8/13/2019 MFP of air

1/8

J . A e r o s o l S c i . Vol , 19, No . 2 , pp. 159-166, 1988. 0021-8502/88 3.00 +0.0 0Pr i n t e d i n Gre a t B r i ta i n. 1988 Pe rga m on Pre ss pk

THE ME N FREE P TH IN IR

S. G. JENNINGSDepartment of Experimental Physics, University College Galway, Ireland

R e c e i o e d 5 M a y 1986;a n d i n f i n a l f o r m 31 J a n u a r y 1987)

A , B , cCK nM aM ~N APPoRTToz(P, T)

e sf P , T )kllo

r11

x ip#

Abstract--Values for the mean free path I in air are presented for temperatures 15, 20, 23 and 25Cfor both dry air and moist air at relative humidities of 50 and 100%, at atmospheric pressure(1.01325 x 105 Pa) and at pressure 1 x 102 Pa. Use is made of an expression which minimizes thedependence on physical constants' for mean free path:

where ;t is the viscosityof air, p is the density of air, P is the pressure and u is a numerical factor equal to0.4987445. Differencesof up to 1.6 % were obtained between he valuesof mean free path at 23C fromthis work and that of other workers. Comparison between the Cunningham slip correction factorfrom the present work and that from other work shows differences up to 1.8 %.

N O M E N C L A T U R ECoefficients in the Knudsen-Weber slip correction equation for Stokes drag forceSlip correction factorMean molecular speed (ms-1)Knudsen number defined as l /rMolecular weight of air ( --- 28.962458)Molecular weight of water (= 18.0152)Avogadro's constant (-- 6.0221358 x 1026 kmole-1)Pressure (Pa)Some 'standard' pressureUniversal gas constant (= 8.31448 x 103 JK- 1kmole-Temperature (K)Some 'standard' temperatureCompressibility factorSaturation vapour pressure of water vapour over a plane surface of pure liquid waterEnhancement factorBoltzmann's constant (= 1.380653 x 10 -e3 JK-1)Mean free path in air (m)Mean free path in air at a 'standard' temperature and pressureParticle radius (micrometers, pan)Numerical factor (= 0.4987445 in this work)Mole fraction for gas constituent iDensity of air (kg m- 3)Viscosity of air (kgm - 1 s - 1

I N T R O D U C T I O NIn aerosol physics, there is often a need to know the terminal fall velocity, mobility, ordiffu sion coefficient (diffusivity) of particles. In the Stokes' regime, all of the above pr opert iesare described by equations which incorporate the so-called Cunningham (1910) slipcorrection C which is depe nden t on the me an free path of air. The slip correcti on C multipliesthe Stokes' fall velocity, the mobility and diffusion coefficient to yield the correct values.Thus, accurate determination of the fall velocity, mobility and diffusion coefficient ofparticles is dependent on accurate knowledge of the mean free path i in air.

As poin ted out by Fuchs (1964), the transfer rates of mome nt um , energy and mass aredetermined by the value of the Knudse n numb er K n = i / r , where r is the parti cle radius. In t helimit ing cases of very small and very large Knu ds en num bers, the theories of these transferprocesses are exact. Thus, for example, in the case of small Knudsen numbers, l / r ~ 0,diffusional growth or evaporation for particles (or droplets) is described by classical

s 1 9 : a - ~ , 159

8/13/2019 MFP of air

2/8

160 S .G . JENNINGSc o n t i n u u m d i f f u s i o n a n d h e a t t r a n s f e r th e o r i e s, a s s u m e d v a l id t o w i t h i n a b o u t o n e m e a n f r e ep a t h d i s ta n c e f r o m t h e p a rt ic le . W h e n t h e m e a n f re e p a t h i s la r ge c o m p a r e d t o t h e s iz e o f a ne va p ora t i n g (o r g rowi ng ) pa r t ic l e , c l a ss i c al k i ne t i c t he ory c a n be a pp l i e d i n t h i s so -c a ll e d f re e -m o l e c u l e re g i m e ( H i d y a n d B r o c k , 1 9 7 0 ) t o d e t e r m i n e t h e e v a p o r a t i o n o r g r o w t h r a t e.P r o g r e s s i n t h e t h e o r y o f p a rt i cl e t r a n s p o r t p r o c e ss e s a t i n t e r m e d i a t e K n va l ue s ha s be e nr e v ie w e d b y F u c h s a n d S u t u g i n ( 19 70 ). T h e y p o i n t o u t t h a t t h e t h e o r y o f t r a n s p o r t t o a e r o s o lp a r t ic u l a t e s c o n s t a n t l y m a k e s u s e o f t h e c o n c e p t o f t h e m e a n f re e p a t h o f g a s m o l e cu l e s .D e s p i t e t h e i m p o r t a n c e o f a c c u r a t e l y k n o w i n g t h e m e a n f re e p a t h l i n a ir , f e w r el ia b l ev a l u e s o f I h av e b e e n p u b l i s h e d a n d i n d e e d d i f f e r e n t m e a n f r ee p a t h v a l u e s h av e a p p e a r e d i nt he l i t e ra tu re fo r t he sa me t e mp e ra t u re . In t h i s pa pe r t he e xpre s s i on fo r t he me a n f re e pa t h i sp re se n t e d f i rs t . In t he n e x t s e c t ion , e xpre s s i ons fo r t he de n s i t y o f d ry a nd mo i s t ai r , r e qu i re d i nt h e e v a l u a t i o n o f m e a n f r ee p a t h , a r e g i v e n t o g e t h e r w i t h p h y s i ca l c o n s t a n t s u s e d t o e v a l u a tet h e a i r d e n s i ty . T h i s is f o l l o w e d b y a s e c ti o n o n t h e v i s c o si ty o f d r y a n d m o i s t a i r . T a b u l a t e dv a l u e s o f t h e v i s c o s it y o f a i r a n d t h e m e a n f r ee p a t h i n a ir a r e p r e s e n t e d f o r t e m p e r a t u r e s o f 1 5,20 , 23 a nd 25C for r e l a t ive hum i d i t i e s o f 0 , 50 a nd 100 K a t p re s sure s 105 a nd 1 .01325x 1 05 P a . F i n a l l y c o m p a r i s o n i s m a d e b e t w e e n t h e n e w l y c o m p u t e d v a l u e o f m e a n f r ee p a t h

a t 2 3 C , 1 .0 1 3 x 105 P a , w i t h t h a t o f o t h e r w o r k e r s . I n a d d i t i o n , C u n n i n g h a m s l ip c o r r e c ti o nv a l u es o n t h e m e a n f r ee p a t h f r o m t h is w o r k a n d f r o m c u r r e n t l y u s e d m e a n f r ee p a t h v a l u e sa r e i n t e r - c o m p a r e d .

T H E O R YThe me a n f re e pa t h , l , i n a i r i s g i ve n by t he e xpre s s i on (C ha pma n a nd C owl i ng , 1970) :

u p( ? (1)whe re p i s t he de ns i t y o f t he a i r mol e c u l e s , C i s t he me a n mol e c u l a r spe e d , a nd / 1 i s t hec oe f f i c ie n t o f v i sc os i ty o f a i r .

The nu me r i c a l f a c t o r u , e qua l t o 0 .4987445 is f r a c t i ona l l y sma l l e r t ha n t he norm a l l y quo t e dv a l u e o f 0 .4 9 9 ( C h a p m a n a n d C o w l i n g , 1 9 7 0 ) d u e t o t h e r e c a lc u l a t io n o f t h e c o e f fi c ie n t o fv i s co s i ty b y P e k e r i s a n d A l t e r m a n n ( 19 57 ), w h o , l ik e C h a p m a n a n d C o w l i n g ( 1 97 0) , m o d e l l e dt he ga s mo l e c u l e s a s r i g i d e l a s t ic sphe re s .

E q u a t i o n (1 ) r e d u c e s t ozt/~ 1 (2)

whe re P i s t he a t mo sph e r i c p re s sure . S i nc e p ~ 1 / T 1/2 un de r t he a s sum pt i o n t ha t t he ga sm o l e c u l e s o b e y t h e e q u a t i o n o f s ta t e w e h a v e

l / l o = ( 1 ~ /# o ) ( P o / P ) ( T / T o )5 12 , (3)w h e r e th e s u b s cr i p t 0 d e n o t e s th e v al u e a t s o m e s t a n d a r d t e m p e r a t u r e a n d p r e s s u re .Eq ua t i o n (3) i s s t r i c t ly a pp l i c a b l e fo r d ry a ir . The m e a n f re e pa t h l a t som e pre s sure a n dt e m p e r a t u r e a t w h i c h t h e v i s c o si ty / ~ i s k n o w n c a n b e d e t e r m i n e d f r o m t h e k n o w n v a l u e o fm e a n f r ee p a t h l o a t a s t a n d a r d p r e s s u r e P o a n d t e m p e r a t u r e T o t h r o u g h u s e o f e q u a t i o n ( 3) .S i nc e t he de ns i t y p i s c on t a i ne d i n t he e xpre s s i on fo r m e a n f re e pa t h I -e qua ti on (2) -], fo rm ul a ef o r t h e d e n s i t y o f d r y a n d m o i s t a i r a r e p r e s e n t e d i n t h e f o l l o w i n g s ec t io n .

D E N S I T Y O F D R Y A N D M O I S T A IRThe de n s i t y o f d ry a i r , p , is g i ve n by :

P M ~P - R T Z ' (4)

whe re P i s t he p re s sure , M a i s t he m ol e c u l a r w e i gh t o f a i r, R i s the un i ve rsa l ga s c on s t a n t a nd

8/13/2019 MFP of air

3/8

The mea n free path in air 161Z is t h e c o m p r e s s i b i l i t y f a c t o r o f t h e n o n - i d e a l g a s ( a ir ) w h i c h i s a f u n c t i o n o f p r e s s u r e P a n dt e m p e r a t u r e T . T a b u l a t e d v a l u e s o f Z f o r a r a n g e o f P a n d T v a l u e s f o r d r y a n d m o i s t a ir a reg i v e n i n t h e Smithsonian Meteorological Tables (1 96 6), b a s e d o n t h e d a t a p r o v i d e d b y G o f fa n d G r a t c h ( 1 9 4 5 ) .

T h e d e n s i t y o f m o i s t a ir is o b t a i n e d f r o m t h e e x p r e s s i o nPMa[- ( 1 M v ~ R H f e s]RTZL1 1 0 0 e J ( 5 )

w h e r e M v i s th e m o l e c u l a r w e i g h t o f w a t e r , R.H. i s t h e r e l a ti v e h u m i d i t y , a n d e s i s t h es a t u r a t i o n v a p o u r p r e s s u r e o f t h e p u r e v a p o u r ( w i t h o u t t h e a d m i x t u r e o f ai r) o v e r a p l a n es u r f ac e o f p u r e l i q u i d w a t er . A c o m p l i c a t e d e x p r e s s i o n f o r t h e e n h a n c e m e n t f a c t o r f i n v o l v i n ga s e ri e s o f v i r ia l c o e f f ic i e n ts w a s d e r i v e d b y H y l a n d a n d W e x l e r ( 1 9 7 3 ) a n d i s u s e d b y H y l a n d( 1 9 75 ) t o o b t a i n e n h a n c e m e n t v a l u e s f o r a i r o v e r w a t e r f r o m - 8 0 C t o 9 0 C a t 1 0 C i n t e rv a l sa n d f o r a w i d e r a n g e o f p r e s s u r e s i n c l u d i n g l 0 s P a ( l b a r) .

V A L U E S O F T H E P A R A M E T E R S U S E D 1 N T H E E X P R E S S I O N F O RD E N S I T Y O F D R Y A N D M O I S T A IR

A n e w p r e l i m i n a r y a d j u s t e d v a l u e f r o m a n e w l e a s t- s q u a re s a d j u s t m e n t c u r r e n t l y m a d e t ot h e u n i v e rs a l ga s c o n s t a n t R ( D r B . N . T a y l o r , N B S , G a i t h e r s b u r g , p r i v a t e c o m m u n i c a t i o n ,1986) i s t a ke n t o be

R = 8 . 3 14 4 8 x 1 0 3 J K - 1 k m o l e - xT h e m o l e c u l a r w e i g h t o f a i r, M a , c a lc u l a t e d f r o m :

M a = ~ x iM i ,i l

w h e r e M i is t h e m o l e c u l a r w e i g h t o f a n i n d i v id u a l c o n s t i t u e n t a n d x i is t h e c o r r e s p o n d i n gm o l e f r a c t i o n , i s t a k e n t o b e :M a = 28 .962458 ,

u s i n g t h e c o n s t i t u e n t v a l u e s g i v e n b y J o n e s ( 19 78 ). T h i s a s s u m e s a m o l e f r a c t i o n o f 0 . 0 0 0 3 3f o r C O 2 i n th e a t m o s p h e r e , i n c l o se a g r e e m e n t w i t h c u r r e n t l y q u o t e d v a l u e s ( K o m h y r et al.,1 98 5 ). V a l u e s o f t h e c o m p r e s s i b i l i t y f a c t o r Z w e r e l in e a r l y e x t r a p o l a t e d f r o m t h e SmithsonianMeteorological Tables (1966).

T h e m o l e c u l a r w e i g h t o f w a t e r , M v , w a s t a k e n t o b e 1 8. 01 52 ( I n t e r n a t i o n a l U n i o n o f P u r ea n d A p p l i e d C h e m i s t r y , 1 9 76 ). T h e s a t u r a t i o n v a p o u r p r e s s u r e e s o f p u r e w a t e r v a p o u r w a st a k e n f r o m W e x l e r ( 1 9 7 6 ) .

T h e e n h a n c e m e n t f a c t o r f w a s t a k e n t o b e 1 .0 0 4 o v e r t h e te m p e r a t u r e r a n g e u s e d h e r e( 1 5 - 2 5 C ) a t p r e s s u r e s 1 05 P a a n d 1 .0 1 32 5 x 1 0 5 P a f r o m a n e x a m i n a t i o n o f t h e t a b u l a t e dv a l u e s o f H y l a n d (1 9 75 ).

T h e c o n s t a n t s u s e d i n t h e d e n s i t y c a l c u l a t i o n s a r e l is t e d in T a b l e 1 . V a l u e s o f d e n s i t y o f d r ya i r u s i n g e q u a t i o n (4 ) a n d o f m o i s t a i r u s i n g e q u a t i o n (5 ) f o r r e l at i v e h u m i d i t i e s o f 5 0 a n d1 0 0 a r e p r e s e n t e d i n T a b l e 2 f o r p r e s s u r e s o f l 0 s P a a n d 1 .0 1 32 5 x 1 05 P a .

V I S C O S I T Y O F A I RA s i n d i c a t e d b y e q u a t i o n ( 2) , t h e m e a n f r e e p a t h i s l i n e a r l y d e p e n d e n t o n t h e v i s c o s i ty o f a ir .

A s u r v e y o f t h e l i t e r a t u r e r e v e a l e d t h a t a m o n g t h e w i d e v a r i a b il i t y o f v i s c o s i ty v a l u e s , t h e b e s ta v a i l a b l e d a t a f o r t h e v i s c o s i t y ( d y n a m i c ) o f a i r a t 1 a t m ( 1 .0 1 3 2 5 x 1 0 5 P a ) i s a s s h o w n i nT a b le 3 .

T h e v a l u es o u t l i n e d i n T a b l e 4 a r e t h o s e a d o p t e d f o r u s e i n t h is w o r k . T h e e f f e c t o fc h a n g i n g p r e s s u r e f r o m 1 x l 0 s P a t o 1 .0 13 x 1 05 P a a t c o n s t a n t t e m p e r a t u r e u p o n t h ev i sc os i t y o f a i r i s ne g l ig ib l e .

8/13/2019 MFP of air

4/8

162 S. G . JENNINGS

T a b l e 1 . P h y s i c a l c o n s t a n t s u s e d i n t h e c a l c u l a t i o n o f d e n s i t yP a r a m e t e r S y m b o l V a l u e U n c e r t a i n t y R e f e re n c e

U n i v e r s a l g a s c o n s t a n t R 8 .3 1 44 8 x 1 03 2 5 p p m N B ~ - p r i v a t eJ K - ~ k m o l e - ~ c o m m u n i c a t i o n b y

D r B . N . T a y l o rM o l e c u l a r w e i g h t o f a i r M a 2 8 . 9 6 2 4 5 8 _+ 4 x 1 0 - 5 J o n e s ( 19 7 8)C o m p r e s s i b i l i t y f a c t o r Z F u n c t i o n o f + 1 .7 x 1 0 5 Smithsonian

t e m p e r a t u r e Meteorologicala n d p r e s s u r e Tables (1966)M o l e c u l a r w e i g h t o f w a t e r M v 1 8 .0 1 5 2 + 0 . 0 0 0 5 I n t e r n a t i o n a lU n i o n o f P u r ea n d A p p l i e dC h e m i s t r y ( 1 9 7 6 )S a t u r a t i o n w a t e r v a p o u r e s D o e s n o t e x c e e d W e x l e r ( 19 76 )p r e s s u r e 6 4 p p m o v e r

t e m p e r a t u r er a n g e 1 5 - 2 5 CEn ha nc em en t f ac to r J 1 . 004 __+0 . 0 0 0 1 3 H y l a n d ( 1 9 7 5 )

T a b l e 2 . D e n s i t y o f d r y a n d h u m i d a i rR e l a t i v e h u m i d i t y ( )

T e m p e r a t u r e P r e s s u r e 0 5 0 1 00( K ) ( P a ) ( k g m - 3288 .15 105 1.2094 1.2055 1.20171.0325 x 105 1.2254 1.2215 1.2177293 . 15 105 1 . t887 1 . 1834 1 . 17831.01325 x 105 1.2045 1.1992 1.1940296 .15 105 1.1766 1.170 4 1.1642

1.01325 x 105 1.1922 1.186 0 1.1798298 . 15 105 1 . 1687 1 . 1617 1 . 15481.01325 x 105 1.1842 1.1772 1.1703

( a ) V i s c o s i t y o f d r y a i r T a b l e 3 . V i s c o s i t y o f a i r

T e m p e r a t u r e M e a s u r e d v a lu e( C ) ( k g m i s - 1 ) U n c e r t a i n t y A u t h o r ( s ) R e f e r e n c e2 0 1 . 8 1 9 2 x 1 0 5 0 . 0 0 0 0 6 B e a r d e n ( 1 9 3 9 ) Phys. Rev. 56 , 10232 0 1 . 81 9 4 x 1 0 - 5 0 . 0 0 0 3 5 K e s t i n a n d L e i d e n f r o s t Physica 2 5 , 1 0 3 3(1959)2 5 1 . 8 4 4 6 x 1 0 - 5 0 . 0 0 0 9 1 K e s t i n a n d L e i d e n f r o s t , Physica 2 5 , 1 0 3 3

(1959)

( b) V i s c o s i t y o f m o i s t a i rT e m p e r a t u r e R e l a t i v e h u m i d i t y ( ~,,)( C ) 5 0 1 0 0 U n c e r t a i n t y A u t h o r ( s ) R e f e r e n c e

2 0 1 .8 1 5 1 . 8 1 2 7 x 1 0 - 5 0 .1 ~ H o c h r a i n e r a n d Sber. ost . Akad. Wiss.x 1 0 - 5 M u n c z a k ( 19 6 6) M a t h . K l a s s e A b t . I I , 1 7 5 ,5 3 92 5 1 . 8 4 2 0 1 . 8 4 0 0 x 1 0 5 + 0 .1 ~ o K e s t i n a n d W h i t e l a w I n t . J . He a t M as s Tr ans fe rx 10 -5 (1964) 7 , 1245

8/13/2019 MFP of air

5/8

The mean free pa th in air 163C a l c u l a t e d v a lu e s o f m e a n f r e e p a t h

V a l u e s f o r t h e m e a n f re e p a t h i n a i r u s i n g v i s c o si t y v a l u e s o f T a b l e 4 a n d d e n s i t y v a l u e sd e r i v e d f r o m e q u a t i o n s ( 4) a n d (5 ) a r e g i v e n i n T a b l e 5 . T h e v a l u e s a re r o u n d e d o f f t o t h ef o u r t h s i g n if i ca n t f ig u re . T h e m e a n f re e p a t h i s i n v e r se l y p r o p o r t i o n a l t o a i r p r e s s u r e f o r d r ya i r a s r e a d i l y v e r i f i a b l e f r o m T a b l e 5 . M e a n f re e p a t h v a l u e s a r e a l so g i v e n a t a s e c o n d p r e s s u r eo f 1 05 P a s in c e th e m e a n f re e p a t h i s n o t q u i t e i n v e r s e ly p r o p o r t i o n a l t o p r e s s u r e f o r m o i s t a i ra t t h e f o u r t h s i g n i fi c a n t f ig u re . ( I n v e rs e p r o p o r t i o n a l i t y d o e s , h o w e v e r , o b t a i n a t t h e t h i r ds i g n i f i c a n t f i g u r e . )

D I S C U S S I O NW h i l e i t i s r e a l iz e d t h a t o n e h a s a s m a n y f re e p a t h s a s t h e re a r e k i n d s o f b i n a r y c o l l i s i o n s

b e t w e e n t h e a i r c o n s t i t u e n t m o l e c u l e s , t h e s i m p l e s t a p p r o a c h is to t r e a t a i r a s a s in g l ec o m p o n e n t g a s p o s s e ss i n g a n a v e r a g e d e n s i ty , v is c o s i t y a n d m e a n f r e e p a t h a t a p a r t i c u l a rt e m p e r a t u r e a n d p r e s s u r e .

R e c e n t u se o f t h e C h a p m a n - E n s k o g t h e o r y a s a p p l i e d to m u l t i - c o m p o n e n t g a se s o f a g asm i x t u r e , b y H i r s c h f e l d e r e t a l . ( 19 6 4) t o g e t h e r w i t h t h e u s e o f t h e l a w o f c o r r e s p o n d i n g s t a t e sa s f o r m u l a t e d b y K e s t i n e t a l . (1 9 72 ) h a s p e r m i t t e d t h e c a l c u l a t i o n o f t r a n s p o r t p r o p e r t i e s( i n c l u d in g v i sc o s i ty ) o f m u l t i - c o m p o n e n t d i l u t e m i x t u r e s o f g a s e s a t d e n s i t i e s w h e r e b i n a r yc o l l i si o n s b e t w e e n t h e c o m p o n e n t s a r e d o m i n a n t . T h e s e t h e o r i e s h a v e b e e n s u c c e ss f u ll ya p p l i e d ( H e l le m a n s e t a l . 1 9 7 3) t o t h e c a l c u l a t i o n o f t h e v i s c o s i t y o f a i r , t r e a t e d a s a t e r n a r y

Table 4. The air viscosity values adopted for use in this work(a) Viscosi ty of dry air

Viscosi ty of a ir a tTemp erature 1.013 x 10~ Pa(C) (kg m - i s - l ) Comm ents20 1.8192 x10 -s Bearden's (1939 ) value, l isted in theAmerican Inst i tu te o f Physics Handbook(1972)25 1.8446 x 10 -5 Kestin and Leidenfrost (1959) value23 1.8353 x 10 -5 Based on Bearden's (1939) value at 20Cand using temperature increment of0.00536 (kg m - a s- 1) /C calculated f romKestin an d Leidenfrost (195 9) measure-mer i ts a t 20 an d 25C.15 1.7924 x 10-5 Using temperature 'decrement' of 0.00536(kg m - t s - ~ /C, tog ether with Bearden'svalue at 20C.

(b) Viscosi ty of m ois t a irViscosity of moist air at1.013 l 0 s P a( kgm - 1 s -1 )Relative hum idityTemp erature ( )(C) 50 100 Com ments

20 1.8154 x 10 -5 1.8 12 7 10 -5 Hochrainer(1966) values25 1.8420 x 10 -s 1.84 00 10 -s

23 1.8314 x 10 -5 1.82 91x 10 -s15 1.7888 x 10 -5 1.7 85 4 10 -5

and MunczakKestin and Whitelaw's (1964)valuesInterpo lat ion o f viscosity valuesat 20 and 25CExtrapolat ion based on vis-cosi ty measurements at 20 and25C

8/13/2019 MFP of air

6/8

164 S. G. JENNINGSTable 5. M ean free path in air

elative humidity~ o )Tem perature Pressure 0* 50 100(K) (Pa) (10 -s m)

288.15 1.01325 x l0 s 6.391 6.389 6.386105 6.476 6.473 6.471293.15 1.01325 x 105 6.543 6.544 6.54810 ~ 6.630 6.631 6.635296.15 1.01325 x 105 6.635 6.638 6.647105 6.723 6.726 6.736298.15 1.01325 l0 s 6.691 6.701 6.71410 s 6.780 6.790 6.803

* Mean free path values for dry air can be determined at othertemperatures and p ressures through the use of equation (3).

m i x t u r e o f n i t r o g e n , o x y g e n a n d a r g o n y i e ld i n g e xc e ll en t a g r e e m e n t b e t w e e n t h e c al c u l at e da n d e x p e r i m e n t a l v a l u e s f o r v i s c o s i t y o f a ir a t 2 5 C ( D i P i p p o a n d K e s t i n , 1 9 68 ). T h i ss t r e n g t h e n s t h e j u s ti f ic a t io n f o r u s e o f a m e a n f re e p a t h b a s e d o n a m e a s u r e d v a l u e f o rv i s c o s it y o f ai r u s i n g e q u a t i o n ( 2) . T h e m e a n f r ee p a t h c a l c u l a t i o n a l s o r el ie s o n t h e u s e o f a na v e r a g e v a l u e f o r t h e d e n s it y o f a ir . T h e p r o c e d u r e f o r o b t a i n i n g a i r d e n s i ty u s i n g t h em o l e c u l a r w e i g h t s o f i n d i v i d u a l a i r c o n s t i t u e n t s b a s e d o n t h e i r m o l e f r a c t i o n s g iv e s g o o da g r e e m e n t w i t h p u b l i s h e d v a l u e s [Sm i t hson i an M e t e oro l og i c a l T ab l e s ( 1 9 6 6 ) ] .

T h e u s e o f e q u a t i o n (2 ) f o r th e m e a n f re e p a t h I = ~ 8 p / u ) l / x / p - P ) h a s t h e m e r i t o fm i n i m i z i n g t h e us e o f p h y si c a l c o n s t a n t s a s c o m p a r e d t o u s in g t h e m o r e u s u a l f o r m u l a t i o nf o r l

l _ P 1u p C 6 )

E q u a t i o n ( 6) u s e s t h e v a l u e o f ( ~ - - 8kT/rcm ) 1/2 w h i c h d e p e n d s o n k , M a a n d N A ,' c o n s t a n t s ' w h i c h u n d e r g o r e a d j u s t m e n t i n v a l u e p e r io d i ca l ly . T h e v a l ue o f t h e n u m e r i c a lf a c t o r u a ri s es f r o m t h e u s e o f t h e m o l e c u l a r m o d e l o f s m o o t h r i g id e la s ti c s p h e r i c a lm o l e c u le s . T h i s is g e n e r a ll y r e g a r d e d a s t h e m o r e c o r r e c t m o d e l f o r a r a ti fi e d h o m o g e n e o u sm o l e c u l a r g a s . A l l e n a n d R a a b e ( 1 9 8 2) b a s e t h e i r m e a n f r e e p a t h d e t e r m i n a t i o n (l =6 .7 3 x 1 0 - s m a t 2 3 C ) o n t h e r e p u l s iv e f o r c e m o l e c u l a r m o d e l w h i c h a s s u m e s t h a t t h e g a sm o l e c u l e s p o s s e s s t h e s a m e e l e c tr i c c h a r g e p o l a r i ty . T h e r e p u l s iv e f o r c e m o l e c u l a r t h e o r yy i e ld s a v a l u e f o r n u m e r i c a l f a c t o r o f 57 t/ 32 ( 0 .4 9 0 8 7 ) in t h e e x p r e s s i o n f o r m e a n f r e e p a t h . T h em e a n f r ee p a t h v a l u e d e r iv e d i n th i s w o r k e q u a l s 6 . 6 3 5 x 1 0 - 8 m a t t e m p e r a t u r e 2 3 C b a s e do n t h e m o r e u s u a l l y p r e f e r r e d r i g i d e l a s t i c s p h e r i c a l m o d e l , w h i c h e m p l o y s t h e n u m e r i c a lf a c t o r o f 1 . 0 1 6 0 3 4 x ( 5 ~ /3 2 ) in t h e e x p r e s s i o n f o r t h e m e a n f r e e p a t h . T h e p e r c e n t a g ed i f f e re n c e o f 1 .4 ~ b e t w e e n t h e t w o m e a n f r ee p a t h v a l u e s is l a r g e ly a c c o u n t e d f o r b y t h ed i f f e re n c e i n t h e v a l u e s o f th e n u m e r i c a l f a c t o rs .

D a v i e s ( 1 9 45 ) u s es a m e a n f re e p a th v a l u e o f 6 .6 0 9 x 1 0 - 8 m a t 2 3 C t h r o u g h e m p l o y m e n to f t h e n u m e r i c a l f a c t o r o f 0 . 4 9 9 ( 5 n / 3 2 x 1 .0 16 ). T h i s m e a n f r e e p a t h v a l u e i s w i t h i n 0 . 4 ~o o ft h e v a l u e o b t a i n e d i n t h is w o r k . F u c h s ( 1 9 6 4) , t h o u g h u s i n g t h e s a m e n u m e r i c a l f a c t o r a sD a v i e s ( 1 9 4 5 ) a b o v e , q u o t e s a v a l u e o f 6 .5 3 x 1 0 _ 8 m . T h i s is c o n s i d e r e d t o b e a nu n d e r e s t i m a t e a n d is m o r e r e p r e s e n t a ti v e o f th e c a lc u l a te d m e a n f re e p a t h a t 2 0 C r a t h e r t h a na t 2 3 C .

H i t h e r t o , t h e m e a n f r e e p a t h i n ai r h a s u s u a l ly b e e n c a l c u l a t e d a t a t e m p e r a t u r e o f 2 3 C , toc o i n c i d e w i t h t h e t e m p e r a t u r e a t w h i c h t h e m a j o r i t y o f M i l l i k a n ' s ( 1 9 2 3) w o r k o n t h e s l ipc o r r e c t i o n c o e f f ic i e n ts r e la t e d t o h i s e l e c tr o n i c c h a r g e w o r k , w a s c a r r i e d o u t . T h i s p r e s e n tw o r k e x t e n d s t h e m e a n f r e e p a t h c a l c u l a t i o n s t o a r a n g e o f t e m p e r a t u r e s f r o m 1 5 t o 2 5 C . I na d d i t i o n , t h e w o r k i n c l u d e s f o r t h e f ir st t i m e a p r e s e n t a t i o n o f m e a n f r e e p a t h v a l u e s f o r m o i s ta i r - - a t r e l at iv e h u m i d i t ie s o f 5 0 a n d 1 0 0 ~o. T h e r e s u lt s i n T a b l e 5 s h o w t h a t r e l a t iv e h u m i d i t y

8/13/2019 MFP of air

7/8

The mean free path in airTable 6. Cunn ingham slip correction factor for d ry air at 23C , 1.013 x 105 Pa

165

Mean free path Cunningham slip Knudsen num berReference (10 -8 m) correctio n formu la 0.01 0.1 1 10Davies (1945) 6.609Fu chs (1964) 6.53Allen and 6.673Raab e (1982)This w ork 6.635

l + K n [ 1 . 2 5 7 + 0 . 4 0 0 e x p ( - ~ - n 0 ) ] 1 . 01 2 6 1 .1 2 57 2 . 3 9 0 1 7.1 53[ { 0 867 ~ 1l + K n [ 1 . 2 4 6 + 0 . 4 1 8 e x p k - ~ L - n ' ) J 1 . 0 12 5 1 . 1 24 6 2 . 4 2 2 1 7.2 93I f 0.596 ~-]l + K n 1 . 1 5 5 + 0 . 4 7 1 e x p k - ~ n - - n : j 1 . 0 1 1 6 1 . 1 1 5 6 2 . 4 1 5 1 6. 98 8

l + K n [ 1 . 2 5 2 + 0 . 3 9 9 e x p ( - ~ n 0 ) l 1 . 0 12 5 1 .1 2 52 2 . 3 8 7 1 7.0 87

h a s g r e a t e s t i n f l u e n c e o n t h e m e a n f r e e p a t h ( ,- - 0 .3 % i n c r e a s e ) a t t h e h i g h e r t e m p e r a t u r e s . I tc a n a l s o b e s e e n t h a t r e l a t i v e h u m i d i t y c a u s e s a s m a l l d e c r e a s e ( ~ 0 . 1 % ) i n th e m e a n f r ee p a t ho v e r t h e v a l u e f o r d r y a i r a t t h e l o w e s t t e m p e r a t u r e ( 1 5 C ) c o n s i d e r e d .

T h e m e a n f r ee p a t h d i r e c t ly af f ec t s t h e s o - c a l l e d C u n n i n g h a m ( 19 1 0) s l ip c o r r e c t i o n f a c t o rC , l a t e r m o d i f i e d b y K n u d s e n a n d W e b e r ( 19 1 1) to t h e f o r m

C = 1 + K n [ A + B e x p ( - c / K n ) ] . (7)Since aerosol propert ies such as the Stokes' fall velocity, particle mobility and diffusivity arelinearly dependent on the value of the slip correction C, accurate determination of theseproperties is dependent on accurate knowledge of the mean free path. Values of mean freepath together with calculated Cunningham slip correction factors C from the work o f Davies(1945), Fuchs (1964), Allen and Raabe (1982) and this work are summarized in Table 6 forselected n values of 0.01, 0.1, 1 and 10. Differences of not more than 0.1 result for thelowest n number as expected. However differences of up to 1.8 in C were obtained forlarger n numbers with the least differences occurring between the values of Davies (1945)and the present work.

A c k n o w l e d o e m e n t s - - T h e auth or w ishes to acknowledge the Nation al Research Council (NRC) for sup port duringthe Research Associateship Program. He also wishes to acknowledge constructive comments on the paper byDr C. N, Davies.R E F E R E N C E S

Allen, M. D. and Raabe, O. G. (1982) d. Aerosol Sci. 16, 57.Ame r i c an I n s t i t u t e o f Phy s i c s Handbook (1972), 3rd edn. McG raw-Hill, New York.Bearden, J. A. (1939) Phys . Rev . 56, 1023.Chapman, S. and Cow ling, T. G. (1970) The Mat he ma t i c a l Theory of Non-uni form Cases . Cambridge UniversityPress, London.Cunningham, E. (1910) Proc. R. Soc. 83, 357.Davies, C. N. (1945) Proc. phys. Soc. 57, 259.DiP ippo, R. an d Kestin, J. (1968) Proc. 4th Symp osium on Thermophysical Properties, Am. S oc. Mec h. Eng., p. 304.Fuchs, N. A. (1964) The Mec hanics o f Aerosols. Pergamon Press, Oxford.Fuchs, N. A . and Sutugin, A. G. (1970) Highly Dispersed Aerosols. Ann Ar bor S cience, London.Goff, J. A. and Gratch , S. (1945) Trans. Am. Soc. Heat. Vent. Engrs 51, 125.Hellemans, J. M., Kestin, J. and Ro, S. T. (1973) Phy s i c a 65, 362.Hidy, G. M. and Brock, J. R. (1970) The Dynam ics o f Aerocollo idal Sys tems. Pergamon Press, Oxford.Hirschfelder, J. O., Curtiss, C. F. an d Bird, R. B., (1964)The M olecular Th eory o f Gases and Liquids. John Wiley, NewYork.Hochrainer, D. and M unczak, F. (1966) Sber. 6 st. Ak ad . Wiss., Math. Klasse Abt. II, 175, 539.Hyland, R. W. (1975) d. Res . natn . Bur . S tand. 79A, 551.Hyland, R. W . and Wexler, A. (1973) J. Res. natn. Bur. Stand. 77A, 133.International Union of Pure and Applied Chemistry (1976) Pur e appl . Chem. 47, 75.Jones, F. E. (1978) d. Res. natn. Bur. Stand. 83, 419.Kestin, J. and Leidenfrost, W, (1959) Phy s i c a 25, 1033.Kestin, J., Ro, S. T. and Wakeham, W. A. (1972) Phy s i c a 58, 165.Kestin, J. and Whitelaw, J. H. (1964) I n t . J . He a t Mas s Tr ans f e r 7, 1245.

8/13/2019 MFP of air

8/8

166 S .G . JENNINGSKo mh yr, W. D., Ga mm on, R. H., Harris, T. B., Waterman, U S., Conway, T. J. , Tay lor, W. R. and Tho ning, K. W.1985) J. geophys. Res. 90 5567.Knudsen, M. and Weber , S . 1911) Ann. Phys . 36, 981.Millikan, R. A. 1923) Phys . Rev . 22, 1.Pekeris, C. L . and Alterm ann , Z. 1957) Proc. natn. A cad. Sci . U.S.A. 43, 998.Smithsonian Mete oro looica l Tables 1966), prepared by List, R. J .) , 6th edn. Smiths onian In st i tut ion, City ofWashington.Wexler , A. 1976) J. Res. hath. Bur. Stand. 80A, 775.