i v e D i f f u s i o n C o e f f i c i e n t of Porous S o l i d s 34 C. Measurement of E l e c t r i c a l R e s i s t i v i t y R a t i o . 35 D. Measurement of P o r o s i t y 36 E. Measurement of Pore S i z e D i s t r i b u t i o n by Mercury P e n e t r a t i o n 36 R e s u l t s 38 D i s c u s s i o n of R e s u l t s 41 Experimental E r r o r s 45. A. E l e c t r i c a l R e s i s t i v i t y R atio 45 B. Measurement of Gas Composition 45 C. Measurement of D i f f u s i o n Rates . 47 D. E f f e c t of Forced Flow 4% Conclusions and Recommendations , 50 SECTION B. THE TEMPERATURE DEPENDENCE OF THE DIFFUSION COEFFICIENT FOR THE HYDROGEN-NITROGEN SYSTEM 51 Apparatus 51 Experimental Procedures . 53 Temperature Dependence of the Binary D i f f u s i o n C o e f f i c i e n t 53 ( v i i ) Table of Contents (Cont.) Page Re s u l t s 56 A. Measurement of Gas Temperatures 56 B. Temperature Dependence of the D i f f u s i o n C o e f f i c i e n t 56 C. Comparison of D i f f u s i o n Rates f o r Hydrogen and Nitrogen 5$ D i s c u s s i o n o f R e s u l t s 63 A. Measurement of Gas Temperatures 63 B. Temperature Dependence of the Bulk D i f f u s i o n C o e f f i c i e n t 63 Conclusions and Recommendations 6? BIBLIOGRAPHY 68 APPENDIX Table T i l l 71 Sample C a l c u l a t i o n - P o r o s i t y . . . 72 Sample C a l c u l a t i o n - E l e c t r i c a l R e s i s t i v i t y R a t i o 73 Sample C a l c u l a t i o n - 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 and D i f f u s i o n R a t i o 74 Sample C a l c u l a t i o n - C a l i b r a t i o n Run 77 Sample C a l c u l a t i o n - D i f f u s i o n C o e f f i c i e n t at Higher Temperatures • 78 Sample C a l c u l a t i o n - D i f f u s i o n C o e f f i c i e n t H i r s c h f e l d e r Equation 80 ( v i i i ) LIST OF ILLUSTRATIONS 1, - Apparatus f o r Measurement of 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 and the Temperature Dependence of the B i n a r y D i f f u s i o n C o e f f i c i e n t 30.as; 2. - Thermal C o n d u c t i v i t y C e l l - W i r i n g Diagram . . . 30. 3 . - D i f f u s i o n C e l l (Diagrammatic) 33.a'. 4. - Evacuation Apparatus - E l e c t r i c a l R e s i s t i v i t y R a t i o 33-."&* 5. - Apparatus - E l e c t r i c a l R e s i s t i v i t y R a t i o . . . 33«c£ 6. - Pore S i z e D i s t r i b u t i o n Chart 4.0 7. - P l o t of R e c i p r o c a l D i f f u s i o n R a t i o and P o r o s i t y 4l©.&. 8. - P l o t of R e c i p r o c a l E l e c t r i c i t y R e s i s t i v i t y R a t i o and P o r o s i t y 4(3 •<?*•• 9.. - P l o t of D i f f u s i o n R a t i o and E l e c t r i c a l R e s i s t i v i t y R a t i o 4G".<fc; • 10. - D i f f u s i o n C e l l 52.av 11. - P l o t of Oven Temperature and C a l c u l a t e d Gas Temperature 55»a;. 12. - Comparison P l o t of D i f f u s i o n C o e f f i c i e n t s from H i r s c h f e l d e r Equation, Other Workers 1 Data and t h i s Work, (Logarithmic) 62.ar. 13. - Thermal C o n d u c t i v i t y C e l l C a l i b r a t i o n P l o t , Output M i l l i v o l t s and Mole F r a c t i o n Nitrogen. . $I.a ;V 14. - Thermal C o n d u c t i v i t y C e l l C a l i b r a t i o n P l o t Output M i l l i v o l t s and Mole F r a c t i o n Hydrogen . §l."b. 15. - C a l i b r a t i o n P l o t - Flowmeter Model 204, S t e e l F l o a t . Standard ccs/min Hydrogen, 70°F, 1 atm $i.c*. 16. - P l o t of Kinematic V i s c o s i t y and Temperature f o r Hydrogen and Nitrogen $l.d4 ( i x ) LIST OF TABLES Page S e c t i o n A. Table I . Table I I . Table I I I . S e c t i o n B. Table IV. Table V. Table VI. Table V I I . Table V I I I , Samples, D e s c r i p t i o n and R e s u l t s . . . . 40 Deviations from the Analogy, and Pore Sizes of the Samples 43 E f f e c t of Forced Flow on Volume of Hydrogen Passed Through Porous Sample . 4® Gas Temperature Measurements 59 D i f f u s i o n C o e f f i c i e n t R e s u l t s , Hydrogen-Nitrogen, 1 atm. 20-300°C 60 D i f f u s i o n C o e f f i c i e n t R e s u l t s of Other I n v e s t i g a t o r s , Hydrogen-Nitrogen, 1 atm. 61 Comparison of D i f f u s i o n Rates f o r Hydrogen and Nitrogen, 1 atm 62 D i f f u s i o n C o e f f i c i e n t s f o r the Hydrogen-Nitrogen System, 1 atm, as a F u n c t i o n of Absolute Temperature using the H i r s c h f e l d e r Equation to the F i r s t Approximation ( I n Appendix) . . 71 1 DIFFUSION OF GASES INTRODUCTION AND THEORY A. D i f f u s i o n i n Porous S o l i d s Many examples of chemical r e a c t i o n s which take place w i t h i n porous s o l i d s can be found i n the broad f i e l d s of heterogeneous c a t a l y s i s and combustion. For these processes, i t i s f r e q u e n t l y necessary to have a knowledge of the f l u x of d i f f u s i n g molecules to and from the boundaries of the porous s o l i d , and w i t h i n the s o l i d i t s e l f . Four main types of gas tran s p o r t are p o s s i b l e i n porous s o l i d s : Knudsen d i f f u s i o n Bulk, or ordinary d i f f u s i o n Surface d i f f u s i o n P o i s e u i l l e or f o r c e d flow Knudsen d i f f u s i o n occurs whenever the mean f r e e path between i n t e r m o l e c u l a r c o l l i s i o n s i s lar g e compared to the pore diameter. This means that a molecule a f t e r having had a c o l l i s i o n w i t h a pore w a l l , w i l l f l y to another w a l l before having a c o l l i s i o n w i t h a second molecule. In gener a l , Knudsen d i f f u s i o n occurs i n o pores about 400 A or l e s s i n diameter f o r most common gases at 2 one atmosphere pressure. The mean free path i n t h i s case i s about 1000A. The second type of d i f f u s i o n p o s s i b l e i# pores i s ordinary "bulk" d i f f u s i o n . Molecules s t r i k e one another much more f r e q u e n t l y than they s t r i k e the pore w a l l i n t h i s case. At 1 atmosphere pressure, the bulk value of the d i f f u s i o n c o e f f i c i e n t holds i n f a i r l y l a r g e pores; 5000 & i n diameter or l a r g e r . A t r a n s i t i o n s t a t e between Knudsen and bulk d i f f u s i o n o „ 0 occurs i n pores from 400 A to 5)000 A. Forced flow through a porous s o l i d occurs when a t o t a l pressure d i f f e r e n c e e x i s t s across the faces of the s o l i d . Whenever the mean fre e path i s larg e compared to the pore diameter, the forced flow w i l l be Knudsen flow which i s i n d i s -t i n g u i s h a b l e from Knudsen d i f f u s i o n . However, when the mean f r e e path i s small compared to the pore diameter, P o i s e u i l l e (stream-line)';' flow w i l l occur. The r a t e of flow i s given by P o i s e u i l l e ' s law, - * * 4 ^ ( 0 T 128 y L where Vf - rat e of flow d = diameter ^ = v i s c o s i t y of gas A p — = pressure drop per u n i t l e n g t h of porous s o l i d . For s o l i d s w i t h pDre diameters i n the r e g i o n of 5)000 1, t h i s 3 4 type of f o r c e d flow i s h i g h l y i n e f f i c i e n t because of the d term i n P o i s e u i l l e ' s law.,(relative t o the r a t e of d i f f u s i o n . ) F i n a l l y , surface d i f f u s i o n can occur. This i s a two-dimensional d i f f u s i o n along the pore w a l l as opposed to ordinary three-dimensional d i f f u s i o n i n the pore void space. The molecules may be p i c t u r e d as 'hopping 1 from one a d s o r p t i o n s i t e to the next. However, i t i s g e n e r a l l y t r u e that t h i s mechanism c o n t r i b u t e s l i t t l e to the t o t a l t r a n s p o r t at temper-atures above the c r i t i c a l temperature of the gas i n question. For a porous s o l i d , an e f f e c t i v e d i f f u s i o n c o e f f i c i e n t , made up of c o n t r i b u t i o n s from the above mechanisms, i s obtained. 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 i s smaller i n value than the bulk d i f f u s i o n c o e f f i c i e n t f o r three main reasons: (a) only a f r a c t i o n of the porous s o l i d i s open s t r u c t u r e , the remaining f r a c t i o n ( 1 - 9 ) being s o l i d matter. (b) due to the deviousness of the pore s t r u c t u r e , the gas molecules must t r a v e l a f u r t h e r distance through the s o l i d than i t s geometric length. (c) i t has r e c e n t l y been shown that p e r i o d i c v a r i a t i o n s i n pore c r o s s - s e c t i o n a l area can s u b s t a n t i a l l y reduce the r a t e of d i f f u s i v e gas t r a n s p o r t through porous media. Wheeler has presented some e m p i r i c a l formulae which permit c a l c u l a t i o n of the e f f e c t i v e d i f f u s i o n c o e f f i c i e n t from a knowledge of the average pore diameter, the p o r o s i t y and the gas system employed. They are, however, based on s i m p l i f i e d models of the pore s t r u c t u r e and allow only q u a l i t a t i v e 4 deductions regarding the e f f e c t s of d i f f u s i o n a l t r a n s p o r t on chemical r e a c t i o n r a t e s . With respect to experimental techniques f o r measuring d i f f u s i o n r a t e s w i t h i n porous s o l i d s , two p r i n c i p a l methods have been reported. (1) The sides of a c y l i n d r i c a l p e l l e t are sealed o f f and the two f l a t faces are contacted w i t h two d i f f e r e n t gases. The r a t e of steady s t a t e d i f f u s i o n through the p e l l e t i s determined by a n a l y s i s of the gas streams f o r "contamination 1 1 by the other component. (2) The second method depends on f i r s t f i l l i n g the pore s t r u c t u r e w i t h one component and then measuring i t s r a t e of e f f l u x i n t o a second component. In t h i s study, the f i r s t of these two methods i s employed. To study the e f f e c t s of bulk d i f f u s i o n alone, i t should be p o s s i b l e to s e l e c t a porous s o l i d , a gas p a i r , and a set of c o n d i t i o n s of temperature and pressure, so that a l l mechanisms except bulk d i f f u s i o n do not c o n t r i b u t e s i g n i f i c a n t l y to mass t r a n s p o r t . From the range of commercially a v a i l a b l e porous s o l i d s , one w i t h a s u i t a b l e average pore s i z e and narrow pore s i z e d i s t r i b u t i o n , can be s e l e c t e d . S i m i l a r l y , a s u i t a b l e gas p a i r can be chosen so that i n co n j u n c t i o n w i t h the porous s o l i d , operation under constant t o t a l pressure c o n d i t i o n s and over f a i r l y wide ranges of temperature and pressure w i l l give a d i f f u s i o n process which i s almost e n t i r e l y due to ordi n a r y or bulk d i f f u s i o n . A d i f f u s i o n r a t i o can be defined: D i f f u s i o n R a t i o = 22- _ _ _ _ ( 2 ) De where, Do = value of the bulk d i f f u s i o n c o e f f i c i e n t at a given temperature and pressure. De = value of 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 f o r the p a r t i c u l a r porous s o l i d at the same conditions of temperature and pressure (obtained experimentally) The d i f f u s i o n r a t i o i s a property of the s o l i d and i s independent of the gas p a i r used i n the d i f f u s i o n measurements provided bulk d i f f u s i o n only takes place. A number of porous s o l i d s were obtained whose p r o p e r t i e s would f u l f i l l the requirements that ensure bulk d i f f u s i o n only occurs. (Table I.) A gas p a i r , hydrogen-nitrogen, was employed i n d i f f u s i o n r a t i o measurements on the above s o l i d s . I n p a r t i c u l a r , with the Selas s o l i d s , us'ing the hydrogen-nit rogen gas p a i r , bulk d i f f u s i o n measure-ments should be p o s s i b l e at pressures up t o 100 atmospheres and temperatures up to any achievable value. Values of the d i f f u s i o n r a t i o , f o r porous s o l i d s at 1 atmosphere pressure and room temperature can range from 2.5>0 -100.0 depending on the type of s o l i d t e s t e d . I t i s important to note that the value of the d i f f u s i o n r a t i o can d i f f e r widely f o r a nearly-unconsolidated s o l i d and a consolidated s o l i d having the same p o r o s i t y value. There i s no present c o r r e l a t i o n of the d i f f u s i o n r a t i o s w i t h other p r o p e r t i e s of a porous s o l i d , such as the p o r o s i t y , the v i s c o u s - f l o w permea-b i l i t y , or the s p e c i f i c surface area. 6 Therefore i t would he extremely u s e f u l to discover some e a s i l y measurable property of a porous s o l i d w i t h which the d i f f u s i o n r a t i o could be c o r r e l a t e d . B. P r o p e r t i e s of Porous S o l i d s Some of the p r o p e r t i e s of a porous s o l i d have already been mentioned. I t i s however necessary to d i s t i n g u i s h between the two main c l a s s e s of p r o p e r t i e s a porous s o l i d can have: D i r e c t p r o p e r t i e s Derived p r o p e r t i e s D i r e c t p r o p e r t i e s are the true p r o p e r t i e s of the porous s o l i d and are independent of the method of measurement. Derived p r o p e r t i e s are those whose magnitude may depend on the method and d e f i n i n g theory of the experimental measurement. Some of the more important d i r e c t p r o p e r t i e s are: Bulk d e n s i t y True d e n s i t y P o r o s i t y or f r a c t i o n a l Void volume P h y s i c a l dimensions, e t c . and some of the more important derived p r o p e r t i e s are: Viscous-flow p e r m e a b i l i t y S p e c i f i c surface area Average pore s i z e Pore s i z e d i s t r i b u t i o n E l e c t r i c a l R e s i s t i v i t y R a t i o D i f f u s i o n R a t i o , e t c . 7 1. P o r o s i t y The pore volume can be determined from a knowledge of the d i f f e r e n c e i n weight between a dry sample and a sample saturated w i t h a l i q u i d of known d e n s i t y . The p o r o s i t y , or the f r a c t i o n a l v o i d volume, i s the r a t i o of the pore volume to the bulk volume, c a l c u l a t e d from the p h y s i c a l dimensions of the sample. 2. Viscous-flow P e r m e a b i l i t y The s p e c i f i c p e r m e a b i l i t y constant, Bo, i s defined by Darcy's Law: where u s v e l o c i t y of gas flow. By regarding a porous s o l i d as being equivalent to a bundle of p a r a l l e l c a p i l l a r i e s , a P o i s e u i l l e law expression can be deriv e d f o r the t o t a l pressure drop, A p _ 32-u L y ^ From these two equations together w i t h a number of s i m p l i f y i n g assumptions, and the d e f i n i t i o n of s p e c i f i c surface as S so = ~i - e - - - - (3) where S i s the surface area per u n i t bulk teolume and © i s the pDCosity i t i s p o s s i b l e to de r i v e the Kozeny equation 8 K'S2 (,-G)2 { Carman (3) gives a value of K ' - 5 . However, f o r b e t t e r accuracy ' „KX • • should be determined experimentally. By determining Bo and e experimentally i t i s p o s s i b l e to c a l c u l a t e a s p e c i f i c surface area, as defined by t h i s equation. By use of another d e r i v a t i o n , i t i s a l s o p o s s i b l e to de f i n e an average pore s i z e f o r the porous s o l i d on the b a s i s of the same assumptions as those used i n d e r i v i n g Kozeny* s -equation:. 3» Pore S i z e D i s t r i b u t i o n A more accurate means of determining the average pore s i z e and the s p e c i f i c surface area i s afforded by a knowledge of the pore s i z e d i s t r i b u t i o n . This can be determined by using the mercury porosimeter method of R i t t e r and Drake. From the pore s i z e d i s t r i b u t i o n c h a r t , the average pore s i z e can be determined by t a k i n g the pore s i z e that l i e s under the peak of the curve. This has been shown to be s t a t i s t i c a l l y (4) c o r r e c t f o r the type of curves obtained by t h i s method. The s p e c i f i c surface area can al s o be c a l c u l a t e d by means of a summation of the surface areas of a u n i t of pores and a knowledge of the pore volume, assuming pores of c y l i n d r i c a l shape. 4, S p e c i f i c Surface Area The most accurate means of determining the s p e c i f i c surface area experimentally i s the Brunauer-Emmett-Teller ^ ) method. This method l o c a t e s the point on the n i t r o g e n adsorption curve where a l l the surface i s covered w i t h a l a y e r of adsorbed gas, one molecule t h i c k . The B-E-T method l o c a t e s t h i s point by extending the Langmuir Isotherm to apply to m u l t i l a y e r adsorption. Unfortunately, only one sample was t e s t e d by t h i s method. This r e s u l t i s given i n Table 5 together w i t h the r e s u l t from mercury p e n e t r a t i o n f o r the p a r t i c u l a r sample t ested. 5. E l e c t r i c a l R e s i s t i v i t y R a t i o Consider a pore space f i l l e d w i t h an e l e c t r i c a l l y conducting l i q u i d . Assuming the s o l i d i s a non-conductor, the c o n d u c t i v i t y of a u n i t volume of the l i q u i d - f i l l e d porous sample w i l l be l e s s than the s p e c i f i c c o n d u c t i v i t y of the l i q u i d . The e l e c t r i c a l r e s i s t i v i t y r a t i o i s then defined as the r a t i o of the apparent s p e c i f i c c o n d u c t i v i t y of the s o l i d to the s p e c i f i c c o n d u c t i v i t y of the l i q u i d . In general f o r a porous sample, of length L, and c r o s s - s e c t i o n a l area A''", ft R Ro To . L/A' 0) 0 where ^ = e l e c t r i c a l r e s i s t i v i t y r a t i o R = r e s i s t a n c e of the saturated sample B 0 = r e c i p r o c a l s p e c i f i c c o n d u c t i v i t y of the s a t u r a t i n g l i q u i d . 10 From a l o g i c a l standpoint, the property most analogous to the d i f f u s i o n r a t i o would seem to be the e l e c t r i c a l r e s i s t i v i t y r a t i o . Much data on t h i s r a t i o can be found i n the l i t e r a t u r e . As i t i s e a s i e r to measure than the d i f f u s i o n r a t i o , such an analogy could u s e f u l l y be employed to p r e d i c t e f f e c t i v e d i f f u s i -v i t i e s f o r porous s o l i d s . From a t h e o r e t i c a l standpoint, the d i f f e r e n t i a l equations f o r the two processes of d i f f u s i o n and e l e c t r i c a l c o n d u c t i v i t y are also seen to be mathematically i d e n t i c a l : NA = - Do grad C A (2) I • - Ko grad V (3) Where N& _ number of moles of A passing through through u n i t c r o s s - s e c t i o n a l area per u n i t time Do - d i f f u s i o n c o e f f i c i e n t C A - c o n c e n t r a t i o n of species A I - current density Ko - s p e c i f i c e l e c t r i c a l c o n d u c t i v i t y V - p o t e n t i a l For a more d e t a i l e d account of the p r o p e r t i e s of porous s o l i d s , the work of Carman (3) should be consulted. C . D i f f u s i o n of Gases 1. Binary D i f f u s i o n C o e f f i c i e n t In a mixture of two gaseous components A and B, the r a t e of t r a n s f e r of A w i l l not only be determined by the r a t e 11 of d i f f u s i o n of A but a l s o by the behaviour of B. The molar r a t e of t r a n s f e r of A , per u n i t area, due to molecular motion i s given by F i c k ' s Law, N A = - DAB- ( O ^ z where = molar r a t e of d i f f u s i o n per u n i t area, A , D A B = the d i f f u s i o n constant of A i n B = the molar c o n c e n t r a t i o n of A Z = the distance i n the d i r e c t i o n of d i f f u s i o n . The corresponding r a t e of d i f f u s i o n of B i s given by:-N.b _ p 8 A . 1 £ ± . ( z ) 3 z I f the t o t a l pressure, and hence the t o t a l molar c o n c e n t r a t i o n i s everywhere constant, ^ £ A , ^ ^ 6 ; * Z. * Z must be equal and opposite and therefore A and B tend to d i f f u s e i n opposite d i r e c t i o n s . Let us consider the case of steady-state d i f f u s i o n which i m p l i e s the continuous supply and removal of the d i f f u s i n g m a t e r i a l . In the general case the t o t a l bulk flow of gas i n the Z- d i r e c t i o n i s (% % ) , the net t r a n s f e r of component A i s the r a t e of d i f f u s i o n plus the t r a n s f e r of A due to bulk f l o w : -NA = -DM.±£i. + ( N j . N . ) - * * - ( 3 ) 12 where P i s the t o t a l pressure. S u b s t i t u t i n g C A = pA/R'T and i n t e g r a t i n g , r e s u l t s i n the f o l l o w i n g equation, according to Sherwood and Pigford.^5) D A 6 P n \-(\+ N C / N A ^ P A Z / P fa »/l>U> » « . / P = N a , N s ^ R'Ti I - 0 + No/N/») PA, IP For p a r a l l e l d i f f u s i o n , N A and Ng w i l l have the same s i g n s ; f o r counter d i f f u s i o n , as i n t h i s work; % and Ng w i l l have opposite s i g n s . Equation (4) has various a p p l i c a t i o n s , but i t s use depends on knowledge of the r e l a t i o n between and Ng. Hoogschagen n a s s h o w n that f o r both Knudsen and bulk steady s t a t e counter d i f f u s i o n i n a constant t o t a l pressure system. N A > / 7 i T • n b H u = c (*) where M A - molecular weight of species A. Mg - molecular weight of species B. or i n general f o r any number of gases Y_ N i / M L - 0 (6) I f i t i s assumed that t h i s r e l a t i o n is t r u e , i t i s p o s s i b l e to reduce equation (4-) t o : N * = P A » — — - — iM (>*fo + P*W. L 7 ) where the s u b s c r i p t s 1, 2, r e f e r to the ends of the d i f f u s i n g path. 2. Use of JPorous S o l i d s to Measure the B inary D i f f u s i o n C o e f f i c i e n t of Gases Few experimental data f o r the v a r i a t i o n of the bulk d i f f u s i o r l c o e f f i c i e n t w i t h temperature are a v a i l a b l e . With a s u i t a b l e apparatus, i t should be p o s s i b l e to i n v e s t i g a t e the v a r i a t i o n of the bulk d i f f u s i o n c o e f f i c i e n t w i t h the absolute temperature. Using a s u i t a b l e porous s o l i d , a chosen gas p a i r , and a range o f t c o n d i t i o n s so that bulk d i f f u s i o n alone occurs w i t h i n the pores, i t should be p o s s i b l e to i n v e s t i g a t e the r a t e of gas d i f f u s i o n at temperatures above room temperature. C a l i b r a t i o n runs would be necessary at room temperature to o b t a i n a c a l i b r a t i o n f a c t o r f o r a given s o l i d sample which would r e l a t e the volume of gas d i f f u s e d to the known bulk d i f f u s i o n c o e f f i c i e n t at room temperature of the gas p a i r used. Accurate determinations of the b i n ary d i f f u s i o n c o e f f i c i e n t s are valuable as they can lead to an e v a l u a t i o n of the e m p i r i c a l , s e m i - t h e o r e t i c a l , and t h e o r e t i c a l equations that have been proposed to p r e d i c t t h i s c o e f f i c i e n t . 3. Temperature Dependence of the Binary D i f f u s i o n C o e f f i c i e n t A number of t h e o r e t i c a l expressions have been published i n the l i t e r a t u r e f o r the v a r i a t i o n of the b i n a r y d i f f u s i o n c o e f f i c i e n t w i t h the absolute temperature. The f o l l o w i n g semi-(7) e m p i r i c a l equation was f i r s t d erived by Maxwell. 14 DAS - A T J / 2 / ' / M A * / / M B CO where T - absolute temperature, °K Tab r c o l l i s i o n diameter CL = numerical constant This was l a t e r modified by G i l l i l a n d ^ to the form:-( V A '6 V 0 ' ^ l 1 P where V^, Vg - molar volume at normal b o i l i n g point (cc per gram-mole.) of A, B. Sutherland (9) p u t for?;ard an expression to ta k e the e f f e c t of temperature more e x a c t l y i n t o account. n aT*) ' / M A ^ ' / n ~ U A S r (z) T A © . P . O * C / T ) where C i s an a d d i t i o n a l numerical constant. A recent expression, by Andrussow i s as f o l l o w s : H - <L TX ( I + / MA • M e ) , . f A 6 . ( 4 ) P. ( V A ' 6 * V a * ) * / /VA.M /3 x i s the index to which the temperature i s to be r a i s e d t o correspond to the data. 15 The most fundamental equation yet developed i s tha t (11) given by H i r s c h f e l d e r et a l . This equation uses the Lennard-Jones 6-12 p o t e n t i a l as the model f o r the i n t e r a c t i o n of u n l i k e molecules i n s p h e r i c a l non-polar gas p&:i'rsV In the f i r s t approximation, the r e s u l t i s : D A 6 = 1 * + ' / M » • (5) where L WAG <" ' , , ) (T * ) ~J i s t h e c o l - L i s i o n i n t e g r a l based on the Lennard-Jones 6-12 p o t e n t i a l , and i s a f u n c t i o n of the group where = Boltzmann's Constant cf^e = maximum energy of a t t r a c t i o n between A, B ^AB = C o l l i s i o n diameter f o r the u n l i k e molecules, A and B. Wo accurate values have yet been determined f o r the maximum energy of a t t r a c t i o n or the c o l l i s i o n diameters i n the case of d i f f u s i o n . In order to c a l c u l a t e them, a s i m p l i f y i n g assumption i s made. I t i s assumed that the simple combining laws (from k i n e t i c theory) apply to the parameters f o r the s i n g l e components: Ae * ; 16 H i r s c h f e l d e r recommends that the re q u i r e d parameters f o r the s i n g l e components be obtained from quantum mechanics, v i s c o s i t y data, or from c l a s s i c a l k i n e t i c theory - i n the order given. For the hydrogen-nitrogen gas p a i r , parameters c a l c u l a t e d from v i s c o s i t y data were employed as no quantum mechanical parameters are a v a i l a b l e as yet. In these s i m p l i f y i n g assumptions and the use of parameters from v i s c o s i t y data l i e s the main disadvantage of the H i r s c h f e l d e r equation. For example, an e r r o r i n the rAB term i s squared when introduced i n t o the H i r s c h f e l d e r equation and so brings about a l a r g e r e r r o r . Values of D^g f o r the hydrogen-nitrogen system at a t o t a l pressure of 1 atmosphere c a l c u l a t e d by means of t h i s equation at 25°K i n t e r v a l s from 273-573°K are presented i n the appendix, TableVTII. A sample c a l c u l a t i o n i s a l s o pre-sented i n the Appendix, page IV . The l i t e r a t u r e survey f o l l o w i n g t h i s s e c t i o n presents 7 experimental t ests of the v a l i d i t y of t h i s equation i n p r e d i c t i n g ."Binary d i f f u s i o n co-e f f i c i e n t s at one atmosphere pressure and d i f f e r e n t temperatures. For gas p a i r s of non-polar s p h e r i c a l molecules, the H i r s c h f e l d e r equation gives r e s u l t s w i t h i n 1 - 2 °/0 of the experimental r e s u l t s . Agreement between experiment and the values c a l c u l a t e d from the H i r s c h f e l d e r equation by the method described when app l i e d to the present study would th e r e f o r e i n d i c a t e that both the equation and the experimental r e s u l t s obtained were >-accep'tabibe;r. 17 LITERATURE SURVEY A. The Binary D i f f u s i o n C o e f f i c i e n t of Gases Experimental values f o r the b i n a r y d i f f u s i o n c o - e f f i c i e n t of two gases were reported as e a r l y as 1907 by both Jackmann^ 1 2^ and Loschmidt^ 1"^ These i n v e s t i g a t o r s published values of the d i f f u s i o n c o e f f i c i e n t f o r p a i r s of gases which were l a t e r reproduced i n I n t e r n a t i o n a l C r i t i c a l Tables and L a n d o l t - B o r n s t e i n Tables. Their apparatus c o n s i s t e d of the n o w - c l a s s i c a l Loschmidt apparatus, i n which two chambers, each c o n t a i n i n g a d i f f e r e n t gas are separated by a s l i d e . The d i f f u s i o n i s allowed to take place when the s l i d e i s h e l d open f o r a given period of time. Their apparatus, by f a r the most commonly used f o r t h i s type of work, operates on the unsteady-s t a t e p r i n c i p l e f o r which the d i f f e r e n t i a l equation f o r d i f f u s i o n has been solved. The c o e f f i c i e n t s published were f o r the most part at atmospheric pressure and f o r temperatures close to room temperature. Their value f o r the gas p a i r hydrogen-nitrogen was: D 0 • 0 .697 cm 2/sec, 1 atm, 0°C. (14) Later i n v e s t i g a t o r s were Waldmann who gave a value f o r the same gas p a i r , D 0 = O.76 cm 2/sec, 1 atm, 20°C. (15) and B-oardman and Wild D 0 = 0.743 cm 2/sec, 1 atm, 14°G. 18 With the type of apparatus used, the i n v e s t i g a t i o n of the d i f f u s i o n c o e f f i c i e n t at temperatures s l i g h t l y above and below room temperature i s p o s s i b l e , but the apparatus i s not s u i t a b l e f o r measurements over wide ranges of temperature. D i f f u s i o n c o e f f i c i e n t s of vapours are most conven-i e n t l y determined by the method developed by Winkelmann. L i q u i d i s allowed to evaporate i n a v e r t i c a l glass tube over the top of which a stream of vapour-free gas i s passed s u f f i c i e n t l y r a p i d l y so that the vapour pressure i n the f l o w i n g stream i s maintained almost at zero. I f the apparatus i s maintained at a constant temperature, mass t r a n s f e r w i l l take place from the l i q u i d surface by d i f f u s i o n alone. The d i f f u s i o n c o e f f i c i e n t can be c a l c u l a t e d from the r a t e of f a l l of the l i q u i d surface and the c o n c e n t r a t i o n gradient. This method i s confined to vapours, and does not r e a d i l y lend i t s e l f to d i f f u s i o n c o e f f i c i e n t measurements over a wide temperature range. B. D i f f u s i o n i n Porous S o l i d s (17) Wicke and Kallenbach were among the e a r l y i n v e s -t i g a t o r s on t h i s subject. They c o u n t e r - d i f f u s e d n i t r o g e n and carbon d i o x i d e through porous glass and carbon p e l l e t s that were cemented i n gla s s tubes. Their r e s u l t s showed that Knudsen d i f f u s i o n took place i n the carbon p e l l e t s and bulk d i f f u s i o n i n the f r l t t e d c g l a s s . / "I Q \ T h i e l e developed mathematical r e l a t i o n s h i p s expressing the r a t e s of d i f f u s i o n i n pores and showing t h e i r e f f e c t on c a t a l y t i c r e a c t i o n r a t e s . 19 H o o g s c h a g e n ^ mounted c a t a l y s t p e l l e t s i n rubber tubing and l e t oxygen from a i r d i f f u s e through them i n t o a closed c i r c u l a t i n g system from which oxygen was continuously being removed. He proposed 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 c o u n t e r - d i f f u s i o n of gases w i t h equal pressures on both sides of the p e l l e t : Experimental evidence presented f o r three gas systems, helium-oxygen, nitrogen-oxygen and carbon-dioxide-oxygen showed th a t Equation (1) held f o r the experimental c o n d i t i o n s used (1 atm and 20°C). The r e s u l t s a l s o showed that i t was d i f f i c u l t to d i f f e r e n t i a t e between Knudsen and bulk d i f f u s i o n i n porous s o l i d s f o r constant t o t a l pressure systems; the t r a n s f e r equation obtained being the same i n both cases. (2) Wheeler v ' i n h i s monograph e n t i t l e d "Reaction Rates and S e l e c t i v i t y i n C a t a l y s t Pores", reviewed some of the past work on gas d i f f u s i o n i n porous p a r t i c l e s and included some of h i s own work on Knudsen d i f f u s i o n i n c a t a l y s t p e l l e t s . He pre-sented a s i m p l i f i e d but u s e f u l working model of the average porous s o l i d and the mechanism by which a gas r e a c t i o n can take place i n s i d e i t . I t was demonstrated that q u a l i t a t i v e deductions from the theory presented were f u l l y confirmed by the few r e l i a b l e data a v a i l a b l e . (19) Weisz v 7 / has r e c e n t l y developed a method u t i l i z i n g a flow-system f o r e f f e c t i v e d i f f u s i o n measurements on porous s o l i d s . He has appl i e d i t to s i l i c a - a l u m i n a c a t a l y s t p e l l e t s 20 using the hydrogen-nitrogen gas system. He chose t h i s system as i t was the most s e n s i t i v e f o r a n a l y t i c a l purposes by means of a thermal c o n d u c t i v i t y c e l l . The r e s u l t s obtained have been used to develop a general d i f f u s i o n a l c r i t e r i o n f o r c a t a l y s t p a r t i c l e s . This c r i t e r i o n makes i t p o s s i b l e to determine whether a given experimental or p r a c t i c a l system w i l l be free of appre-c i a b l e d i f f u s i o n a l e f f e c t s , independent of the r e a c t i o n k i n e t i c s . P e t e r s e n ^ ^ has developed a mathematical theory of pore c o n s t r i c t i o n s to account f o r high experimental values of the d i f f u s i o n r a t i o obtained i n t a b l e t e d or extruded porous media. D i f f u s i o n i n a pore of var y i n g c r o s s - s e c t i o n i s compared w i t h that o c c u r r i n g i n an {equivalent' c y l i n d r i c a l pore. The d e r i v a t i o n shows that p e r i o d i c c o n s t r i c t i o n s i n the pore channel may account f o r the f a c t that r a t e of d i f f u s i v e t r a n s p o r t i n the 'equivalent' c y l i n d r i c a l pore can be as much as three times greater than that i n the c o n s t r i c t e d pore. This may t h e r e f o r e provide the b a s i s f o r a reasonable explanation of the high d i f f u s i o n r a t i o s obtained i n p r a c t i c e , but no s a t i s f a c t o r y experimental technique has yet been derived to t e s t the above theory. In the present work, a method s i m i l a r to that of (19) Weisz has been used to measure the e f f e c t i v e d i f f u s i o n co-e f f i c i e n t s f o r various types of porous media. A modified arrange-ment of the same method was used to measure the temperature dependence of the d i f f u s i o n c o e f f i c i e n t f o r the hydrogen-nitrogen system i n the range 293-573°K. 21 C. The E l e c t r i c a l R e s i s t i v i t y R a t i o Values of the e l e c t r i c a l r e s i s t i v i t y r a t i o , or forma-t i o n f a c t o r as i t i s sometimes c a l l e d , have been measured f o r various non-conducting porous s o l i d s over a wide range of p o r o s i t i e s . D i f f e r e n t workers have t r i e d to r e l a t e t h i s r a t i o to other p r o p e r t i e s of the s o l i d , most commonly, the p o r o s i t y . One of the e a r l y workers i n t h i s f i e l d was the eminent p h y s i c i s t (20) C l e r k Maxwell. He derived an expression f o r the e l e c t r i c a l r e s i s t i v i t y r a t i o based on a cubic assemblage of spheres. R/R 0 = (3 - e/J/2© - - - ( l ) Some experimental data were presented to support the above r e l a t i o n s h i p but i n general, values p r e d i c t e d from i t are lower (21) than those obtained experimentally. (22) A t h e o r e t i c a l expression by S l a w i n s k i v * f o r an aggregate of spheres i s , R/R0 = (1.3219 - 0 . 3 2 1 9 6 ) 2 /G - (2) More experimental evidence was put forward than p r e v i o u s l y , and (21) the S l a w i n s k i formula t e s t e d on other work shows that the r a t i o s p r e d i c t e d are too low at p o r o s i t i e s l e s s than 20$. (2^) Archie on the b a s i s of measuring the e l e c t r i c a l r e s i s t i v i t y r a t i o s of o i l - b e a r i n g sands, put forward the e m p i r i c a l formula, R/R0 = e ~ 1 , 3 - - - - (3) f o r unconsolidated porous media. This was the equation of the best s t r a i g h t l i n e through h i s experimental p o i n t s . Later work (20) found the AriShie r e l a t i o n s h i p f i t t e d the experimental data w e l l f o r e = 10 - 25 %. (21) W y l l i e and Gregory v ' reviewed a l l of t h i s past work and decided t h a t scant a t t e n t i o n had been paid to the systematic experimental determination of e l e c t r i c a l r e s i s t i v i t y r a t i o s f o r unconsolidated porous media. They embarked on an experimental determination of these r a t i o s f o r aggregates of spheres, cubes, c y l i n d e r s , d i s c s and t r i a n g u l a r prisms i n the p o r o s i t y range 12 - 56 %. Their r e s u l t s showed that f o r unconsolidated spheres and sands, the t h e o r e t i c a l expressions are obeyed i n c e r t a i n p o r o s i t y ranges. For a r t i f i c i a l l y - c o n s o l i d a t e d packings the r e l a t i o n s h i p , R/R0 c 6 " (4) where C, K are constants i s obeyed. The constants C, K depend on the type of cementation present. C o r n e l l and K a t z ^ 2 4 ^ have determined e l e c t r i c a l r e s i s t i v i t y r a t i o s f o r sandstones, limestones and dolomites i n t h e i r work on the tur b u l e n t flow of gases through porous media. The length of the flow path through the medium was evaluated by means of the r a t i o i n order to p r e d i c t the nature of t u r b u l e n t flow through the pores. The r e s u l t s obtained f o r the porous rocks showed f a i r s c a t t e r when p l o t t e d against the p e r m e a b i l i t y and the p o r o s i t y . (25) F i n a l l y , McMullin and Muccini y determined the e l e c t r i c a l r e s i s t i v i t y r a t i o s f o r a number of v a r i e t i e s of porous s o l i d s , both conducting and non-conducting, i n order to e s t a b l i s h a c o r r e l a t i o n betv^een the h y d r a u l i c r a d i u s , the vis c o u s - f l o w p e r m e a b i l i t y and the e l e c t r i c a l r e s i s t i v i t y r a t i o , T heir r e s u l t s are presented i n the form of a p l o t of the McMullin equation. m2 = k r. P.« R/R0 - - - - - - ( 5 ) where m - h y d r a u l i c radius p 1 - vi s c o u s - f l o w p e r m e a b i l i t y k' - constant f o r which t h e i r data f a l l s on a reasonably good s t r a i g h t l i n e when p l o t t e d l o g a r i t h m i c a l l y . D. The Analogy between D i f f u s i o n and E l e c t r i c a l C o n d u c t i v i t y  i n Porous S o l i d s For a porous non-conducting s o l i d , the value of the d i f f u s i o n r a t i o might be expected to be the same as the value of the e l e c t r i c a l r e s i s t i v i t y r a t i o . ) and K l i n k e n b e r g ( 2 6 ) fc (27) This i s stated by Carman^ l i n k e n b e r g ^ ^ f o r gas d i f f u s i o n i n porous s o l i d s . S c h o f i e l d and Dakskinamurti have v e r i f i e d t h i s statement f o r l i q u i d d i f f u s i o n i n porous sands. Klinkenberg presents only one experimental d e t e r -mination, the main p o r t i o n of h i s work being devoted to the comparison of d i f f u s i o n r a t i o s and e l e c t r i c a l r e s i s t i v i t y r a t i o s published by other i n v e s t i g a t o r s f o r s i m i l a r porous media. For 24 unconsolidated media, the r a t i o of e i t h e r the e l e c t r i c a l r e s i s t i v i t y r a t i o or the d i f f u s i o n r a t i o to the p o r o s i t y was found to range from 1.40 to 2.60. Prom the above comparison and a c o n s i d e r a t i o n of the d i f f e r e n t i a l equations governing both d i f f u s i o n and e l e c t r i c a l c o n d u c t i v i t y , he concluded that the above analogy should hold t r u e . (27) Among the references c i t e d by Klinkenberg are (oft) e a r l y workers such as Buckingham^ who i n v e s t i g a t e d the d i f f u s i o n of carbon d i o x i d e through c l a y s and l a t e r workers >ta (29) (2Q) (30) such as Penman^ 7 and Van Bavel. From the r e s u l t s obtained f o r the d i f f u s i o n of carbon d i o x i d e , through s o i l , Penman put forward the r e l a t i o n s h i p _ — * = o.6 e . . . (i) Do / De L a t e r , Van B a v e l ^ ^ ) modified t h i s r e l a t i o n t o , 0.66 6 - - - (2) Do / De f o r l oose, unconsolidated beds. As the average d e v i a t i o n of the experimental data used i s of the order of 5 %i there i s not much to choose between e i t h e r of these two r e l a t i o n s . In the p l o t s presented l a t e r , the f i r s t r e l a t i o n i s shown. E. The Temperature Dependence of the Binary D i f f u s i o n C o e f f i c i e n t H i r s c h f e l d e r , et a l ^ " ^ w r i t e : In the f i r s t approximation to the d i f f u s i o n c o e f f i c i e n t i t i s only the for c e s between u n l i k e molecules which occur. This means that the c o e f f i c i e n t of d i f f u s i o n as 25 a f u n c t i o n of temperature a f f o r d s the best method f o r o b t a i n i n g the f o r c e constants c h a r a c t e r i s t i c of the i n t e r -a c t i o n between u n l i k e molecules. At the present time, u n f o r t u n a t e l y , such experimental data are not a v a i l a b l e . Hence we can see the importance and need f o r the experimental determination of the temperature dependence of the bin a r y d i f f u s i o n c o e f f i c i e n t . Recent work not mentioned i n the above p u b l i c a t i o n has been done by Schafer et a l . J They f i r s t i n v e s t i g a t e d the temperature dependence of the d i f f u s i o n c o e f f i c i e n t s f o r three gas p a i r s ; hydrogen-nitrogen, carbon d i o x i d e - n i t r o g e n and hydrogen-carbon d i o x i d e , at 1 atmosphere and from 197-330°K, using a modified Loschmidt type apparatus. T h e i r experimental r e s u l t s showed that the d i f f u s i o n c o e f f i c i e n t , f o r the hydrogen-nitrogen system, was d i r e c t l y p r o p o r t i o n a l to the absolute temperature r a i s e d t o the 1.81 power. The r e s u l t s were then compared w i t h values of the d i f f u s i o n c o e f f i c i e n t f o r the same gas p a i r as pred i c t e d from the s e m i - t h e o r e t i c a l equation of Andrussow. The agreement reached was good ( w i t h i n 3#)» despite the f a c t that the Andrussow equation has no term t o take i n t o account any conc e n t r a t i o n dependence of the d i f f u s i o n c o e f f i c i e n t . (32) In a succeeding paper J , by the same authors; d i f f u s i o n c o e f f i c i e n t s were measured from 200 - 400°K, and at 1 atmosphere, f o r the gas systems hydrogen-nitrogen, argon-helium and argon-n i t r o g e n . The temperature exponent f o r the hydrogen-nitrogen system appears to be 1.61 f o r the r e s u l t s presented i n t h i s l a t t e r paper. The authors o f f e r no explanation f o r the d i f f e r e n t v a l u e . 26 (33) In t h e i r l a t e s t published paper, Schafer et a l , have extended t h e i r work to i n v e s t i g a t i n g systems of r a r e gas pains. The d i f f u s i o n c o e f f i c i e n t of neon-argon was measured between 90°K and 473°K, and t h a t of argon-krypton between 200°K and 473°K at 1 atmosphere. For the f i r s t system, the temperature dependence of the d i f f u s i o n c o e f f i c i e n t was found to be to the 1.79 power, w h i l s t f o r the second system, i t was found to be to the 1.74 power. From t h i s work they concluded that the ' r i g i d sphere' c o l l i s i o n diameters, (simple k i n e t i c theory) d e r i v e d from d i f f u s i o n measurements not only are smaller than those derived from v i s c o s i t y data but they a l s o show a much smaller temperature dependence. In the work described i n the f o l l o w i n g paragraphs, measurements of the binary d i f f u s i o n c o e f f i c i e n t have been made f o r various gas p a i r s . In a l l cases, these measurements were compared to values of the d i f f u s i o n c o e f f i c i e n t p r e d i c t e d from the t h e o r e t i c a l equation of H i r s c h f e l d e r et a l ^ " ^ taken to the f i r s t approximation. The most common a p p l i c a t i o n of t h i s equation attempts to p r e d i c t the binary d i f f u s i o n c o e f f i c i e n t s of gas p a i r s from parameters obtained from v i s c o s i t y measurements of the pure components and simple combining r u l e s . The p o t e n t i a l f u n c t i o n u s u a l l y used i n t h i s equation to evaluate the c o l l i s i o n i n t e g r a l i s the Lennard-Jones (6/12) p o t e n t i a l , although other functions ;e.g. the Stockmayer p o t e n t i a l , are suggested f o r p o l a r or long molecules. Amdur and S c h a t z k i ^ ^ ) have measured the d i f f u s i o n co-e f f i c i e n t s f o r the system xenon-xenon and argon-xenon over a 27 temperature range from 195 - 378°K w i t h a claimed experimental accuracy of 1 %. Their l a c k of agreementv;wilb.h p r e d i c t i o n s of the d i f f u s i o n c o e f f i c i e n t from the H i r s c h f e l d e r equation i s of the order of 0.5%. This discrepancy i s a s c r i b e d , by the authors, to the inadequacy of the Lennard-Jones 6/12 p o t e n t i a l i n the ev a l u a t i o n of the c o l l i s i o n i n t e g r a l s . Kilbanova et a l ^ 5 ) measured the temperature dependence of the d i f f u s i o n c o e f f i c i e n t f o r the gas p a i r s carbon d i o x i d e - a i r and water vapour-air from 293°K - 1500°K by means of a modified Loschmidt technique. The accuracy a t t a i n a b l e i n t h e i r work has been sta t e d to be qu i t e poor, however. (36) w i l k e and Lee^36) who reviewed t h e i r work s t a t e that i f the authors' value f o r the d i f f u s i o n c o e f f i c i e n t at 293°K i s assumed to be c o r r e c t , then c l o s e agreement, 2$, i s reached w i t h p r e d i c t i o n s from the H i r s c h f e l d e r equation. This agreement i s a l s o stated to be w i t h i n the l i m i t s of experimental p r e c i s i o n . The systems helium-argon, hydrogen-argon, hydrogen-n butane and hydrogen-sulphur h e x a f l u o r i d e were i n v e s t i g a t e d by S t r e t c l o w ^ 8 ) 0 v e r the temperature range 245 - 420°K using a modified Loschmidt type c e l l to determine the d i f f u s i o n co-e f f i c i e n t s . The r e s u l t s , f o r the f i r s t three systems named, showed only f a i r agreement when compared w i t h p r e d i c t i o n s from the H i r s c h f e l d e r equation. (5 - 10%). The r e s u l t s f o r the d i f f u s i o n c o e f f i c i e n t s of the hydrogen-sulphur h e x a f l u o r i d e system were the only ones that agreed c l o s e l y w i t h the above p r e d i c t i o n s . In t h i s case, the agreement was w i t h i n 1%. The 28 disagreement w i t h theory was st a t e d by the author to be due to the i n a b i l i t y of the t h e o r e t i c a l p o t e n t i a l to p r e d i c t a c t u a l behaviour. The l a t e s t work on t h i s subject has been done by Walker and Westenberg. 0 In a new method c a l l e d the "point source" technique, they i n j e c t e d a trace gas from a f i n e , hypodermic tube, ( e s s e n t i a l l y a "Point source") i n t o a slow, laminar stream of a second ( c a r r i e r ) gas. Measurements of the trac e gas con c e n t r a t i o n downstream of the source by means of p r e c i s e gas sampling permitted the bi n a r y d i f f u s i o n c o e f f i c i e n t to be determined. By heating the c a r r i e r gas, the measurements were extended to high temperatures (1150°K). R e s u l t s , w i t h a claimed p r e c i s i o n of ± 1%, over the temperature range 300 -1150°K, and at 1 atmosphere, were presented f o r the systems helium-nitrogen and carbon d i o x i d e - n i t r o g e n . The experimental data were discussed i n terms of int e r m o l e c u l a r p o t e n t i a l energies and f o r the gas p a i r helium-n i t r o g e n a purely r e p u l s i v e p o t e n t i a l was found to be s a t i s f a c t o r y . The data were al s o f i t t e d w i t h a Lennard-Jones 6-12 p o t e n t i a l w i t h e q u a l l y good r e s u l t s . In summary, i t may be concluded that data on the temperature dependence of the binary d i f f u s i o n c o e f f i c i e n t i s not p l e n t i f u l , and much of i t i s l a c k i n g i n accuracy or p r e c i s i o n , or both. The best data f o r non polar sample molecules appears to agree to w i t h i n 2 - 3% w i t h that p r e d i c t e d by the H i r s c h f e l d e r equation as u s u a l l y a p p l i e d . F u r t h e r , the temperature dependence pr e d i c t e d by t h i s same equation f o r p a i r s of simple molecules i s v e r i f i e d w i t h an accuracy of about % by the a v a i l a b l e data. 29 SECTION A. DIFFUSION IN POROUS SOLIDS APPARATUS A. D i f f u s i o n Apparatus The apparatus i s shown i n F i g u r e ( 1 ) . I t c o n s i s t e d of two gas t r a i n s , one f o r hydrogen and the other f o r n i t r o g e n which then contacted opposite faces of the c y l i n d r i c a l s o l i d sample. The hydrogen used was standard commercial grade (99*8$ p u r i t y ^ , and the n i t r o g e n was premium q u a l i t y (99.9$ p u r i t y ) . Let us consider the hydrogen gas t r a i n i n d e t a i l . The gas from the storage c y l i n d e r was passed through a pressure r e g u l a t o r and a Deoxo tube which employed a c a t a l y s t to remove any oxygen present. A f t e r t h i s p u r i f i c a t i o n , the gas was passed through a c a p i l l a r y tube (2" long s e c t i o n of thermometer tubing) to provide a back pressure of approximately 10 inches of mercury to smooth out the flow. In comparison, the pressure drop through the r e s t of the system, a f t e r the c a p i l l a r y tube, was of the order of 1 i n c h of water. The hydrogen, a f t e r being d r i e d by s i l i c a g e l i n a 10 i n c h long, 1 i n c h diameter tube was metered i n glass tube flowmeters of the rotameter type. These flowmeters were Matheson Corporation U n i v e r s a l flowmeters, nos. 202, 203 and 204. By s e l e c t i n g a s u i t a b l e flowmeter s i z e and f l o a t m a t e r i a l a hydrogen flow ranging from 100 - 5000 ccs/min. or a n i t r o g e n flow ranging from 40 - 2000 ccs/min. at room temperature and atmospheric pressure could be measured. From the flowmeters, the gas passed i n t o the d i f f u s i o n c e l l where i t contacted one face of the 30 c y l i n d r i c a l sample a f t e r which i t passed to atmosphere, or to a thermal c o n d u c t i v i t y c e l l , through an o u t l e t stop-cpck. The n i t r o g e n stream followed the same sequence as the hydrogen, except that no oxygen removal step was i n c l u d e d . I d e n t i c a l pressures on each side of the sample were maintained by a d j u s t i n g the hydrogen o u t l e t stop-cock to show a zero pressure on the d i f f e r e n t i a l draught gauge. An E l l i s o n draught gauge was employed w i t h a range of 0 - 1 i n c h of water over a s c a l e length of 12 inches. D i f f e r e n t i a l pressure readings could be made w i t h an accuracy of + 0 .005 i n c h water peessure. The use of hydrogen as one of the gases l e d to a very s e n s i t i v e d e t e c t i o n system when using a thermal c o n d u c t i v i t y c e l l as a d e t e c t o r . This c e l l was placed downstream from the d i f f u s i o n c e l l and could be used to analyse f o r hydrogen i n the n i t r o g e n stream or v i c e - v e r s a . The thermal c o n d u c t i v i t y c e l l used was a Gow-Mac, Model NIS recorder type, four f i l a m e n t u n i t which was operated w i t h a filament current of 120 m A. The f i l a m e n t s were s i t u a t e d i n deep d i f f u s i o n passages which gave n e g l i g i b l e s e n s i t i v i t y t o flow r a t e changes provided the flow r a t e d i d not f a l l below 100 ccs/min. The w i r i n g diagram f o r the thermal c o n d u c t i v i t y c e l l i s shown i n F i g u r e 2. A V a r i a n Associates G-10 recorder and a Leeds and Northrup portable p r e c i s i o n potentionmeter were used to measure the output m i l l i v o l t s i g n a l of the thermal conduc-t i v i t y c e l l b r i d g e . MANOMETER-, MANOMETER—I FLOWMETER -INLET COIL OVEN DIFFUSION CELL DIFFERENTIAL D R A U G H T G A U G E M > ± T H E R M A L CONDUCTIVITY C E L L T O ATMOSPHERE F i g u r e 1. Apparatus f o r Measurement of 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 and the Temperature Dependence of the  Bin a r y D i f f u s i o n C o e f f i c i e n t 30.&. TO RECORDER — W W V N A A SENSI TIVI CONTROL TY TOO OHMS 20 OHMS ZERO ADJUST -A/WWW 2 OHM POT AA/WWWWv C E L L 4 X 30 OHMS \ SAMPLE \ REFERENCE \ I—AAMAAy—»-7WWv\ I © 6V © TO POTENTIOMETER Figure 2. Thermal. Conductivity/ (Stem - W i r i n g Dlagrami 31 B. D e s c r i p t i o n of Samples The samples used i n t h i s study were obtained i n the form of rods of approximately 1" diameter or p l a t e s 1" t h i c k and formed i n t o c y l i n d e r s which were then cut to the r e q u i r e d l e n g t h on a diamond saw. The Selas 01 , 015, 03-1? 03-2 samples were suppli e d by the Selas Corporation. They were microporous s y n t h e t i c ceramic rods obtained i n 1 i n c h diameter, 6 i n c h lengths. The Limestone 36 and 63 samples were obtained from o i l w e l l cores. They were a n a t u r a l l y o c c u r r i n g A l b e r t a limestone and were donated by the Imperial O i l Research Department, Calgary. The Sandstone, SI sample was obtained from the same source and was a n a t u r a l l y o c c u r r i n g sandstone from the Pembina O i l f i e l d i n A l b e r t a . The Alundum samples came i n the form of a standard p l a t e 12" x 12" x 1" s u p p l i e d by A.P. Green F i r e B r i c k Co. L t d . , the Canadian r e p r e s e n t a t i v e of the Norton Co. which manufactured the Alundum. I t was an e x t r a f i n e , hard s y n t h e t i c ceramic s o l i d manufactured f o r a e r a t i o n purposes. The F i l t r o s sample a l s o came i n p l a t e form 12" x 12" x l ^ r " s u p p l i e d by F i l t r o s Inc. The grade was e x t r a dense and i t c o n s i s t e d e s s e n t i a l l y of s i l i c a (quartz sand bonded by v i t r e o u s s i l i c a ) . The Carbon sample was supplied by the N a t i o n a l Carbon Co. and a r r i v e d i n the form of a 2 i n c h diameter rod, 24 inches long. The sample was a porous gra p h i t e , Manufacturer's grade 60 C. 32 C. D i f f u s i o n C e l l The d i f f u s i o n c e l l i s shown i n Fi g u r e s (3) and (10). I t c o n s i s t e d of two i d e n t i c a l one inc h diameter, bell-shaped halves having i n l e t and o u t l e t tubes, 8 mm i n diameter, e n t e r i n g by means of r i n g s e a l s . The i n l e t tube entered at r i g h t angles to the body of the c e l l and then was bent 90° down the center l i n e of the c e l l to w i t h i n 1/4 i n c h of the s o l i d sample face, terminating i n a s l i g h t b e l l . The o u t l e t tube takes the o u t l e t gas d i r e c t l y from the upper p o r t i o n of the c e l l . Two methods of sample mounting were employed. In the f i r s t i n s t a n c e , a Gooch rubber sleeve was drawn over the sample and the ends of the c e l l h a l v e s . In t h i s manner the d i f f u s i o n r a t e determinations at room temperature were c a r r i e d out. For r a t e measurements at higher temperatures an A r a l d i t e AN 130 r e s i n sleeve bonded the sample to the glas s ends of the d i f f u s i o n c e l l . This r e s i n withstood temperatures up to 300°C. D. E l e c t r i c a l R e s i s t i v i t y R a t i o Apparatus The e l e c t r o l y t e used f o r c o n d u c t i v i t y measurements i n t h i s work was 0.1003-7 N KC1 s o l u t i o n . E l e c t r i c a l c o n d u c t i v i t i e s f o r KC1 s o l u t i o n s have been very a c c u r a t e l y measured. The A/C bridge used was an I n d u s t r i a l Instruments type RC C o n d u c t i v i t y Bridge reading d i r e c t l y i n ohms. A l l measurements of r e s i s t a n c e were made at 1000 cycles/second. The s p e c i f i c r e s i s t i v i t y of the e l e c t r o l y t e used was measured w i t h a d i p - c e l l having a c e l l constant equal to 1.0. 33 F i g u r e (4) shows the apparatus f o r evacuating and s a t u r a t i n g the sample w i t h the O.IN KC1 e l e c t r o l y t e . The sample was held between the two bell-shaped glass halves by means of a Gooch rubber sleeve. The upper h a l f of the bell-shaped ends was connected to a vacuum system w h i l s t the lower h a l f was connected by a tube w i t h a stop-cock to the e l e c t r o l y t i c s o l u t i o n . The r e s i s t a n c e of the sample was measured by means of the apparatus i l l u s t r a t e d i n F i g u r e ( 5 ) . The saturated sample was held between two assemblages each c o n s i s t i n g of a platinum d i s c backed by a saturated c o t t o n wool compress, a copper e l e c t r o d e and a l u c i t e p l a t e . A com-pre s s i v e f o r c e of approximately 10 pounds was a p p l i e d to the whole and the r e s i s t a n c e of the sample was meaaured on the AC bridge. A s i m i l a r apparatus to that used by McMullin and M u c c i n i ^ 2 ^ was constructed o r i g i n a l l y . T h e i r method was based on the p r i n c i p l e of using a s i n g l e piece of apparatus to both evacuate and measure the r e s i s t a n c e of a sample. This method was found u n s a t i s f a c t o r y due to poor vacuum s e a l i n g p r o p e r t i e s and poor e l e c t r i c a l contact. O U T L E T S A M P L E 6 0 0 C H RUB-BER S L E E V E INLET N L E T O U T L E T Figure 3. D i f f u s i o n C e l l (diagrammatic) 33..b:. TO VACUUM PUMP SAMPLE GOOCH RUBBER SLEEVE 3 WAY COCK O.I N KCI -SOL'N 0 ^ n u VACUUM GAUGE Figure 4. Evacuation Apparatus - E l e c t r i c a l R e s i s t i v i t y Ratio: 3 3 . cV COMPRESSIVE FORCE ( 1 0 LBS APPROX ) SAMPLE GOOCH RUBBER SLEEVE PLATINUM GAUZE SATURATED COTTON WOOL COPPER LUCI T E TO A/C BRIDGE Figure 5» Apparatus - E l e c t r i c a l R e s i s t i v i t y R atio 34 EXPERIMENTAL PROCEDURES A. Thermal C o n d u c t i v i t y C e l l C a l i b r a t i o n An experimental c a l i b r a t i o n had to be made before the thermal c o n d u c t i v i t y c e l l could be used. This was done by making up gas mixtures of known composition and passing them through the c e l l . The method employed was to pass the two streams at constant known flow r a t e s i n t o a tee branch mixer and thence to the thermal c o n d u c t i v i t y c e l l f o r a n a l y s i s . The composition of the mixture was c a l c u l a t e d from the flowmeter readings. The output m i l l i v o l t s were measured on the p o t e n t i -meter a f t e r a constant reading had been obtained on the m i l l i v o l t r ecorder. The filament current was set at 120 m i l l i a m p s . In t h i s way a s e r i e s of p o i n t s , mole f r a c t i o n against m i l l i v o l t s was obtained." To measure the hydrogen flow r a t e more a c c u r a t e l y , a r i s i n g soap bubble flowmeter was used. This c o n s i s t e d of a v e r t i c a l b u r e t t e tube through which the gas flow was passed. The flow r a t e was d i r e c t l y obtained by t i m i n g the rat e of r i s e of soap f i l m s up the b u r e t t e . The c a l i b r a t i o n r e s u l t s are shown i n the appendix, Figures (13) and (14). B. Measurement of E f f e c t i v e D i f f u s i o n C o e f f i c i e n t of Porous  S o l i d s To s t a r t a run, the sample was f i r s t mounted i n p o s i t i o n i n the d i f f u s i o n c e l l . The gas r e g u l a t i n g valves were then set to give steady flowmeter readings at the r e q u i r e d flow r a t e s . 35 The thermal c o n d u c t i v i t y c e l l was switched on, p r e v i o u s l y having been zeroed. The d r a f t gage was then set to zero d i f f e r e n t i a l pressure by a d j u s t i n g the o u t l e t hydrogen stream cock. When cond i t i o n s were steady, a constant reading was obtained on the recorder c h a r t . The output m i l l i v o l t a g e of thermal c o n d u c t i v i t y c e l l was then a c c u r a t e l y measured on the potentimeter connected i n p a r a l l e l w i t h the recorder. The f o l l o w i n g data was recorded f o r each run:-Run Number Date of Run Type of sample, l e n g t h , other c h a r a c t e r i s t i c s Nitrogen flow r a t e , flowmeter reading, flowmeter number Hydrogen " 11 » " » " Ambient temperature Reading on d r a f t gage. (Usually zero) Output m i l l i v o l t s - Thermal c o n d u c t i v i t y c e l l . From t h i s data i t was p o s s i b l e to c a l c u l a t e 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 , De, f o r the sample t e s t e d . A sample c a l c u l a t i o n i s given i n Appendix p. (710. Se v e r a l runs, at d i f f e r e n t gas flow r a t e s , from 300 -1000 cc/minute f o r both gases were performed on each sample w i t h the average r e s u l t being taken f o r c a l c u l a t i o n purposes-C. Measurement of E l e c t r i c a l R e s i s t i v i t y R a t i o This r a t i o was determined f o r a l l the samples whose d i f f u s i o n r a t i o s were obtained. The technique used was to evacuate the sample which was enclosed i n a Gooch rubber sleeve, to a pressure of 1 mm f o r one or two hours. E l e c t r o l y t e s o l u t i o n was then admitted to the bottom of the sample and allowed to saturate i t . A f t e r s a t u r a t i o n , the sample was withdrawn from the evacuation apparatus and put between the electrodes as shown i n Fig u r e ( 5 ) . The r e s i s t a n c e , R, of the sample was then measured on the A.C. bridge. A d i p - c e l l measured the s p e c i f i c r e s i s t i v i t y , r 0 , of s o l u t i o n that was used to saturate the sample. Hence the r a t i o R/R0 could be c a l c u l a t e d . A f t e r the r e s i s t a n c e of the sample had been determined, the sample was placed i n a beaker of d i s t i l l e d water f o r 24 hours i n order to le a c h out the s o l u t i o n w i t h i n i t . Determinations were a l s o c a r r i e d out using d i f f e r e n t lengths of sample to i n v e s t i g a t e the magnitude of end e f f e c t s . D. Measurement of P o r o s i t y The v o i d volume of the sample was found from the weights of the sample before and a f t e r s a t u r a t i o n of the s o l i d w i t h d i s t i l l e d water f o l l o w i n g evacuation. The v o i d volume was taken as equal to the d i f f e r e n c e i n the weighings. C a l i p e r measurements gave the bulk volume of the sample, the p o r o s i t y was then the r a t i o of the vo i d volume to the bulk volume. E. Measurement of Pore S i z e D i s t r i b u t i o n by Mercury P e n e t r a t i o n The pore s i z e d i s t r i b u t i o n was determined by mercury p e n e t r a t i o n f o r c e r t a i n of the samples by the method of 37 (4) R i t t e r and Drake and were done at the I m p e r i a l O i l Research l a b o r a t o r i e s at Calgary, A l t a . B a s i c a l l y the method c o n s i s t e d of p l a c i n g the sample i n a pressure v e s s e l of known volume which i s evacuated and then flooded w i t h a non-wetting l i q u i d , mercury. As the pressure i s increased, i t i s p o s s i b l e to record f o r each pressure increment a corresponding amount of mercury absorbed i n t o the sample. In t h i s way, knowing the pore volume, i t i s p o s s i b l e to o b t a i n a p l o t of pressure against pore volume f i l l e d at each pressure. By a pressure balance on an i n d i v i d u a l pore, the f o l l o w i n g equation i s obtained. U . 0 ) where P c = c a p i l l a r y pressure to i n j e c t a non-w e t t i n g l i q u i d ^ - i n t e r f a c i a l surface t e n s i o n R c sr c a p i l l a r y radius - angle of contact between the mercury and the s o l i d . T his equation has a d i r e c t r e l a t i o n s h i p between pressure and pore r a d i u s . Hence pressures ean be converted to equivalent pore r a d i i and p l o t t e d against the percentage pore volume f i l l e d . Such a p l o t i s shown i n F i g u r e (6) f o r the Selas 03-2 sample, which was the one used i n the measurement of binary d i f f u s i o n c o e f f i c i e n t s at elevated temperatures. 38 RESULTS The ten porous s o l i d s p r e v i o u s l y described were c h a r a c t e r i z e d by p o r o s i t y , p e r m e a b i l i t y , e l e c t r i c a l r e s i s t i v i t y r a t i o and d i f f u s i o n r a t i o measurements where p o s s i b l e . Of the above mentioned t e s t s the f i r s t two were done mainly by other w o r k e r s ^ 3 ^ and the l a t t e r two were determined i n t h i s work. In a d d i t i o n , some s p e c i f i c t e s t s were made f o r average pore diameter (by a bubble point method, and mercury p e n e t r a t i o n method) and f o r pore s i z e d i s t r i b u t i o n (by mercury p e n e t r a t i o n o n l y ) . From the p e r m e a b i l i t y , "bubble point and mercury pene-t r a t i o n t e s t s , other q u a n t i t i e s such as the s p e c i f i c surface area and the average pore diameter were c a l c u l a t e d . Both the measured and the c a l c u l a t e d r e s u l t s are ta b u l a t e d i n Table I . The mercury p e n e t r a t i o n method was p a r t i c u l a r l y valuable i n g i v i n g a c l e a r idea of the s i z e and u n i f o r m i t y of the pores of a porous s o l i d , and the nature of the pore s i z e d i s t r i b u t i o n . A t y p i c a l pore s i z e d i s t r i b u t i o n p l o t f o r the Se.las 03-2 sample i s shown i n Figu r e ( 6 ) . This p l o t shows the uniform s t r u c t u r e of t h i s p a r t i c u l a r s o l i d very c l e a r l y . F i g u r e (9) shows a p l o t of the d i f f u s i o n r a t i o , . Do/De, against the e l e c t r i c a l r e s i s t i v i t y r a t i o , R/Ro, f o r seven of the s o l i d s t e s t e d . The f u l l l i n e denotes equivalence of the r a t i o s . The r e s u l t s f o r the limestone samples have been omitted from t h i s p l o t as i t was f e l t that the values f o r the e l e c t r i c a l r e s i s t i v i t y r a t i o were i n serious e r r o r . No e l e c t r i c a l r e s i s t i v i t y r e s u l t s 39 were obtained f o r the carbon sample as the apparatus could not be used to t e s t conducting porous s o l i d s . The r e c i p r o c a l s of these r a t i o s are a l s o p l o t t e d i n d i v i d u a l l y against the p o r o s i t y , 0. F i g u r e (fO shows such a p l o t of the r e c i p r o c a l d i f f u s i o n r a t i o and the poro.sity, w h i l e F i g u r e (8) shows that of the r e c i p r o c a l e l e c t r i c a l r e s i s t i v i t y (29) r a t i o and the porosity.. The t h e o r e t i c a l l i n e proposed by Penman l 1 > = o. 6 e i s shown on these p l o t s as a f u l l l i n e . Sample c a l c u l a t i o n s are also given i n the Appendix to show the method of o b t a i n i n g p o r o s i t i e s , e l e c t r i c a l r e s i s t i v i t y r a t i o s and d i f f u s i o n r a t i o s from the experimental data. TABLE I Samples, Description and Results Sample Description Porosity, Permeability Average Specific Effective Diffusion E l e c t r i c a l © cgs units x Pore Size Surface Diffusion Ratio Resistivity 108 microns Area Coefficient Do/De Ratio, a, m2/cm3 De R/Ro cm2/sec - -Selas 01 Microporous synthetic ceramic 0.590 2.424 4.50 b 0.575 b 1.100 d 0.248 3.08 2.61 Selas 015;. F i l t e r s 0.659 0.717 2.33 b 1.452 b 0.239 3.19 2.90 Selas 03-1 tt 0.345 0.124 1.31 b 1.336 b 0.109 7.00 6.15 Selas 03-2 11 0.286 0.159 1.52 b 1.168 b O.O67 11.44 8.55 Limestone 36 Natural limestone 0.251 0.0081 0.466 b 1.313 b 0.0213 35.90 11.50 Limestone 63 (Alberta) 0.204 0.0025 0.239 b 2.820 b 0.0114 66.90 8.15 Sandstone SI Natural Sand-stone ((Pembina, Alberta 0.123 0.0039 0.398 b 0.966 b 0.0532 13.35 11.60 Alundum Al Synthetic ceramic aeration solid (Norton Co) extra fine 0.403 15.30 14.50 c 0.0289c 0.127 6 .02 5.65 Filtr o s F . l . n (Filtros Corp3 extra fine p 0.401 5.53 11.60 c 0.0340 c 0.149 5.12 5.10 Carbon CI Porous Graphite Grade 60 (National Carbon Co.) 0.159 0.0257 0.0588 c 0.0128 59.60 a. Results from References (39) and (40) b. By Mercury penetration c» By Kozeny Equation. d. By B-E-T Method o 0 1 2 3 4 5 PORE RADIUS (MICRONS) Fl&ure 6. Pore S i z e D i s t r i b u t i o n Chart 41 DISCUSSION OF RESULTS - SECTION A The r e s u l t s obtained f o r the p o r o s i t y , the e l e c t r i c a l r e s i s t i v i t y r a t i o , and the d i f f u s i o n r a t i o were found to be c o n s i s t e n t and reproducible to w i t h i n 1-2$. The e l e c t r i c a l r e s i s t i v i t y r a t i o s and the d i f f u s i o n r a t i o s were a l s o found to be independent of sample l e n g t h . The values of these r a t i o s , although not d i r e c t l y comparable to the values of any previous workers, were of the same order of magnitude as those found f o r somewhat s i m i l a r s o l i d s . The average pore s i z e s obtained from the mercury p e n e t r a t i o n t e s t s were considered more accurate than those c a l c u l a t e d from p e r m e a b i l i t y t e s t s even though the P o i s e u i l l e equation f o r viscous gas flow was c l o s e l y obeyed i n a l l of the s o l i d s t e s t e d . T£e- cori??lati"<3m p l o t , F i g u r e (9)> of the e l e c t r i c a l r e s i s t i v i t y r a t i o and the d i f f u s i o n r a t i o show d e v i a t i o n s from exact equivalence of up to 25$ f o r some samples. The limestone samples show very l a r g e d e v i a t i o n s that are out of the range of t h i s p l o t . However, i n t h i s l a t t e r case, t h i s i s f e l t to be experimental e r r o r due to the s o l u b i l i t y and hence c o n d u c t i v i t y of these samples. I f the analogy between the d i f f u s i o n r a t i o and the e l e c t r i c a l r e s i s t i v i t y r a t i o i s to be t r u l y u s e f u l , i t should hold f o r a l l i n e r t , non-conducting porous s o l i d s and should be independent of the average pore s i z e , the pore s i z e d i s t r i b u t i o n and the pore s t r u c t u r e . An explanation f o r those cases i n which 42 lar<je d e v i a t i o n s occur i s hard to o f f e r . Values of the pore s i z e , the (pore s i z e ) and the percentage d e v i a t i o n of the two r a t i o s from equivalence are presented i n Table I I . These data i n d i c a t e c l e a r l y that the greatest d e v i a t i o n s occurred i n those samples having the smallest average pore s i z e s . With the data at hand, i t i s not p o s s i b l e to say whether the l a c k of c o r r e l a t i o n i n these samples i s due e n t i r e l y to some form of experimental e r r o r oeeunrling i n pore s i z e s below a c e r t a i n value or whether i t i s caused by some unsuspected n a t u r a l e f f e c t . (41) F u r t h e r evidence, from the work of Selby, shows that r e s u l t s f o r the e l e c t r i c a l r e s i s t i v i t y r a t i o , obtained on s e v e r a l of the same samples under more c a r e f u l experimental c o n d i t i o n s , have n e g l i g i b l e d i f f e r e n c e from those found i n t h i s study. This wauld suggest t h a t i t i s u n l i k e l y that experimental e r r o r i s a cause of the d e v i a t i o n s observed. F i g u r e s (7) and (8) show the i n d i v i d u a l r e c i p r o c a l e l e c t r i c a l r e s i s t i v i t y and d i f f u s i o n r a t i o s p l o t t e d against the p o r o s i t y . No d e f i n i t e r e l a t i o n s h i p i s observed f o r the s o l i d s used i n t h i s study; the r e s u l t s , on the average, f a l l i n g below (29) the l i n e p r e d i c t e d by Penman f o r unconsolidated s o l i d s . Again, the s o l i d s having the c l o s e s t agreement w i t h the Penman l i n e are those w i t h the l a r g e s t average pore s i z e s . I t may be p o s s i b l e t h a t as the average pore s i z e decreases, m a t e r i a l s of the ki n d used i n our i n v e s t i g a t i o n s ( f i n e p a r t i c l e s fused or cemented together to form the porous s o l i d ) change i n character from mainly unconsolidated to more 43 TABLE II D e v i a t i o n s from the Analogy, and Pore S i z e s of the Samples Pore S i z e , (Pore S i z e ) * D e v i a t i o n $ D e v i a t i o n Sample Microns Selas 01 4 .50 20.25 0.47 16.5 Selas 015 2.33 5.42 0.29 9.5 Selas 03-1 1.31 1.72 0.85 12.9 Selas 03-2 1.52 2 .30 2.89 25*. 2 Limestone 36 0.466 0.218 24.40 — Limestone 63 0.239 0.057 58.80 — Sandstone 51 O.398 0.158 2.74 21.2 Alumdum A l 14.50 210.0 0.37 6.3 F i l t r o s F l 11.60 134.2 0.02 0.4 44' h i g h l y c o n s o l i d a t e d . This may be accompanied by a change i n the behavior of the t r a n s p o r t p r o p e r t i e s w i t h i n the pores of the s o l i d . 4# EXPERIMENTAL ERRORS A. E l e c t r i c a l R e s i s t i v i t y R a t i o I t i s f e l t t hat the maximum e r r o r s made i n determining the p o r o s i t y and the e l e c t r i c a l r e s i s t i v i t y r a t i o are l e s s than £ 1$. The p o r o s i t y can be determined a c c u r a t e l y by the method used here, and the e r r o r s i n the e l e c t r i c a l r e s i s t i v i t y r a t i o , p r i m a r i l y the value of the c o n d u c t i v i t y of the e l e c t r o l y t e s o l u t i o n and the value of the sample r e s i s t a n c e , are f e l t to be s m a l l . The e l e c t r o l y t e s o l u t i o n was made up by a c c u r a t e l y weighing out the r e q u i r e d amount of pure KC1 (A.C.S. s p e c i f i c a -t i o n s ) and d i s s o l v i n g i t i n a 2 l i t r e v o l u metric f l a s k . The s p e c i f i c r e s i s t i v i t y of t h i s s o l u t i o n was measured independently by a dip-type c o n d u c t i v i t y c e l l and was found to check w i t h published v a l u e s . Subsequent r e s u l t s . f o r the e l e c t r i c a l r e s i s t i v i t y r a t i o , obtained by R. S e l b y ^ 4 ^ on some of the samples, showed n e g l i g i b l e d i f f e r e n c e when compared w i t h the r e s u l t s presented here. This l a t e r work was performed under more c a r e f u l experimental c o n d i t i o n s , the samples were evacuated to a pressure of 0 .1 mm pf mercury as compared to 1 mm of mercury i n t h i s work. B. Measurement of Gas Composition The c a l i b r a t i o n p l o t f o r the hydrogen and n i t r o g e n gas compositions (expressed as mole f r a c t i o n s ) against the thermal 4<? c o n d u c t i v i t y c e l l m i l l i v o l t readings were used i n the c a l c u -l a t i o n of the d i f f u s i o n r a t e s . The c a l i b r a t i o n p l o t f o r the hydrogen gas composition i s estimated to give the composition of the stream w i t h a maximum e r r o r of ± 2% and an average e r r o r of 1% and the c a l i b r a t i o n p l o t f o r the n i t r o g e n gas composition i s estimated to give the composition of t h i s stream w i t h a maximum e r r o r of £ ^% and an average e r r o r of 2%. The main v a r i a b l e s were the accurate c o n t r o l and measurement of the q u a n t i t i e s of the two gas streams that made up the mixture of known composition f o r c a l i b r a t i o n purposes. The smaller of the two streams forming the mixture could be held accurate to w i t h i n 2% by manipulation of the c y l i n d e r reducing v a l v e . This stream was measured d i r e c t l y by the r i s i n g s o a p - f i l m method. The l a r g e r stream could be kept f l o w i n g at a constant and steady r a t e , the e r r o r i n t h i s case was the measurement of the flow r a t e using the Matheson gas rotameters. The b a l l f l o a t i n these rotameters could be read w i t h an accuracy of 1%. ( i . e . , 5ml/min i n a t o t a l flow of 500 ml/min at room temperature and one atmosphere pressure.) The e l e c t r i c a l measuring c i r c u i t and the instruments used w i t h i t were f e l t t o have a much smaller e r r o r . The l a r g e r e r r o r s i n the case of the n i t r o g e n a n a l y s i s were due to the r e l a t i v e i n s e n s i t i v i t y of the thermal c o n d u c t i v i t y c e l l to small amounts of n i t r o g e n i n a hydrogen stream as com-pared to the s e n s i t i v i t y w i t h hydrogen i n !&. nifrtrogen stream. 47 C. Measurement of D i f f u s i o n Rates In the a c t u a l c a l c u l a t i o n of the experimental d i f f u s i o n r a t i o s , a s i g n i f i c a n t source of e r r o r was the accurate determination of the flow rates of the two streams of gas, hydrogen and n i t r o g e n , one on e i t h e r s ide of the porous sample, i n order to c a l c u l a t e the volume of gas d i f f u s e d . As shown above, the f l o w - r a t e measurement e r r o r f o r large flow r a t e s (400-1000 ml/min) was estimated to be of the order of 1%. The other main e r r o r was introduced by the use of the composition c a l i b r a t i o n chart to determine the mole f r a c t i o n of one of the gases i n the other gas stream. In the case of hydrogen, the maximum e r r o r was of the order of £ 2% as pointed out p r e v i o u s l y . In the case of n i t r o g e n the maximum e r r o r was l a r g e r , £ 4%. As the d i f f u s i o n r a t i o was always c a l c u l a t e d on the ba s i s of hydrogen as the d i f f u s i n g gas, the magnitude of the maximum e r r o r i n v o l v e d was of the order of 3$ ; the average e r r o r b f i l n d l M d u a l . d e t e r m i n a t i o n s of d i f f u s i o n r a t e being about 1 .5$. D. E f f e c t of Forced Flow C l e a r l y i t i s necessary t o keep the d i f f e r e n t i a l pressure drop at the zero mark (equal pressure on both sides of the sample) during the course of each experimental run. I n the present work, t h i s d i f f e r e n t i a l could be maintained at l e s s . than 0 .005 i n c h of water pressure. The r e s u l t s given i n 4& TABLE III E f f e c t of Forced Flow on the Volume of Hydrogen Passed Through a Sample Sample: Selas 03-2, 1" 294°K, 1 atmosphere. Pressure Drop In inches of water Volume Hydrogen passed, cm3/sec. D e v i a t i o n ( D % D e v i a t i o n + 0 . 3 0.299 0.044 17 .3 + 0.2 0.278 0 .023 9.0 t 0.1 0 .266 0.011 4 . 3 0.0 0 .255 — - 0.1 0.245 0.010 3.9 - 0.2 0.229 0 .026 10.2 - 0 . 3 0.220 0 .035 13.7 (N.B. P o s i t i v e Sign - Hydrogen Side Pressure.) ( S i n g l e Values, not average of s e v e r a l runs) (1. A d d i t i o n a l s a s flow, cm 5/sec, due t o d i f f e r e n t i a l pressure drop) 49 Table I I I i n d i c a t e that at t h i s l e v e l , the e r r o r due to absolute pressure d i f f e r e n c e s should be n e g l i g i b l e . To determine the e f f e c t of a d i f f e r e n t i a l pressure across the s o l i d sample on the transport through the porous sample, a d e f i n i t e e x t e r n a l pressure was a p p l i e d i n t u r n t o both the hydrogen side of the porous sample and to the n i t r o g e n s i d e . These pressures were set by t h r o t t l i n g the o u t l e t stop-cocks of the stream i n q u e s t i o n so as to i n d i c a t e a constant pressure on the d i f f e r e n t i a l draught gauge. The r e s u l t s , shown i n Table I I I , i n d i c a t e a d e f i n i t e increase or decrease i n the volume of gas d i f f u s e d f o r as l i t t l e as a t e n t h of an i n c h of waterppressure. This e f f e c t of the pressure i s due t o the presence of P o i s e u i l l e flow i n the d i r e c t i o n of the a p p l i e d pressure. CONCLUSIONS AND RECOMMENDATIONS - SECTION A The analogy between the e l e c t r i c a l r e s i s t i v i t y r a t i o and the d i f f u s i o n r a t i o holds f o r the s o l i d s t e s t e d i n t h i s study to w i t h i n 20%, The l a r g e s t d e v i a t i o n s are observed i n those s o l i d s having the sma l l e s t mean pore s i z e s . I t i s not p o s s i b l e , at present, to e x p l a i n why these l a r g e d e v i a t i o n s e x i s t i n the smaller pore s i z e s . F u rther work on s o l i d s w i t h both l a r g e r and smaller mean pore diameters than those used i n t h i s study (0.2 - 11.6 micron diameter) would be worthwhile. In p a r t i c u l a r , the smaller s i z e s are of i n t e r e s t i n c a t a l y t i c work. The hydrogen-nitrogen system appears to be q u i t e s a t i s f a c t o r y as a t e s t system f o r the measurement of the e f f e c t i v e d i f f u s i o n c o e f f i c i e n t of porous s o l i d s . 5\ SECTION B THE TEMPERATURE DEPENDENCE OF THE DIFFUSION COEFFICIENT FOR THE HYDROGEN-NITROGEN SYSTEM APPARATUS The apparatus i s shown s c h e m a t i c a l l y i n F i g u r e ( 1 ) . The d i f f u s i o n r a t i o apparatus used e a r l i e r was modified by p l a c i n g the d i f f u s i o n c e l l and i n l e t preheater c o i l s i n s i d e a constant temperature oven. A new method of sample mounting was a l s o necessary as the Gooch rubber sleeve used p r e v i o u s l y f a i l s above 100°C. The oven used was a F i s h e r Essftemp oven w i t h a 1000 watt heater c a p a c i t y c o n t r o l l e d by a Cenco b i m e t a l l i c thermo-r e g u l a t o r . A f a n mounted i n the upper part of the oven c i r c u l a t e d the a i r w i t h i n i t . This arrangement gave a ± 1°C temperature c o n t r o l up to t h e highest temperature reached. (300°C). A s p e c i a l door c o n s i s t i n g of two sheets of 1/4 i n c h t h i c k b o i l e r p l a t e b o l t e d together 1 i n c h apart w i t h i n s u l a t i o n between was constructed f o r t he oven. This door was designed so that the d i f f u s i o n c e l l was mounted i n p o s i t i o n on the i n s i d e face of the door and a l l accesa l i n e s f o r the i n l e t and o u t l e t gas streams and the pressure connections were made through the door. This c o n s t r u c t i o n allowed the e n t i r e assembly to be e a s i l y removed from the oven. The i n l e t c o i l s were two 50 foot lengths of 1/8 i n c h copper tubing wound on a frame p r o j e c t i n g from the i n s i d e face of the door. They were j o i n e d to the d i f f u s i o n c e l l by means 52 of machined T e f l o n s l e e v e s , 1 i n c h long. A f u r t h e r 50 f o o t l e n g t h of 1/4 i n c h copper tubing was added t o the o u t l e t n i t r o g e n stream to c o o l the gas to room temperature before i t entered the thermal c o n d u c t i v i t y c e l l . A 2 inches long by 1 i n c h diameter nominal s i z e c y l i n d e r of the 03-2 Selas sample was s e l e c t e d f o r the temperature work. I t had a narrow range of pore s i z e s (from 0 . 5 microns to 2 .0 microns diameter) as shown i n F i g u r e ( 6 ) . Other pro-p e r t i e s of t h i s s o l i d are given i n Table I . I t a l s o gave con-v e n i e n t l y measurable gas concentrations at the maximum temperature. (300°C). Fi g u r e (10) shows the method of sample mounting f o r the temperature dependence work. A l a y e r of A r a l d i t e epoxy r e s i n AN 130 was f i r s t bonded onto the sample. This l a y e r was then joined to the glass ends of the d i f f u s i o n c e l l by a "butt weld" of the same r e s i n . The A r a l d i t e s l e e v e , which was a i r -t i g h t , allowed the work to be c a f r i e d out t o a maximum temper-ature of 300°C at which p o i n t the r e s i n s t a r t e d to decompose. 52. ex. EXPERIMENTAL PROCEDURES Temperature Dependence of the Binary D i f f u s i o n C o e f f i c i e n t The runs at hig h temperatures were done i n much the same manner as the runs at room temperature. Before the gases were allowed to flow the oven was maintained at a f i x e d temperature f o r at l e a s t 15 minutes. Due to the l a r g e pressure drop caused by the 50 f e e t of i n l e t preheater t u b i n g , (approx-imately 2-4 inches mercury pressure) the flowmeters were c a l i b r a t e d at a pressure of 6 inches of mercury. This pressure was maintained by r e g u l a t i n g a f i n e needle valve s i t u a t e d down-stream from the flowmeters. Some d i f f i c u l t y was experienced i n the f i e l d of gas temperature measurement. Due to r a d i a t i o n e r r o r and the very low gas flow r a t e s , normal methods of gas temperature measure-ment such aB thermocouples, s h i e l d e d thermocouples, e t c . , proved i n e f f e c t i v e . In the end, the gas temperature was obtained by measuring the pressure drop across a l e n g t h of c a p i l l a r y tubing through which the gas flowed. A 1 i n c h l e n g t h of quartz c a p i l l a r y of approximately 1 mm bore was chosen as i t had a low c o e f f i c i e n t of thermal expansion and thus would be l e s s subject to dimensional changes as temperatures v a r i e d . For viscous flow through a c a p i l l a r y , the Ha g e n - P o i s e u i l l e equation a p p l i e s , and provided the r a t e of flow i s kept constant, the pressure drop i s d i r e c t l y p r o p o r t i o n a l to the kinematic v i s c o s i t y , / V f of the gas. 54 The v a r i a t i o n of the kinematic v i s c o s i t y w i t h the temperature f o r gases i s obtained from standard references. Hence i t i s p o s s i b l e to gauge the temperature of a gas ffom a knowledge of i t s pressure drop through a c a p i l l a r y . Values of the necessary p r o p e r t i e s f o r n i t r o g e n and hydrogen are p l o t t e d i n Appendix, F i g u r e 16. This method of temperature measurement was found to be accurate enough f o r our purposes, t h a t i s to f 3°C The gas temperature was always found to be w i t h i n ¥ 3°c of the oven temperature, as shown i n F i g u r e 11. Therefore, i t can be con-cluded that the le n g t h of heat t r a n s f e r t ubing used (50* of 1/8" copper tubing) was adequate to heat the gases t o oven temperature. The oven temperature i t s e l f was known w i t h an accuracy of + 1°C. The maximum temperature at which the runs were c a r r i e d out was 300°C. At t h i s temperature, the A r a l d i t e r e s i n , used to cement the sample and the d i f f u s i o n c e l l t ogether, showed the f i r s t signs of decomposition. C a l i b r a t i o n runs were al s o c a r r i e d , out at room temper-ature since the geometric (L/A) f a c t o r f o r the cemented sample might be expected to be d i f f e r e n t to that of the sample when (5) sheathed i n a Gooch rubber sleeve. The Sherwood a n d - P i g f o r d w / equation, already d e r i v e d (p. 12) can be w r i t t e n : 0 R'j.L-k he c ¥ ) he ( — c A 0) 55 Assuming the p e r f e c t gas laws apply N H , = ™ ± Z R 'T (2) Equation (1) can now be r e w r i t t e n : ^ AS ~C —T~ - " (o) or D A 6 « * . - . . . (4) where o< i s a c a l i b r a t i o n f a c t o r i n c o r p o r a t i n g C, L, and A. The c a l i b r a t i o n f a c t o r , o< , i s experimentally determined f o r a given porous sample at room temperature. I t can then be used to determine, D^B? at a higher temperature, provided L, A remain constant. An example i l l u s t r a t i n g t h i s method of c a l c u l a t i n g the b i n a r y d i f f u s i o n c o e f f i c i e n t at higher temperatures i s given i n the Sample C a l c u l a t i o n s p.75. Measurements were a l s o c a r r i e d out at room temperature to determine the comparative rates of d i f f u s i o n of hydrogen and n i t r o g e n . A Selas 0 3 - 2 , 1 i n c h diameter, 1 i n c h long sample was used f o r the t e s t . Both o u t l e t gas streams were analyzed by means of thermal c o n d u c t i v i t y c e l l s . Two c e l l s were r e q u i r e d , one c a l i b r a t e d f o r hydrogen i n n i t r o g e n and the other c a l i b r a t e d f o r n i t r o g e n i n hydrogen. 52.af; 0 200 400 600 OVEN TEMPERATURE T T ° F . Figure 11. P l o t of Oven Temperature and C a l c u l a t e d  Gas Temperature 56 RESULTS - SECTION B A. Measurement of Gas Temperatures F i g u r e (11) shows the v a r i a t i o n of the gas temper-ature w i t h the oven temperature f o r both gases. The gas temperatures were measured by means of the c a p i l l a r y thermometer tha t has p r e v i o u s l y been described. The p l o t shows that the d i f f e r e n c e between the gas temperature and the oven temperature d i d not exceed 3°C, even at the maximum temperature (300°C). The readings obtained are given i n Table IV. B. Temperature Dependence of the D i f f u s i o n C o e f f i c i e n t Values of the b i n a r y d i f f u s i o n c o e f f i c i e n t f o r the hydrogen-nitrogen system ane p l o t t e d l o g a r i t h m i c a l l y against the absolute temperature i n F i g u r e (12) and shown -In T a b l e d . The b i n a r y d i f f u s i o n c o e f f i c i e n t at higher temperatures was obtained from the volume of hydrogen that d i f f u s e d through the sample and a c a l i b r a t i o n f a c t o r t h a t was e s t a b l i s h e d at room temperature. A sample c a l c u l a t i o n showing the procedure i s i n the Appendix P. 7 5 . Using the method of l e a s t squares, the temperature dependence l i n e , f o r the d i f f u s i o n c o e f f i c i e n t i s found to have a slope of 1 .6821, the standard percentage d e v i a t i o n of the p o i n t s from t h i s l i n e being 1.91 percent. A summary of a l l the published d i f f u s i o n c o e f f i c i e n t data f o r the hydrogen-nitrogen system at 1 atmosphere t o t a l 57 pressure and various temperatures i s given i n Table VI. These data are p l o t t e d i n F i g u r e (12); the s o l i d l i n e represents the d i f f u s i o n c o e f f i c i e n t f o r the hydrogen-nitrogen system c a l -c u l a t e d by means of the H i r s c h f e l d e r equation taken to the f i r s t approximation, using f o r c e constants d e r i v e d from v i s c o s i t y data. A sample c a l c u l a t i o n i l l u s t r a t i n g t h i s a p p l i c a t i o n of the H i r s c h f e l d e r equation i s given i n the Appendix p. 7 7 . C. Comparison of D i f f u s i o n Rates f o r Hydrogen and Nitrogen The r e l a t i o n s h i p f o r equal pressure counter_-diffusion (6) of gases across an i n t e r f a c e given by Hoogschagen and mentioned e a r l i e r i n the theory s e c t i o n was t e s t e d . This r e l a t i o n s h i p s t a t e s that the r a t e s of the c o u n t e r - d i f f u s i o n of two gases at constant t o t a l . , pressure through a porous s o l i d are i n v e r s e l y p r o p o r t i o n a l to the square r o o t s of t h e i r molecular weights. The Selas 03-2 sample was s e l e c t e d f o r the t e s t and the r e s u l t s obtained are presented i n Table V I I . The values of a number of i n d i v i d u a l runs are given, together w i t h the average value. I t can be seen that the observed r a t i o of the r a t e s of c o u n t e r - d i f f u s i o n of hydrogen and n i t r o g e n through the Selas 03-2 sample agrees c l o s e l y w i t h t h a t p r e d i c t e d by the Hoogschagen r e l a t i o n s h i p . With the experimental e r r o r p o s s i b l e i n the d e t e r -mination of the volume of n i t r o g e n d i f f u s e d (discussed i n Se c t i o n A) i t i s f a i r to say that the r e l a t i o n s h i p 58 holds w e l l f o r t h i s gas system, at l e a s t at ord i n a r y temperatures and pressures. In the o r i g i n a l work performed by Hoogschagen, di s c r e p a n c i e s of s i m i l a r and greater magnitude were reported i n comparing the r a t e s of c o u n t e r - d i f f u s i o n of other b i n a r y gas systems through porous s o l i d s . TABLE IV GAS TEMPERATURE MEASUREMENTS Oven Temperature T xoven <°F) HYDROGEN Ca l c u l a t e d Temperature c a l c (°F) Kinematic V i s c o s i t y y / q cm 2/sec Oven Temperature T o v e n (°F) NITROGEN C a l c u l a t e d Temperature ^ c a l c (Op) Kinematic V i s c o s i t y r /€ cm 2/sec 70 70 1.065 70 70 0.162 120 117 1.20 160 159 0.195 190 185 1.48 250 245 0.242 256 254 1.76 260 257 0.256 306 304 1.94 342 335 0.310 360 350 2.13 390 391 0.346 392 406 2.32 430 4 3 2 0.377 420 422 2.40 470 461 0.397 460 464 2.68 526 527 0.445 490 490 2.82 510 515 2.96 60 TABLE V Diffusion Coefficient Results, Hydrogen-Nitrogen, 1 atm, 20-300°C.  Absolute Temperature T (OK) Diffusion Coefficient, DAB cm2/sec Diffusion Coefficient, cm2/sec. jCHirschfelder Eqn.)  Percentage Deviation 294 0.763 0.763 296.5 O .78I 0.776 0.64 322 0.903 0.891 + 1.26 355 1.051 1.050 + 0.03 373 1.161 1.142 1.66 398 I .289 1.273 + 1.28 411 1.370 1.343 1.97 422 1.384 1.404 — 1.43 450 1.541 1.542 - 0.06 455 1.547 1.594 - 2.97 483 1.751 1.763 — 0.66 506 I .883 1.906 - 1.21 508 1.909 1.918 - 0.47 536 2.120 2.097 1.10 539 2.171 2 . H 9 2.42 573 2.417 2.346 + 3.03 61 TABLE VI D i f f u s i o n C o e f f i c i e n t R e s u l t s of Other I n v e s t i g a t o r s , Hydrogen-Nitrogen, 1 atm. Absolute Temperature T (°K)  273 293 287 2^2 273 290 298 308 195 234 282 331 355 398 D i f f u s i o n C o e f f i c i e n t cm 2/sec 0.697 O.76O 0.743 0.620 0.697 O.78O 0.815 0 .858 0.355 0.507 O.676 0 . 8 7 1 1.000 1.175 I n v e s t i g a t o r Jackmann (12) Waldmann (14) Boardman and Wild (15) Schafer, Corte and Moesta ( 3 D 11 n Schafer and Moesta (33) TABLE V I I Comparison of D i f f u s i o n Rates f o r Hydrogen and Nitrogen, 1 atm. Sample; Selas 0.3-2, 1 i n c h T (°K) Volume Hydrogen d i f f u s e d cm3/sec Volume Nitrogen d i f f u s e d cm3/sec R a t i o T h e o r e t i c a l R a t i o 294 0.258 0.075 3.44 3.742 294 0.250 0.062 4 . 0 3 3.742 294 0.255 0.068 ..3*75 , 3.742 Average 3.77 (N.B. S i n g l e Values) 62.a, 40 ,20 0.0 o O o CD O V 1 8 0 1.60 • < • • -/ •-SCHA ©"SCHA ©"JACK FER,M OE FER.C0R1 MANN STA . "E, MOESTA f 3-WALf O- 0OA O-THIS )MANN RDMAN,V\ WORK. ILD. • e 12. Cc LINE-H mparison: I IRSCHFEL E T AL. lOt Of Hl2 DER •schfelder • & uation. ot her workei •a1 riata 1 ar d t h i s wo2 k. (logarj thmic) 2.20 2.40 _ O u r 2 . 6 0 LOG to 1 K 2.80 63 DISCUSSION OF RESULTS A. Measurement of Gas Temperatures The c a l c u l a t e d gas temperature, f o r both of the gases, i s p l o t t e d against the measured oven temperature i n Fig u r e (11). The r e s u l t s can be seen to f a l l both above and below the 45° l i n e , w i t h an average divergence of + 5° (ST. These divergences would appear to be due t o experimental e r r o r i n gauging the pressure drop, and i n maintaining a constant flow r a t e through the c a p i l l a r y . The o i l manometer used f o r the pressure drop measurements had a reading e r r o r of approximately + 1»5$» A l a r g e r e r r o r i n t h i s method of temperature measurement i s due to the d i f f i c u l t y i n maintaining a flow r a t e through the preheater c o i l s at a value which v a r i e s by l e s s than yfo. This method w i l l give the absolute temperature w i t h a probable accuracy of £ 3^ C». In view of the r e s u l t s obtained, i t was concluded that the gas temperature was w i t h i n 1 or 2° C of the oven temperature reached. Therefore, the oven temperature was taken to be the temperature at which the d i f f u s i o n took p l a c e . B. Temperature Dependence of the Bulk D i f f u s i o n C o e f f i c i e n t The trtB d i f f u s i o n c o e f f i c i e n t was c a l c u l a t e d from the runs at h i g h temperature using the c a l i b r a t i o n f a c t o r determined at room temperature as described e a r l i e r . 64 The experimental values of the d i f f u s i o n c o e f f i c i e n t obtained i n t h i s way f o r the hydrogen-nitrogen system over a temperature range from 20°C to 300°C are l i s t e d i n Table V together w i t h the c a l c u l a t e d values from the H i r s c h f e l d e r equation. A l e a s t squares l i n e drawn through these p o i n t s when p l o t t e d l o g a r i t h m i c a l l y would have a slope equal to 1 .6821. The standard d e v i a t i o n of the po i n t s from the l i n e i s 0.026 cm 2/sec which would represent a percentage standard d e v i a t i o n equal to 1.91 percent. Very good agreement i s found between the experimental values obtained and the values p r e d i c t e d from the H i r s c h f e l d e r equation. The maximum percentage d e v i a t i o n i s approximately 3$ which i s small i n view of the p o s s i b l e estimated e r r o r . E r r o r s i n v o l v e d i n the measurement of d i f f u s i o n r a t e s have already been discussed i n d e t a i l . In a d d i t i o n there i s a p o s s i b l e e r r o r i n these r e s u l t s due t o a s m a l l u n c e r t a i n t y i n temperature measurement. This i s estimated to cause a maximum er r o r of no more than ± 2% i n the value of Do c a l c u l a t e d . The estimated maximum e r r o r at the highest temperature reached would amount to approximately 5$ w i t h a l l these f a c t o r s being taken i n t o c o n s i d e r a t i o n . The r e s u l t s given by the H i r s c h f e l d e r equation are tabulated i n Table V I I I , Appendix, and the experimental data obtained by other workers f o r t h i s p a r t i c u l a r system are shown i n Table VI. A l l these values are p l o t t e d l o g a r i t h m i c a l l y i n Fig u r e ( 1 2 ) , the f u l l l i n e being the p l o t of the values from 65 the H i r s c h f e l d e r equation. This l i n e has a slope of 1,6815. I t i s t h e r e f o r e seen that the data obtained i n t h i s work correspond c l o s e l y to the H i r s c h f e l d e r equation to the f i r s t approximation. Comparison w i t h the data of previous workers i s a l s o f a i r l y good. I s o l a t e d values which have been obtained e x p e r i m e n t a l l y by Jaekmann, Waldmann, and Boardman and W i l d also agree w i t h p r e d i c t i o n s from the H i r s c h f e l d e r equation. Other temperature work has already been mentioned i n the l i t e r a t u r e survey. I n t h e i r f i r s t paper, Schafer, Corte and M o e s t a ^ D have i n v e s t i g a t e d the d i f f u s i o n c o e f f i c i e n t at f i v e d i f f e r e n t temperatures from - 80° to 60°C. Their p o i n t s show a temperature dependence of 1.81 f o r the d i f f u s i o n of a 56$ hydrogen-nitrogen mixture. In l a t e r work, Schafer and Moesta^^) i n v e s t i g a t e d the temperature dependence f o r hydrogen-nitfogen mixtures d i f f u s i n g at d i f f e r e n t c o n c e n t r a t i o n s . These p o i n t s were averaged f o r the purpose of comparison i n Figu r e (12) by t a k i n g the mean of t h e i r d i f f u s i o n s c o e f f i c i e n t s at 0, 20, 40, 60, 80, and 100$ mole f r a c t i o n of n i t r o g e n and these average values are p l o t t e d i n Fig u r e (12). C l o s e r exam-i n a t i o n shows these p o i n t s to have lower values than those of the previous work. The temperature dependence, i n the l a t t e r work, shows an exponent of 1.61. Schafer's work t h e r e f o r e has two sets of r e s u l t s which f a l l on e i t h e r s i d e of the H i r s c h f e l d e r l i n e , and of the data obtained i n t h i s study. This i s good evidence that the present data are at 66 l e a s t as accurate as those of Schafer and h i s co-workers. The H i r s c h f e l d e r equation, w i t h f o r c e constants from v i s c o s i t y -data and taken to the f i r s t approximation, p r e d i c t s the d i f f u s i o n c o e f f i c i e n t f o r non-polar s p h e r i c a l gas p a i r s up to 300°C w i t h an accuracy of 1-2$, Q'h). Our data, which shows a temperature dependence slope of 1,6821 compared to the slope of 1.6815 from the H i r s c h f e l d e r l i n e , may t h e r e f o r e give the d i f f u s i o n c o e f f i c i e n t of the hydrogen-nitrogen system to as great a degree of accuracy as the t h e o r e t i c a l equation. This would i n d i c a t e that the c a l i b r a t i o n f a c t o r obtained at room temperature i s s u f f i c i e n t l y accurate at l e a s t to 300°C. 67 CONCLUSIONS AND RECOMMENDATIONS The r e s u l t s from the temperature dependence study of the d i f f u s i o n c o e f f i c i e n t of the hydrogen-nitrogen system appear to be encouraging. I t would t h e r e f o r e be i n t e r e s t i n g to i n v e s t i g a t e other systems of both p o l a r and non-polar gas p a i r s to see whether the H i r s c h f e l d e r equation to the f i r s t approximation i s obeyed e q u a l l y w e l l over the same or greater temperature range. Should a cement w i t h b e t t e r heat r e s i s t i n g p r o p e r t i e s than the A r a l d i t e AN 130 be found, i t could be used to extend the i n v e s t i g a t i o n to temperatures higher than those reached i n t h i s work (300°C). A l t e r n a t i v e l y , i t may be p o s s i b l e t o enclose the porous s o l i d i n g l a s s . The apparatus and method used appear to be s u f f i c i e n t l y r e l i a b l e to give good r e s u l t s . The hydrogen-nitrogen system would be e n t i r e l y s u i t a b l e f o r the c a l i b r a t i o n of a d i f f u s i o n sample, which could then be used on other gas p a i r s w i t h undetermined d i f f u s i o n c o e f f i c i e n t s . 68 BIBLIOGRAPHY 1. Petersen, E.E., A.I.Ch.E. J o u r n a l . , 4, 343, (1958) 2. Wheeler, A., i n "Advances i n C a t a l y s i s and Related Subjects", V o l . 3 , 249-327, Academic Press, New York, 1951. 3 . Carman, P.G., "Flow of Gases through Porous S o l i d s " , Academic Press, New York, 1956. 4. R i t t e r , H.L., and Drake, I.C., Ind. Eng. Chem., Anal Ed., 12, 787, (1945). 5. Sherwood, T.K., and P i g f o r d , R.L. "Absorption and E x t r a c t i o n " , 2nd Ed., McGraw-Hill, New York, 1952. 6. Hoogschagen, J . , Ind. Eng. Chem., 4£ 906 (1955). 7. Maxwell, J.C., " S c i e n t i f i c Papers", V o l . 2, Cambridge U n i v e r s i t y P r e s s , New York, 1890, 8. G i l l i l a n d , E.R., Ind. Eng. Chem., 26, 681 (1934). 9. Sutherland, A., P h i l . Mag., ^ 6 , 507 (1893) . 10. Indrussow, L., Z. Elektrochem., £4, 566 (1950). 11. H i r s c h f e l d e r , J.O., C u r t i s s , C.F., and B i r d , B. "Molecular Theory of Gases and L i q u i d s " , Wiley, N.Y., 1954. 12. Jackmann, I n t e r n a t i o n a l C r i t i c a l Tables, V o l . 5» 1928, p. 6 2 , McGraw-Hill, New York. 13. Loschmidt, I n t e r n a t i o n a l C r i t i c a l Tables, V o l . 5» 1928, p. 62, McGraw-Hill, New York. 14. Waldmann, L., Naturwissenschaften, ^2, 223 (1944). 15. Boardman, L.E., and W i l d , N.E., Proc. Roy. Sec. (London), A162, 511(1937). 16. Winkelmann, A., Ann. Physik., 22, 1 , 152, (1994) 17. Wicke, E., and Kallenbach, R., K o l l o i d , Z., 2 Z , 135 (1941). 18. T h i e l e , E.W., Ind. Eng. Chem., ^ 1 , 916 (1939). 69 B i b l i o g r a p h y (Cont.) 19. Weisz, P.B., Z. Phys. Chem., 11 Band, Heft 1/2, 1, (1957). 20. Maxwell, see F r i c k e , H., and Morse, S., Phy. Rev., 25, 361 (1925). 21. W y l l i e , M., and Gregory, A. Trans AIME., 198, 103, (1953). 22. S l a w i n s k i , A.J. Chem. Phys., 23, 710 (1926). 23. A r c h i e , G., Trans AIME., 146, 54 (1942). 24. C o r n e l l , D., and Katz, D.L. Ind. Eng. Chem. 4£, 2145 (1953). 25. McMullin. R., and M u c c i n i , G. A.I.Ch.E. J o u r n a l ; , 2 393 (1956) . 26. Klinkenberg, L . J . , B u l l Geol Soc. Amer., 6 2 , 559 ( 1 9 5 D . 27. S c h o f i e l d , R.K., and Dakshinamurti, C., Disc Faraday S o c , 3 56 (1948). 28. Buckingham, E., U.S. Dept. A g r i . , Bur. S o i l s , B u l l . 22, (1904). 29. Penman, H.L., J . A g r i c . S c i . , 3 0 , 437, 570, (1940). 30. Van Ba v e l , C.H.M., S o i l S c i . , Zli 91 , (1942). 31. Schafer, K., Corte, H., and Moesta, H.,Z. Elektrochem, 52, 662, ( 1 9 5 D . 32. Schafer, K., and Moesta, H., Z. Electrochem., %8 743 (1954). 33. Schafer, K., and Schumann, K., Z. Elektrochem., 61 247 (1957) . 34. Amdur, I . , and S c h a t z k i , T.F., J . Chem. Phys., 2Z, 1049, (1957). 35* Kilbanova, Pomerantsev, and Frank-Kamenetskii, J. Tech. Phys., (U.S.S.R.) 12, 14, (1942). 36. Walker, R.E., and Westenberg, A.A., J . Chem. Phys., 22, 1139, (1958). B i b l i o g r a p h y (Cont.) 70 37. W i l k e , C.R., and Lee, C.Y., Ind. Eng. Chem., 4£, 1253, (1955) . 3 8 . Strehlow, R.A., J . Chem. Phys., 21 , 2101 (1953). 39 . Bertram, D., B.A.Sc. Thesis i n Chemical Engineering, U n i v e r s i t y of B r i t i s h Columbia, 1958. 40. Novak, G., B.A.Sc. Thesis i n Chemical Engineering, U n i v e r s i t y of B r i t i s h Columbia, 1957. 41. Selby, R., P r i v a t e Communication to Dr. D.S. S c o t t , Department of Chemical Engineering, U n i v e r s i t y of B r i t i s h Columbia, 1958. A P P E N D I X TABLE V I I I D i f f u s i o n C o e f f i c i e n t f o r the Hydrogen-Nitrogen System, 1 atm, as a Fun c t i o n of Temperature using the H i r s c h f e l d e r Equation to the F i r s t Approximation  T (°C) T (°K) (T) 3/2 T 12 W ( i ) ( D Do, c m 2 / 8 0 273 4511 4 . 9 6 0 .8430 0 .6742 25 298 5144 5 .42 O.83OO 0 .7814 50 323 5805 5 .87 0 .8167 0.8961 75 358 6492 6.32 0.8056 1.0159 100 373 7204 6.78 0.7951 1 .1423 125 398 7941 7.23 0 .7857 1 .2741 150 423 8700 7 .69 0 .7773 1 .4111 175 458 9482 8 .14 0 .7693 1 .5540 200 473 10290 8.60 0.7622 1 .7019 225 498 11113 9 .05 0 .7556 1 .8542 250 523 11960 9.51 0.7490 2.0131 275 558 12827 9 .96 0 .7424 2 .1783 166 573 13716 1 0 . 4 1 0.7372 2 .3460 SAMPLE CALCULATION 1. P o r o s i t y Sample Selas 0 3 . 2 Length 2.47 cm Average Diameter 2.62 cm Average Cross S e c t i o n a l Area 5«39 cm 2 Bulk Volume 13.31 cm^ Weight wet (s a t u r a t e d w i t h d i s t i l l e d water 26.60 g Weight dry 22.80 g .*. D i f f e r e n c e 3 .80 gms. Assuming a d e n s i t y of 1 gm/cm^ f o r water, v o i d volume - 3*80 cm^ P o r o s i t y * 3 .80 = .2855 13.31 73> SAMPLE CALCULATION 2. E l e c t r i c a l R e s i s t i v i t y R a t i o Sample Length Average Cross S e c t i o n a l Area Length/Average Cross S e c t i o n a l Area r0» s p e c i f i c r e s i s t i v i t y of s a t u r a t i n g s o l u t i o n Ro> = r 0 . L / A = R, r e s i s t a n c e of saturated sample Selas 03 .2 2.47 cm 5.39 omc 0.458 cm -1 82.0 ohm cm 37.56 ohms. 321.0 ohms E l e c t r i c a l R e s i s t i v i t y R a t i o = 321.0 • 8 .55 37.5'6 74 SAMPLE CALCULATION 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 and D i f f u s i o n R a t i o Sample Selas 03.2 Average Cross S e c t i o n a l Area, A, 5*39 cm 2 Length, L, 2.47 cm ( L /A) 0.458 cm" 1 Temperature 294 °K D i f f e r e n t i a l pressure drop (draught gauge) 0 .00 i n c h water Nitrogen flow r a t e , EN2, 9 .15 cm3/sec Output mV 29.45 mV Hydrogen mole f r a c t i o n i n o u t l e t N2 stream, X H2 J 0.0273 mf C a l c u l a t i o n : By means of a m a t e r i a l balance, we can c a l c u l a t e the volume of hydrogen d i f f u s e d , V H2 (This m a t e r i a l balance takes the form of a molar or volume balance f o r a constant pressure system.) A m a t e r i a l balance on the n i t r o g e n stream g i v e s : F N z - v „ 7 = ( f * x < , » x ) . 3c M Z ( 1 ) V « 2 = ( ^ i t H . ) . X H i ( 2) • * H , = 1 (3) To solve these equations one more r e l a t i o n i s r e q u i r e d . We make use of the t h e o r e t i c a l expression of Hoogschagen f o r d i f f u s i o n through a porous s o l i d : In our case, F N 2 • 9.15 cmVsec XH 2 = 0.0273 m.f. XN2 - 0 .972f m.f. VN 2 - 0.2682 O. Z6 Q2. VH2 So l v i n g f o r VH2> we get: 9.15 - VH2 x 0.2682 = Vffp x 0.9727 0.0273 VH2 = 0.255 cm3/sec We can now apply the Sherwood and P i g f o r d equation developed i n the I n t r o d u c t i o n and Theory s e c t i o n , p. 12. 76 /?TL ^ /-c ( f>JU ) , assuming the p e r f e c t gas laws apply, nh2 * and N ^ s / f to \ 4i we have, O. 73/8 = 0.0 27 3 a,hn ( t±Lx ) , - O ^ ^ 4 4 Gut-m DE = 0.7318 x 0 .255 x 0.458 172807 DE = 0.0667 cm 2/sec t a k i n g Do = O.763 cm 2/sec at 294°K _2P_ - 0^763 = 11.44 ° E 075557 • SAMPLE CALCULATION C a l i b r a t i o n Run at 294°K D i f f e r e n t i a l Pressure Drop (draught gauge) 0 .00 i n c h water Nitrogen flow r a t e 5*45 cmVsec Output m i l l i v o l t s 28.35 Hydrogen mole f r a c t i o n 0.0263 m.f. V H 2 I S determined by a m a t e r i a l balance s i m i l a r to that c a r r i e d out i n the previous s e c t i o n . V H 2 = 0 . 1 4 6 cm3/sec The Sherwood and P i g f o r d equation can be w r i t t e n i f A,L remain constants I- c ( where o ( i s a c a l i b r a t i o n f a c t o r . (PH2)2 = 0.0263 ( P H 2)l - 0.9950 and In 1^0^318 x (0.026,3) = 1.283 1-0.7318 x (0.9950) Do at 294°K = O.763 cm 2/sec o< z Do VH2/1.283 * = 0 . 7 6 3 x 1 . 2 8 3 0.146 <* = 6 . 7 1 0 78 SAMPLE CALCULATION D i f f u s i o n C o e f f i c i e n t s at Higher Temperatures; Sample 03.2 Length 5.19 cm Room Temperature 294°K Oven Temperature 573°K D i f f e r e n t i a l Pressure Drop (draught gauge) 0 .00 i n c h water Nitrogen flow r a t e -> (at room temp.) 6 .50 cmvsec Output m i l l i v o l t s 39.15 mV Hydrogen mole f r a c t i o n i n o u t l e t N2 stream (at room temp.) 0.0359 m.f. The volume of hydrogen d i f f u s e d i s measured at room temperature. This volume, as c a l c u l a t e d from the above data, must be corrected to the true c o n d i t i o n s of d i f f u s i o n . VH2 (294°K) = 0.2396 c m V s e c . Assuming th a t the p e r f e c t gas laws apply, V a t T j _ T X V a t T 2 ~ o r VH2 (573°K) r 573 = 1.949 VH2 (294°K) 294 s o VH2 (573°K) = 0.4670 cmVsec We can now apply the Sherwood and P i g f o r d equation; w i t h <K - 6.71 Do (573°K) = 6.71 C^73°K) - C (f»jL«- ) CPH2)l 0.9917 (PH2)2 = 0.0359 In ( 1-0.7318 , 1-0.7318 (0.9917) ; Do = 6.71 x 0.467 172^7 Do z (573°K) 2.473 cm2/sec, SAMPLE CALCULATION D i f f u s i o n ! C o e f f i c i e n t , H i r s c h f e l d e r Equation', Data on f o r c e constants from Hirschfelder,. JtCurt;iss L e t l a l , "Molecular Theory of Gases and L i q u i d s . "  (11) The c o e f f i c i e n t of d i f f u s i o n of a b i n a r y gas mixture may be obtained from the f o l l o w i n g equation which i s taken to the f i r s t approximation: where [ D i 2 l , = d i f f u s i o n c o e f f i c i e n t i n cm 2/sec p s pressure i n atmospheres T - temperature i n °K T i 2 ^ . k T /«f 12 M^, M 2 = molecular weights of species 1, 2 ; £ = molecular p o t e n t i a l energy parameters c h a r a c t e r i s t i c of 1-2 i n t e r a c t i o n i n 2. and 9K r e s p e c t i v e l y . To o b t a i n the c o e f f i c i e n t of d i f f u s i o n f o r the gas p a i r H 2 - N 2 at 273°K and 1 atm, we use the above formula. From Table IA, we f i n d ( l e t t i n g H 2 be component 1) t h a t : = 3*9.0- ° K , *z /* s 79. 2 'K , These parameters are obtained from v i s c o s i t y data From the simple combining r u l e s ; we o b t a i n , ^ z 3.332 A # = 55.07 °K For the temperature 273°K, - r * — = _ m = 4 . '11 - 5 F T 0 7 The molecular weights are H 2 - 2.016 N 2 - 14.02 From Table ( I M), we o b t a i n W u<''" (4.U) z °'8436 The f i r s t approximation t o £ P i i jl i computed from Equation (1) i s 0.6742 cm 2/sec. 2 4 MOLE FRACTION N 2 , (%) Thermal C o n d u c t i v i t y G e l l l C a l i b r a t i o n • P l o t  Output M i l l i v o l t s and Mole F r a c t i o n Nitrogen 0 I 2 3 4 5 6 MOLE FRACTION H 2 , (%) 25 20 10 0 2000 4000 6000 8000 H 2 , S T D CC / Ml N , I ATM , 7 0 ° F . F i g u r e 1^. C a l i b r a t i o n P l o t . Flowmeter Model 20k. S t e e l F l o a t . Standard ccs/min Hydrogen. 70°F. 1 atiru •GO 63 SiB.dv 600 400 LJ tr. < 01 LU a. LxJ h-200 CO < CD O - H 2 ©- N 2 f .15 1.0 .20 1.5 25 2.0 .30 25 .35 3.0 .40 3.5 N H. 4/e KINEMATIC VISCOSITY/ C M 2 / S E C . Figure 16. P l o t of: Kinematic V i s c o s i t y and Temperature  f o r Hydrogen and Nitrogen 

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