helium properties

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« Riso Report No. 224

5o,

« D anish A tom ic Energy Com m ission

.1

* Research Establishment Ris6

The Properties of Helium : D ensity,

Specific Heats, Viscosity, and Therm al

C on ductivity at Pressures from 1 to

100 bar and fro m R oom Tem perature

to about 1800 K

by Helge Petersen

September, 1970

Sain dUiributort: JUL GjiUtrap, 17, » W pd t, D K-U07 Capubaftn K, D nau rk



U . D . Cm»!:62U34.3t



September, 1 970 Rise Report No. 224

The Properties of Helium: Density, Specific Heats. Viscosity, andThermal Conductivity at Pressures from 1 to 100 bar and from

Room Temperature to about 1800 K

by

Helge Petersen

Danish Atomic Energy Commission

Research Establishment RiseEngineering Department

Section of Experimental Technology

Abstract

An estimation of the properties of helium is carried out on the basisof a literatu re su rvey. The ranges of pressu re and temperature chosena re applicable to heliu m -cooled a tomic rea ctor d esign. A brief outline ofvhe theory for the properties is incorporated, and comparisons of the recommended data with the data calculated from intermolecular potentialfunctions are presented.



I3BI 87 550 0035 5



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C O N T E N T S

P a g e

1 . Introduct ion5

2 . S u m m a r y o f R e c o m m e n d e d D a t a , , 5

3 . G a s L a w 7

4 . S p ec i f i c H e a t s ' 2

5 . C o r r e l a t i o n F o r m u l a e f o r V i s c o s i t y a n d C o n d u c ti vi ty ' 3

6 . V i s c o s i t y D a t a ' 2 C

7 . T h e rm a l C o n d u c t iv i ty D a t a 2 9

8 . P ra n d t l N u m b er 3 0

9 . A ck n o w l ed g em en t s a n d N o t e s 3 2

1 0 . R e f e r e n c e s 3 3

1 1 . . T a b l e s o f R e c o m m e n d e d D a t a 3 7



- 5 -

1 . I N T R O D U C T I O N

C a l c u l a t io n s o f t h e h e a t t r a n s f e r a n d t h e t e m p e r a t u r e in th e c o r e a n d

t h e h e a t e x c h a n g e r s o f a h e l i u m - c o o l e d r e a c t o r r e q u i r e k n o w l e dg e of s o m e

of t h e th e r m o p h y s i c a l p r o p e r t i e s o f t h e h e l i u m g a s , a n d th e s a m e a p p l i e s

t o t h e t r e a t m e n t o f o b s e r v e d d a t a i n h e a t t r a n s f e r e x p e r i m e n t s .

T h e p r o p e r t i e s t r e a t e d i n t h i s r e p o r t a r e th e d e n s i t y , t h e s p e c i fi c h e a t s ,t h e t h e r m a l c o n d u c t i v it y , a n d t h e v i s c o s i t y .

T h e g a s p r o p e r t i e s a r e o n ly k no w n w it h a c e r t a i n a c c u r a c y , a n d i t i s

t h e r e f o r e n e c e s s a r y t o s u r v e y t h e e xi s ti n g e x p e r i m e n t a l an d t h e o r e t i c a l d a t a

i n t h e l i t e r a t u r e t o e s t a b l i s h a s e t of r e c o m m e n d a b l e d a t a . T h e d a t a r e s u l t

i n g f r o m t h i s s u r v e y a r e p r e s e n t e d a s e q u a t io n s in o r d e r t o f a c i l i t a t e t h e

u s e o f c o m p u t o r s , a n d i t i s s h o w n th a t v e r y s i m p l e e q u a t i o n s s u ff ic e t o e x

p r e s s t h e d a t a w i th a m p l e a c c u r a c y . Tw o e q u a ti o n s a r e p r e s e n t e d f o r e a c h

p r o p e r t y , o n e i n t e r m s of t h e a b s o l u t e t e m p e r a t u r e , T , i n K e lv in a nd t h e

o t h e r i n t e r m s of t h e r a t i o T / T , T b e i n g t h e a b s o l u t e t e m p e r a t u r e a tz e r o d e g r e e s C e l s i u s , e q u a l t o 2 7 3. 16 K . T h e r a n g e s o f p r e s s u r e a n d

t e m p e r a t u r e c o n s i d e r e d f o r t h e d a ta a r e 1 t o 1 00 b a r f o r p r e s s u r e a n d 2 73

t o 1 80 0 K f o r t e m p e r a t u r e .

T h e P r a n d t l n u m b e r i s a c o m b i n a t i o n o f t h e p r o p e r t i e s w h i c h i s o f te n

u s e d in t h e r m o d y n a m i c c a l c u l a t i o n s , a n d f o r t h a t r e a s o n e q u a t io n s a r e a l s o

p r e s e n t e d f o r t h i s q u a n t i t y ,

2 . S U M M A R Y O F R E C O M M E N D E D D A T A F O R H E L I U M

A T 1 T O 1 00 BAR AND 273 T O 1 800 K

k g u n i t o f m a s s , k i l o g r a m

m u n i t o f l e n g t h , m e t r e

s u n i t o f t i m e , s e c o n d

N u n i t o f f o r c e , n e w t o n , k g * m / s

J u n i t o f e n e r g y , j o u l e , N * m

W u n i t o f p o w e r , w a t t , j / s

P p r e s s u r e , b a r , 1 0 N / m • 0 .9 8 6 9 p h y s . a t m

T a b s o l u t e t e m p e r a t u r e , K

T a b s . t e m p e r a t u r e a t 0 ° C • 2 73 .1 6 K



- 6 -

G as Law of Helium

P • 10"*2077.S* p • T * Z • (3-2),(3-3)

where p is the density, and Z is the compressibility factor:

Z = 1 + 0.4446 - £ , = 1 + 0.530 • 10"3 £» - » . (3-18)r-' {T/T O )'- ,J

The standard deviation, o , is about 0.03% at a pressu re of 1 ba r and0.3% at 100 bar, i. e. a - 0. 03 • VP%. (3-20)

Ma ss D ensity of Helium

P = 48.14 f [ l + 0.4446 - J 75 J (3-19)

• 0.17623 ^ [ , • 0.53 • l O - 3 - ^ ] " 1 ^ m \

The standard deviation is as for Z.

Specific Heata

c p = 5195. J /kg • K, (4-4)

c y = 3117. J /k ? - K, (4-5)

1 ' c p / ° v = K 6 6 6 7 - <4"

6>

The standa rd deviation, 9, i s at 273 K about 0.05% a t a pressu re of1 bar and 0.5% at 100 bar, d ecreasing to 0.05% at high temperature at a llpres su res, i . e . • « 0 .0 5 p ' 0 , 6 " " - ' ( T / T ^ ( 4 _ 7 )

C oefficient of D ynamic Viscosity

M- 3 .674- 10" 7 T ° ' T - 1 .865- 10" 5 (T /T o ) 0 ' 7 k g /m-S . (6 -1 )

The standard deviation, t, ia about 0.4% at 273 K and 2.7% at 1800 K,i . e . e • 0.0015 T%. (6-3 )



- 7 -

Coefficient of Thermal Conductivity

k « 2.682 • 10" 3(1 + 1.123 - 10" 3 P ) T < 0 - 7 1 ( 1 " 2 ' 1 0 P J ) ( 7 - 1 )

= 0.144(1 + 2.7 • 10"*P) (T/T o )( 0

-7 1

0 "2

" '<>" ?)) w / m . JJ.

The standard deviation, o, is about 1 % at 273 K and 6% at 1800 K ,i . e . o = 0.0035 T%. (7-4)

Prandtl Number

Pr = c p {

1 +

, 0.6728 ( T . T j - (0 .01 - 1 .42 • 10~ 4P)

1 + 2 .7 • 10 " 4 P °

p,. . 0.711 7 T - (0 .01 - 1 .42 • 10" 4 P ) ( 8 _ , }

1 + 1.123 • 10" 3 P

The stand ard deviation is o • 0.004 T V (8-2)

3. GAS LAW

The id ea l ga s law or the id ea l equation of sta te is the p- v-T relationfor an idea l ga s:

p • v = K • T (3-1)

2p, pressure , N/mQ

T, specif ic molal volume, m /kg-moleR, u niversal ga s constant, 8314.5 J /k g-m ole • KT, a bsolute temperature, K.

For a noble gas such as helium the real p-v-T relation deviates littlefrom the real gas law at the pressure and the temperature in a reactor

core. The deviation increa ses numerically with pres su re and d ecrea seswith tempera tu re. If the idea l gaa law ia applied to helium at a p ress u reof 60 bar and a temperatu re of 350 K, the deviation from the real gas lawi a l . S H .

In calculations where »uch d**tttkm» a rt allowable, tta id W '! •* * * * - 'can therefore kV applied; Ttte wtottAd ar weight d f bettts* t» 4V » w » Y a t f «



- 8 -

this g ives the fo l lowing gas law for hel ium when considered an ideal gas:

„„„ P - ' ° S ,„ * J (3 -2 )2077.3 • p • T l '

P , pr ess u re , b ar (1 bar = 10 5 N/m = 0 .9863 phys . a tm)p , m a s s d e ns it y , k g /m .

In some ca lcu la t ions i t might , however , be des irable to operate wi th

an express ion which g ives a greater accuracy .

This is norm a lly done by introdu ct ion of the com pr ess i bi l i ty fa ctor , Z,

in the gas law:

irrr = z • < 3 - 3 >

The com press ib i l i ty fac tor , Z , can be d e termined by cons id era t ion o f

the virial equation of state.

An equat ion o f s ta te for a gas i s a re la t ionship be tween the charac ter

i s t i c force , the cha ra c ter i s t ic conf igura t ion and the tempera ture . For an

ide a l g a s t he pr e s s u r e 1 B the only cha ra c ter i s t ic force , the vo lum e be ing the

charac ter i s t ic conf igurat ion , but o ther forces must be taken into accountfor a rea l ga s i f the pr ess u re is not ve ry low. Th is can be d one by ad d ing

force -descr ib ing terms to the idea l equat ion o f s ta te .

Vir ia l equat ions are power expans ions in terms o f vo lume or pressure

respec t ive ly

p . v = A(l + £- + Sj . . . ) (3-4)v

2p - v = A + B - p + £ i 2 - p 2 + ( 3 - 5 )

The coe f f i c ients A , B , C and so on a re nam ed v ir ia l coe f f i c ients , i . e .

they a re forc e coe f f i c ients . The v ir ia l coe f f i c ients a re obta ined exp er i

menta l ly by f i t t ing o f the power ser ie s to measured i so therms o f the gas .

This g ives a d i f f erent se t o f coe f f i c ients for each I so therm, and thus the

v ir ia l coe f f i c ients a re tempera ture -d epend ent .

Equat ion (3 -5 ) converges l e s s rapid ly than eq . (S -4 ) and thus requiresmor e terms for equa l pre cis io n. Thia ie of minor importan ce for d ensity

d etermina tion, for which only the f ir st corr ect ion t(r m , the secon d v iria l

coef f icient , B, U »ignif lcant except at extre m ely high pr sa su r* . The

v iria l equation o f Sta t* su f f i c ient for dans i ty ca leu la i ioa o f U jht noble | i u u



- 9 -

i s then:

p - v " A + B • p . ( 3 - 6 )

A s y s t e m o f u n i t s o f t en u s e d in t h i s e qua t ion i s t he A m a g a t u n i t s . I nt h i s s y s t e m t h e p r e s s u r e p a i s in a tm, and the vo lume v Q i s e x p r e s s e d i n

t he no r m a l m o la l v o lum e a s a un i t , w r i t ing

p a - v n = A + B - p a = A Q £ + B - p a , (3 -7 )o

A i s t he r a t i o o f p • v a t z e r o p r e s s u r e t o p • v a t t he pr e s s u r e

of 1 a tm , or

A 0 - 1 - B 0 . ( 3 - 8 )

B i s the c o e f f i c i e n t B a t 0 ° C , m e a s u r e d t o be be t w e e n 0 . 5 • 1 0 "° - 3

and 0 . 54 • 10 . Equ at ion (1 -7 ) b ec om es

P a * v n ° 0 - B0

) £ + B -P f t

, ( 3 - 9 )o

and eq . (1 -9 ) ca n be rea rra nge d to g iv e the fo l lowing form :

P. • »_ • TK a n io

O n t he l e f t - ha nd s i d e the c o r r e c t ing t e r m , B Q , s ho u ld no t be o m i t t e d ,

bu t o n t he r ig ht - ha nd s i d e i t i s ne g l i g ib l e , s o

p • v • T T

E q . (1 -9 ) then b ec om es

Z - 1 + -$• B • p a . ( 3 - U ) k

It 1« obvious ilml i£ U can be expressed wi th su f f i c ient accuracy by

B • B, (-£)". ( » - ' 3 )



- 10 -

B . being a constant , then a very s imple equat ion for Z is obtained:

Z = 1 + B , • p a ( , j i - ) . (3 - 14)

In the fo l lowing such an equat ion is establ ished, and by means o f this as imple equat ion for the density is der ived.

Reported measurements o f i so therms o f he l ium are most f requent ly

presen ted with the second v ir ia l coef f ic ient , B, a s a funct ion of temp era tu re .

The la te s t m ea su rem ents , Stroud , M i l ler and Brand t , 1960 , how ever ,

g ive Z d irect ly , but at about room tem pera tu re only . I t i s notab le that

(Z - 1) is found to be s tr ict ly proport ional to the pressure over the ent ire

pr ess u re range invest iga ted , 10 to 275 atm . Fo r com pa rison with other

measurements the resu l t s are conver ted into va lues o f B by means o f eq .(1 -1 2) and plot ted in f ig . 1 toge ther wi th o lde r mea su rem ents a s fo l low s .

The shaded area in f ig . 1 represents measurements carr ied out unt i l

1 9 4 0 , d i s c us s e d by K e e s o m , 1 942 , who adopts the do t ted l ine as representa

t ive . Mea surem ents reported by Schne ider and D uf f ie , 1 949, and Yntema

and Schne ider , 1950 , a re a l so inc luded .

Although the reproducibi l i ty for each invest igat ion is within 1 %, the

resu l t s dev ia te by 4% at room tempera tu re . Bea r ing in mind , how ever , thatth i s correspond s to d ev ia t ions o f only 0 .3% in Z a t 100 a tm , i t s ee m s m os t

l ike ly that such deviat ions can occur between two di f ferent invest igat ions or

from one method o f measurement to another .

The two cu rves b and c in f ig . 1 a re the resu l t s of ca lcu la t ion s ba sed

on intermolecu lar potent ia l s , a s wi l l be dea l t wi th la ter .

The cu rve adopted for this work is the fu l l l ine in the f igu re . This fu nc

tion is

B • 0 . 5 3 7 - 1 0 " 3 ( £ ) - 0 ' 2 . (3 -15 )o

Eq. (3-14) is then

Z - 1 + 0 . 5 3 7 - 1 0 - 3 p a ( ^ . ) - K Z . (3 -16 )o

W ith -the pr ess u re , F , in bar the ad opted funct ion is

Z ' 1 + 0 .53 - 10" 3 F . H . (3-17)( T / T / 7 ^



O 200 400 600 800 1000 1200 M O O 1600

• ' i 1 I I I i

-200 0 200 400 600 BOO 1000 1200

' ' i Imperat ive *C

F i g . 1

a A do pte d t y K e o a o m , 1 M 2

b Ma eon an- i R ie* , 1 9M . "Exp -8" potent ia l

c A m du r a id Ma s o n , 1 958. Inverae power pot .

A Ad opted for thi« work , B • 0. J37 - 1 0" 3(T/T ( )J a f e '*



- 1 2 -

Then the equat ion of s tate (eq. (3 -2)) for hel ium becomes

mi.*?,, it •« + o.w. 10_s

rø~rrr • <3

" '8

>

Ma s s De ns i t y

Rearrangement o f eq . (3 -18 ) g ives the dens i ty o f he l ium.

p = 48 .14 f [ ' + ° - 4 4 4 6 T T s ] < 3 " 1 9 '

" ° -, 7 6 2 J

TT7T? [ '+

o -53*

,0~

3^ T 7 ? ] "

1W™-

P , p r e s s u r e , b a r ,

T , a bs . t emp eratu re , K ,

T Q - 273 .16 K .

I f the f i r s t -order term a lone i s used , and the condi t ions are l imi ted to

t e m pe r a t ur e s a bo v e 5 0 0 K a nd pr e s s u r e s up to 6 0 ba r , t he a c c u r a c y i s

better than 1 .5%. Fo r the fu l l equa t ion the s tand ard d eviat ion, » , i s abou t

0 .03% at a pressure o f 1 bar and 0 . 3% a t 100 ba r , i . e . c = 0 . 03 / P % .

( 3 - 2 0 )

Va lues of p ca lcu la ted from eq. (3-1 9 ) a gre e very c lo se l y wi th the

va lu es ca lcu la ted and tabula ted by W i l son, 1960 . C heck s o f th i s agr eem ent

were made at 1 , 20 and 100 atm, at 40 °F and 1 6 0 0 ° F . T he m a x im um

de v ia t io n w a s 0 .2 % . C he c ks of a g r e e m e nt w e r e a l s o m a d e w i th v a lu e s

tabula ted by Hol ley , W orl ton and Zi eg le r , 1959 , a nd the se showed d ev ia

t ions up to 0 .5% at 100 ba r . An a na lys i s of the report d i sc lo sed , however ,that the authors were aware o f the dev ia t ion from measured va lues a t that

pr ess u re , and that the dev ia t ion may have been a l lowed wi th the a im o f

e s ta b l i s h ing a f o rm u la w hic h c o v e r e d p r e s s u r e s up to 1 0 00 a t m .

4 . SPECIFIC HEATS

At the zer o condi t ion , p - • 0 , the mo la l spe c i f i c h ea ts , C a t cons tant

p r e s s u r e a n d C y at cons tant vo lume , for an idea l monatemic gas wi th themolecular we ight M can be shown to be



- • a -

C = 2 . 5 R / M J / k g - m o l e - K . ( 4 - 1)

C y = 1 . 5 R / M J / k g - m o l e • K . ( 4 - 2 )

T h e r a ti o C ^ C y i s c o r r e s po nd ing ly

Y = V C v * 5f3' (4"3)

F o r a r e a l g a s t h e s p e c i f i c h e a t s c a n b e d e t e r m i n e d f r o m m e a s u r e m e n t s

o f t h e i s e n t h a l p s a nd C f r o m d i r e c t m e a s u r e m e n t s , w h i le Y c a n b e d e r i ve d

ind i r e c t l y f r o m m e a s u r e m e n t s o f t he s pe e d o f s o und . In t he c a s e of he l i u m

s u c h m e a s u r e m e n t s d o n o t a p p e a r t o h a v e b e e n p e r f o r m e d w i t h a n a c c u r a c y

be t t e r t ha n t he a c t u a l d e v i a t i o n f r o m t he a bo v e - m e nt io ne d b a s i c f i g u r e s , t hed e v i a t i o n s b e i n g v e r y s m a l l .

C , C a nd Y c a n a l s o be d e r iv e d f r o m t he v i r i a l equa t io n o f s t a t e .-V ~ v > 2 D / ! 1 _ 2In s u c h e x p r e s s io n s t he qu a n t i ti e s de pe nd o n d B /d T a nd a B / d T , a nd

t he s e de r iv a t i v e s w i l l d i f f e r s ubs t a nt i a l l y f r o m o ne f unc t io n t o a no t he r f o r

t he s e c o nd v i r i a l c o e f f i c i e n t , B , s ho w n in f i g . 1 . C a l c u l a t i o ns ind i c a t e t ha t

t he de v i a t i o ns f r o m t he ba s i c f i g ur e s a t 1 0 0 ba r a r e a bo ut 0 . 5 % a t r o o m

t e m p e r a t u r e a nd a bo ut 0 . 0 5% a t 1 0 00 K . Suc h d e v i a t i o ns a r e ins ig n i f i c a ntf o r e n g i n e e ri n g p u r p o s e s f o r w h ic h r e a s o n C , C a n d Y c a n b e r eg a r d e d

a s c o n s t a n t s a n d t h e d e v i a t i o n s r e g a r d e d a s t h e s t a n d a r d d e v i a t i o n s .

I n pr a c t i c a l un i t s t he s pe c i f i c he a t s o f he l i um a r e :

( 4 - 4 )

( 4 - 5 )

c - 519 5.

c y - 3117 .

a nd

• • * - cp /

cv '

J / k g • K ,

J / k g • K ,

1 . 6 6 6 7 . ( 4 - 6 )

T h e s t a n d a r d d e v i a t i o n i s a - 0 . 0 5 p ' 0 - 6 " °" ' * T ' T o % .

5 . C ORREL ATION FORM U L AE FOR VISCOSITY AND C OND UC TIVITY

For the purpose o f the fo l lowing par t o f the report , which i* to e s tab-l i eh the mol t rea l i s t i c va lues for the transport proper t ie s : v i scos i ty and

therma l cond u ct iv i ty . It » ou ld be qui te sa t i s fa c tor y only to exa min e the

e x pe r im e nt a l da ta if s u f f i c i e n t a nd r e l i a b l e da t a w e r e a v a i l a b l e . A ( r s a t

m a ny m e a s ur e m e nt s o n t he s ub j e c t a r e r e po r t e d , m o s t o f t h* m < * t t e m pe r a

ture* be low 1 000 K, and only a f ew a t t l lgner temperatur«« . *J>» tesi^Hs,



- 14 -

h o w e v e r , d e v i a t e m u c h m o r e fr o m o n e I n v e s t ig a t i o n t o a n o t h e r t h a n t h e

c l a i m e d a c c u r a c y o f e a c h i n v e s t i g a t io n a c c o u n t s fo r , f o r w h i c h r e a s o n i t I s

n e c e s s a r y t o p e r f o r m a j ud g e m e n t i n o r d e r to s e l e c t th e m o s t p r o b a b l e

v a l u e s . F o r g u i d a n c e in p e r f o r m i n g s u c h j u d g e m e n t a n d e s p e c i a l l y fo r

j ud g e m e n t of t h e v a l u e s a t h i g h e r t e m p e r a t u r e s i t i s r e l e v a n t t o e x a m i n e

t h e r e s u l t s e v a lu a te d b y m e a n s o f t h e s t a t i s t i c a l m e c h a n i c a l t h e o r y o f g a s e s .

I t m u s t , h o w e v e r , b e p o i n t e d o u t t h a t e ve n a d v a n c e d t h e o r i e s c a n n o t a t

p r e s e n t p r e d i c t v a l u e s of |i a n d k u n l e s s s o m e v a l u e s a r e a l r e a d y k n o w n

f ro m e x p e r i m e n t s . T h e t h e o r e t i c a l f o r m u l a e d e r iv e d i n s u c h a w a y c a n b e

u s e d t o i n t e r p o l a t e b e tw e e n t h e k no w n v a l u e s , b u t c a n n o t i m p r o v e t h e a c

c u r a c y de t e r m i n e d by t h e e x p e r i m e n t a l v a l u e s , a n d e x tr a p o l a t i o n to h i g h e r

t e m p e r a t u r e s i s bo un d to b e v e r y u n c e r t a i n s i n c e w h o l e f a m i l i e s o f fu n c

t i o n s c a n b e b r o u g h t t o fi t t h e m e d i u m t e m p e r a t u r e d a t a , g i v in g g r e a t l y

d if fe r in g e x t r a p o l a t e d v a l u e s a t h i g h t e m p e r a t u r e .

In t h e f o ll o w in g a b r i e f a n d n o t a t a l l c o m p l e t e e v a l u a t i o n o f t h e d e p e n

d e n c i e s of t h e v i s c o s i t y , c o n d u c t iv i ty a n d F r a n d t l n u m b e r o n t h e p r e s s u r e ,

t e m p e r a t u r e a nd m o l e c u l a r q u a n t i t i e s i s o u t l i n e d .

T h e c o n d u c t iv i t y a n d t h e v i s c o s i t y of t h e g a s d e s c r i b e t h e o v e r a l l t r a n s

p o r t r a t e s o f h e a t a nd m o m e n t u m t r a n s f e r r e d w i th i n t h e g a s b y m o l e c u l a r

t r a n s p o r t .

T h e c o e f fi c i e n t o f v i s c o s i t y , a , i s d e fi n e d b y t h e e q u a t i o n f o r t h e

m o m e n t u m c u r r e n t d e n s i t y , t h e a m o u n t o f c o n v e c t i ve m o m e n t u m p a r a l l e l

t o t h e y - a x l s t r a n s f e r r e d p e r un i t t i m e a c r o s c a u n i t a r e a p e r p e n d i c u l a r t o

t h e d ir e c t i o n in wh i ch t h e c o n v e c t i v e ve l o c i ty c h a n g e s , t h e x - a x i s ,

, uv

i • - u - f ^ . ( 5- 1)

T h e c o e f f i c i e n t o f c o n d u c t i v i t y , k , i s d e fi n e d b y t h e l a w ot F o u r i e r a s

q = -k - g . (5-2)

q i s th e e n e r g y c u r r e n t d e n s i t y d u e t o t e m p e r a t u r e g r a d i e n t , i . e . t h e

e n e r g y w h i c h c r o s s e s p e r u n i t t i m e t h r o u g h a u n i t a r e a p e r p e n d i c u l a r t o t h e

d i r e c t i o n i n wh i ch t h e e n e r g y f l o w s , t he x - a x i s .

T h e s i m p l e s t m o d e l t h a t i s a p p l i e d f or t h e o r e t i c a l c o n s i d e r a t i o n s o f t h et r a n s p o r t p r o p e r t i e s of g a s e s l a th e s o - c a l l e d b i l l i a r d - b a l l m o d e l , t h e m o l e

cu le s be ing regarded as r ig id spheres wi th the d iameter « , mov ing free ly

among each o ther .

F r o m t h e d e f i n i t i o n a l equat ions < 6-1) and (5-2) it follows that



- 1 5 -

1 - d u v 1] - - y u u v V - j J . g i v in g u * • n m v X ,

a nd

« " - 3 »C

m o l 'X

l i 8 iv i n

ek

" 7n C

m o l *X

-

w h e r e n i s the num be r o f m o le c u l e s pe r un i t v o lu m e , m t he m o le c u l a rm a s s , C j the spec i f i c hea t per m ole cu le , v the spe ed which i s propor -t i ona l t o Y k T /m , a nd X. t he m e a n f r e e pa th pr o po r t io na l t o l / ( n x o ) . k i s

B o l t i m a n n ' s c o n s t a n t .

T he pr o pe r t i e s c a n t he r e f o r e be e x pr e s s e d a s

/ s m k T ( 5 - 3 )

% (5-4)

T he r ig o r o us t he o r y f o r r ig id - s phe r e m o le c u l e s pr e d i c t s t ha t t he

va lu es o f a and p° a re

_ 5 . - _ 25a - n and P - •JJ .

As a resu l t o f th i s theory inser t ion o f a and p g ives the fo l lowing

e qua t io ns , e x pr e s s e d in pr a c t i c a l un i t s ( o i s in A ) :

a « 2 . 6 6 9 3 - I 0 " 6 H ^ k g / m • s , ( 5 - 5 )

k = 2. 5 |i • c y W / m • K . ( 5 - 6 )

From this i t fo l lows that the Prandt l number i s

» • c 2c ,

V

These express ions show that bo th v**cu8 i ty and conduct iv i ty mm e a s e

w i th t e m p e r a t u r e a s YT" a nd a r e inde pende nt o f t he pr e s s u r e . E x p e r im e nt ss h o w t h a t t h e c o n d u c t i v i ty i* s l i g h t l y d e p e n d e n t o .i p r e s s u r e , w h i c h c a n b e

v e r l i i e u by a m o r e e i a b o m t * t h e o r y , a r,d t h a t t h e t e m p e r a t u r e d e p e n d e n c e

i s a c t u a l l y c o n B i de r a b ly g r e a t e r . F o r explanat ion of t h i « a m o r e a dv a nc e d

m o l e c u l a r m o 4 * l &u*\ be axAgjined, taking t h e j u t e n n o l o c v i a / f o r c e ^ n t o ,

c o n s i d e r a t i o n .



- 16 -

T h e u s u a l a p p r o a c h t o t h e p r o b l e m a s s u m e s a n a n a l y t i c a l f o r m f o r t h e

i n t e r m o l e c u l a r po t e n t i a l , a n d a f t e r t h e n e c e s s a r y c a l c u l a t i o n s th e r e s u l t i s

c o m p a r e d w i th a v a i l a b l e d a ta f o r s e l e c t i o n o f t h e p a r a m e t e r s o f t h e p o t e n t i a l

i n o r d e r to v e r if y t h e a s s u m e d m o d e l . S uc h a p o t e n t i a l i s s k e t c h e d i n f ig . 2 .

v ( r ) i s t h e p o t e n t i a l e n e r g y of t wo m o l e c u l e s a t a s e p a r a t i n g d i s t a n c e , r , c

i s t h e d e p t h o f t h e p o t e n t i a l e n e r g y m i n i m u m , r i s t h e v a l u e o f r f o r t h e

m i n i m u m , a n d o i s t h e v a l u e o f r a t w h i c h ? ( r ) i s z e r o .

F i g . 2

P o t e n t i a l e n e r g i e s o f a p a i r of h e l i u m a t o m s

a s a fu n ct io n of t h e s e p a r a t i n g d i s t a n c e , r ,

b e tw e e n t h e c e n t r e s o f t h e tw o a t o m s .

T h e g e n e r a l e qu a ti o n fo r t h e v i s c o s i t y d e du c e d f r o m a n i n t e r m o l e c u l a r

p o t e n t i a l f un c t i o n i s

, , - 2 . 0 6 9 3 . V S"6

g V ! , , V ^ (5-8)

« 5 . 34 • 10 n

where M i s inser ted wi th i t s va lu e for he l ium , 4 .0 0 3 . Fu r ther



- 1 7 -

T i s t he r e d u c e d t e m p e r a t u r e , T k / « ,1 6k , Bo l tzm an n's cons tant = 1.3805 • 1 0" e r g /K ,

t , the potent ia l pa ra m eter , erg , and

a i s the potent ia l pa ra me ter , A .

A s i t w i l l be s e e n u . i s de r iv e d f r o m t he v i s c o s i t y pr e d i c t e d by t he

r igorou s r ig id - s ph er e theory by d iv i s ion by the co l l i s i on integ ra l f l' ' ' (T*) .

T he r e qu i r e d f o r m ula e f o r c a l c u l a t i o n o f t he c o l l i s i o n in t e g r a l s a r e

d i s c u s s e d by H ir s c h f e ld e r , C u r t i s s a nd B ir d , 1 9 5 4 , "M o le c u l a r T he o r y o f

G a s e s a n d L i q u i d s " .

In the 3rd approx imat ion, which i s normal ly used , the coe f f i c ient o fv i s c o s i t y i s g iv e n by

H3 = f , ^ • (5 - 9)

Corresponding ly the coe f f i c ient o f thermal conduct iv i ty i s g iven by

, ( 3 )

k

3 *

2

-

5

-

c

v - * 3 7T3V

( 5

-

, 0

>|i

Th e ra t io f.* ' / f ' ' i s c l o se to u nity .

T he e qua t io ns a r e no t c o r r e c t e d f o r qua ntum e f f e c t s . Fo r he l i um s u c h

a c o r r e c t i o n w o u ld a m o unt a t t he m o s t t o 0 . 3 % a t r o o m t e m pe r a t u r e a nd be

v a n i s h ing a t h ig he r t e m pe r a t u r e s , a nd i t c a n t he r e f o r e be ne g l e c t e d .

T a b le s f o r t he c o l l i s i o n in t e g r a l a nd t he f a c t o r s I a nd f ^ a r e r e a d i l y

ava i lable for many potent ia l func t ions .

Fo r t he pr e s e nt w o r k a v e r y s im ple f unc t io n i s c ho s e n f o r c o r r e l a t i o no f the exper imenta l da ta o f the v i scos i ty :

u - a - T b . (5 -11 )

T he c o nduc t iv i t y da t a a r e c o r r e l a t e d w i t h a s im i l a r e qua t io n .

Such an equat ion i s deduced from the inverse power potent ia l func t ion

, ( r ) . e ( o / r ) \ (5- 1 i j

for which the coll is ion integral 1«;



- 18 -

The v iscos ity is then:

- 6 T ' / » T ^= 5. 34 • 10

- e f t / « ) 2 / 6T d / 2 + 2 / 6 )

= 5 . 3 4 • 1 0 " iS i« ^ ± _ _ . . ( 5 - 1 4 )o (»)

The quantity W . .* only depends on o . I ts va lu e ha s been ca lcu late d by

s e v e r a l a u t hor s, l a t e s t by L e Fe v r e , 1 958 . It i s u nity at b = oo, and 1.0557

a t 6= 2.

As wil l be seen, the equat ion has the form of eq. (5 -11) , p s a * T .

For most gases th i s i s no t a very good corre la t ion formula for thetransport prop ert ies at about room temp era tu re , while i t i s v ery good a t

high temperature . Fo r he l ium , however , cha ra c ter ized by i t s ex trem ely

sm a l l a t trac t ive intermolecu lar forc es , the equat ion i s su per ior to mos tothers as wi l l be demonstrated in the next sect ion.

As an i l lustrat ion of the development in the knowledge of the transport

propert ies o f he l ium through the years , f ig . 3 is presented.

Curve I represents measurements o f the v i scos i ty o f he l ium per formed

up to 1 940, su rveyed by K eesom, 1942. C u rves 2 and 3 a re pred ic ted ex tra po la t ions ca lcu la ted by Amdur and Mason, 1958 , and Lack and Em m ons ,1965 , respec t ive ly . C urve 4 i s the resu l t o f the presen t l i t era tu re sur veywhich wi l l be d i scu ssed la ter . As wi l l be se en from cu rves 1 and 4 there

i s a t rend to accept h igher va lues o f v i sc os i t y in recen t ye a rs . C urve 5

represents measurements o f the thermal conduct iv i ty o f he l ium t i l l the

beginning of (he f i f t ies reported by Hilsenra th and Tou louk ian, 1954 . Amd ur

and Ma son (curve 2) k eep the Pra nd t l nu mber consta nt at 0 .66 6, and L ienand Emmons (curve 3) take the Prandt l number equal to 0 .6718 for the

temperatu re range shown in the f igure . C u rve 6 wa s ca lcu la ted by Mann,1960, on the bas i s o f measurements carr ied out by Mann and Bla i s , 1959 .

C urve 7 i s the resu l t of the present work . As in the ca se of v i s cos i ty ther ehas been an increase in the accepted va lues o f thermal conduct iv i ty through

the years , but the two increases have not been s imul taneous , and th i s has

caused much trouble for those try ing to f ind an intermolecular potent ia l

that might su it v is co s it y as w el l a s condu ct iv ity d ata . In the fou rt ies therat io o f v i scos i ty to condu ct iv i ty wa s incr ea s ing wi th tempera ture , and th i sindicated that the Pra nd t l num ber would in cr ea se wi th temp era tu re . In the

f i f t i e s h igher va lues o f conduct iv i ty data were reported , indica t ing decreas

ing Prand t l nu mb ers . However , mo re recen t me a su rem ents of the v i sc os i ty



273

« • Ab *, fempcrotur* T K

500 1000

- 1 9 -

2000 5000 8000

273 50 0 1000

* Ab». Ttmperotur* T K

Fig . 3

2000 5000 • 0 0 0

V is c o s i t y a a d c c nda c t t v i t y o f he l i um a t 1 b a r

1 - K t t s o m , 1 942

2 - Am du r aod M ason, I 958

- L i c k a a d Fm m o r « , 1 965

th t» » « * . • • 1. » 5 • f e " 5 ( T / T 0 ) 0 - 7 0

HLUenrata aad Ton' .oqkian, I 964

II

7 - tb> røfc, * tt , 1 4 * W / T 0 ) '0.71



- 20 -

over an extended temperature range show good agreement with the theory

when treated together with newer conductivity data as pointed out in an

ar t ic le " D iscrepanc ies Between Visc os i ty D ata for S imple G a ses" by H. J . M.

Hanley and G. E. C hi lds , 1968.

6. VISCOSITY D ATA

The following equation, which is of the form of eq. (5-11), is adopted

for the present work as being the best express ion for the v iscos ity data o f

hel ium:

H = 3.674 • 10~ 7 T 0 - 7 « 1.865 • 1 O"5 ( T / T o ) 0 - 7 k g / m - s ( 6 - 1)

Fig. 4 is a d evi3tion plot ba sed on this equ ation. The grea t nu mb er ofmeasurements conducted in the course o f t ime are not shown in the deviat ion

plot , s ince i t i s fe l t that the reveiws c i ted incorporate a l l known measurements in such a way that de ta i l ed cons iderat ion o f the o lder indiv idua l meas u rements see m s super f luous . Esp ec ia l ly the an a lys i s under taken by pro fe s

sor J . K est in, o f Brown U nivers i ty , U. S. A . , es ta bl is hes the v is cos ity a troom temperature and some hundred degrees above wi th a great accuracy .

Therefore only the fo l lowing observat ions and interpolat ion formulae areincluded in f ig , 3 , one o f them, the o ldest , most ly for his tor ical reasons .The numbers refer to the numbers in the graph.

1 - K e e s o m , 1 942 , corre la te s the measurements per formed t i l l that

time at temperatures up to 1100 K with the equation:

u = 1 .894 • 10~ 6 ( T / T 0 ) 0 - 6 4 7 k g / m • s .

2 - Mason and Bice , 1 954, correlate the measurements t i l l then withthe "Exp-6" potent ia l and f ind the parameters:

a = 12 .4 , r m = 3 . 1 3 5 Å a nd « / k « 9 . 1 6 K ,

and compare them with the Lennard-Jones (6 -12) potent ia l parameters:r m * 2 - 3 6 9 Å and t /k • 10 .22 , d e termined by d e Boe r and M iche ls , 1939 ,

and revised by Lundbeck, 1951, compiled by Hirschfe lder , Curt ies andB ir d , T 934. Both poten tials were shown to give good agre em ent to secondvirial coefficient« and to the data of the viscosity of helium available at thatt ime .



^

" - ^\fø*

V

\

"*5

^ 2

7 * ^

^

<

— * 3 ~

V

\

273400 600 800 1000 1200

T K

1200 KOOT K

1600 1800 2000

F i g ,

D yna mic v i s cos i ty »i o f He lium a i 1 b a r

De v ia t i o n« in % trom n » 1 . 8 6 6 10"? ( T /T 6 ) 0 - 7

1 - K e e s o m , 1 S42

2 - Ma s o n a n d R i c e , ! SM

3 - Ma nn, I l 6 0

i - K w t t n W U J i d e n f r b it , 1 » 54

5 - D I P ip p o a n d K e s t i n , 1 96 8

6 - G u e v a r a , M c l n ta e r a nd W a j e m a n ,

7 - K. : lka r aa 4 K H U C , » n »



- 22 -

The two mentioned intermolecular potent ia l funct ions have the fo l lowing

exp ress i on s . The L enna rd -J ones (6 -1 2) potent ia l i s the (6 -1 2) form of the

(6-n) family o f potent ia ls .

"Exp-6" potential:

^) = r r ? o M [ ! e - ( - ( ' - t ) ) - ( ^ ) 6 ] ' r>r—

»w * ». r<rmaX' >

6-

2>

where a i s a d imens ionless parameter .

The (6 -n) family:

, ( r ) = c - J L g . ( » ) 6 / C - 6 ) [ (£)" - ( | ) 6 ] . , 6 - 3 )

Fo r the notation se e f ig . 2 .

3 - Mann. 1 960 , deduc ts the v i scos i ty f rom measurements o f thermalconduct iv ity carr ied out by Mane and Blais , 1 959 , a t h igh temperatures .

1 200 - 2000 K, and uses the exponential potential:

, ( r ) = £ • e

o ( 1 _ r

/

r

c > . ( 6 - 4 )

This potent ia l ma y g ive the best f it to experimen tal v i sco s it y d a ta . An

alternat ive form, employed by Amdur and Mason, 1 958, and by Monchick,

1959, i s

<P(r) = e o • t " r / p .

These potent ia ls unfortunate ly do not g ive a s imple equat ion for theviscos ity suitable for engineering purposes as does the inverse powerpotential chosen for this work.

4 - Kest in and Leidenfrost , 1 959 , per form measurements by means o f

the osc i l la t ing d i sk method. The resu l t s are ma rked wi th ova l s . Theycorre la te the data toge ther wi th ex i s t ing measurements a t h igher temperatu res with the "Exp-6" potentia l obtaining the pa ra m eters : o • 1 2 .4 , r •

3 .225 A and s /k - 6 .482 K. For the L eona rd -J ones pa ra meter they f ind:

r • 2 .482 and « / k - 69 .08 .m ' -

5 - D iPippo and K eet in, 1968, re f ine the osc i l la t ing disk method tos t i l l greater accuracy , a prec i s ion o f 0 .05%, and obta in the resu l t s markedwith c i rc le s . They cor rela te with the "Exp-B" potent ia l , w ith the pa ra m eters :



- 23 -

a = 1 2 . 4 , r m = 2. 7254 and c/k • 36 . J 2 . F o r t he L en n a rd - Jo n es p o t en t ia l

t he y d e t e r m i n e th e p a r a m e t e r s r = 2 . 4 2 2 0 a nd c / k = 8 6 . 2 0 .

6 - G u e v a r a , M c l n t e e r a n d W a g em a n , 19 6 9 . T h e s e v i s c o s i t y m e a s u r e

m en t s w ere p er f o rm ed b y t h e ca p i l l a ry t u b e m et h o d i n t h e t em p era t u re ra n g e

1100 to 2150 IC re l a t i v e t o t h e v i s co s i t y a t t h e re f eren ce t em p era t u re , 2 8 3 K .T h e m ea su rem en t s a re o f g rea t r ep ro d u c i b il it y , 0 . 1 %, a n d a c cu ra cy , 0 .4 %.

7 - K a l e lk a r a n d K e s t i n , 1 9 7 0 . In t h e s e m e a s u r e m e n t s t he t e m p e r a

tu re ra nge o f the osc i l l a t ing d i sk method i s extended to 1100 K. The r es u l t s

a re r ep re sen t e d b y a ( 6 - 9 . 5 ) p o t en ti a l m o d e l w it h t h e p a ra m e t ers o = 2 . 2 15 A

a n d s / k = 7 3.2 1 K .

Th e d ev ia t ion p lot , f ig . 4 , ind ica tes tha t the s t and ard dev ia t ion o f the

v i sco s i t y d a t a i s a b o u t 0 . 4 % a t 2 7 3 K a n d 2 . 7% a t 1 800 K, i . e.

o = 0 .001 5 T%. (6 - 5)

The potent ia l parameters appl ied for f ig . 4 are l i s t ed in the fo l lowing

ta b le . Only the pa ra m ete rs of potent ia l s for which c and a h a v e t h e p h y s

i ca l m ea n i n g a scr i b ed t o t h em i n f i g . 2 a re t a b u l a t ed , a n d t h i s h ere a p p l i e s

to the "Ex p-6" and the (6 -n ) pote nt ia l s . The exponent ia l and the inv er se

p o w er p o t en t i a l s d o n o t d e scr i b e a t t ra c t i v e i n t erm o l ecu l a r f o rces , a n d f o rt h i s r ea so n c e t c . f o r t h e se p o t en t ia l s h a v e m a t h em a t i ca l s i g n i f i c a n ce

on ly. In the ta ble r m i s r ep l a ced b y o , t h e ra t i o ° / r i s 0 . 8 7 9 2 f o r t h e

"E xp -6 " , o « 1 2 .4 , p otent ia l and 0 .89 09 for the (6 -12 ) potent ia l

P o t e n t i a l

" E x p - 6 " , o * 1 2* 4

"i t

(6-1 2)i i

_ n

( 6 - 9 . 5 )

R e f .

a b o v e

2

4

5

2

4

5

7

S o u r c e

Ma son and R ice , 1954

K es t i n a nd L e i d en f ro s t , 1 9 5 9

D i P ip p o a n d K es t i n , 1 9 6 9

M a so n a n d R i ce , 1 9 5 4

K es t i n a n d L e i d en f ro s t , 1 9 5 9

D i P ip p o a n d K es t in , 1 8 6 9

K a l e l k a r a n d K es t i n , 1 9 7 0

c / k K

9 . 1 6

6 . 4 8

36 12

1 0 . 2 2

6 9 . 0 B

8 6 . 2 0

7 3 . 2 1

9

o A

2 . 7 5 7

2 . 8 3 6

2 . 3 9 6

2 . 6 5 6

2 .211

2 . 1 5 8

2 . 2 1 5

T h e t a b l e in d i ca t e« t h » i ' f i e n ew er t n « a » u r* » « « w o f to r vitaomity at ,

h cU u m ca n b e rep resen t ed b y t h e p o t en t i a l * a en U en ed or,\jiljmit*±y*)f

h igh # a l u « of f / k a r e u s e s , 'J h » w * * » s tt rv * n k r M o f 4 u t e c « * M O t r .



- 24 -

pared with the older values of about 1 0 K, which a re , however , based on a

mu ch wid er temperatu re ran ge than the newer on es . Bu ckingham, D a viesand D a vies , 1957, fu rther d isc u ss the f indings o f M ason and R ice and oth ers .From the ir de ta i l ed ana lyses i t fo l lows that for v i scos i ty measurements a t

2 to 260 K the value of c /k is 9 .7 to 1 0 .3 , d er ived f rom a mod if ied "Exp -6"potential with a • 1 3 . 5 , proper ly correc ted for quantum e f f ec t s . The potent ia l a l so f i t s the o lder mea su reme nts o f v i scos i ty a t 300 to 1000 K that a re

now sup posed to be inc orr ect, but si n ce th e pa rt of the potentia l in thevic inity o f the energy minimum has very l i t t 'e bearing on the calculat ionof the v isco s ity at these tem pera tu res , i t d oe- not im ply that c / k = to is

incorrect , merely that the "Exp-6" as wel l as the (6 -n) potent ia l are unreal is t ic for calculat ions when high as wel l as low temperatures are to be

covered by one potential .The two potent ia ls u sed by Ma son and Ric e ar e sk etched in f ig . 5A. I t

wil l be seen that quant itat ive ly they di f fer rather much in spite o f the ir equalabi l ity to express the mea su red va lu es . I t i s interes t in g to compare thesear t i f i c ia l or emper ica l ly de termined potent ia l s wi th interac t ion energy func

t ions calcu lated on the ba s is of atom ic ph ysics a lone . Su ch funct ions f orhe l ium are shown in f ig . 5B; the curves correspond to d i f f erent as sumpt ions for the calcu lat ion. The a greem ent i s good a s fa r a s the ord er of

magnitude o f the energy minim u m i s concern ed . Fi g . 5 is sk etched fromgraphs in the monograph "Theory of Interm olecu la r F or ce s" by H. Ma rgenaua ndN . R . K e s t ne r , 1 9 6 9 .

The problem as to which part o f a potent ia l i s s igni f icant for calculat ion o f the v i scos i ty a t a g iven temperature has been d i scussed by Amdur

and R os s , 1958 , and by K ale lkar and K es t in , 1 970 . Amd ur and Ro ss presenta very s imple so lut ion to the problem based on the inverse power potent ia l ,

mentioned ea r l ie r . Equat ing the quant ity W in the co l l is io n integra l withuni ty, and introduc ing the equiva lent ha rd -sph ere d iam eter , o o , which at ag iven temperature g ives the same va lue o f the v i scos i ty as does the ac tua linverse power potent ia l , we get

k T = q>(o0) erg ,

where k - 1 .381 • I0"' 6 e r g / K .

These cons iderat ions are va l id a t t emperatures that are h igh comparedwith t ii e "te jnperuturo" c / t . S iM i th i s i s only 1 0K , i . e _ t « . PJHI14- 10"erg, for hel ium, i t fo l lows that the express ion is accurate enough for hel ium

alread y from room tempera ture and u p. For the ran ge o f temp eratu res



- 25 -

• 1

r • i

sZ - 2p

,

\ \

V

/ /

25 3.0 15 40— rÅ

F i g . 5

F i g . 5 A . T h e L e n n a r d - J u n e sPotent ia l , 1 , and t h e " E x p - 6 "

potent ia l , 2 , for he l iu m. M ason

a nd R i c e , 1 954 .

i o

r 2

•

- 4

\

L ^ S

^

/

U 10 U 40- r i

F i g . 5 B . T h eo re t i ca l l yev a l u a t ed i n t era c t i o n en erg y

of he l ium, Margenau and

Kes t n er , 1 9 6 9 .

con s id ere d in th i s work , na m ely 273 •• 1800 K, the range o f en erg ies i s- 1 2 - 1 2

0. 04 • 10 to 0 . 25 • 10 er g, a ss u m ing that only one point of the po ten t i a l d e t erm i n es t h e v i s co s i t y . T h is a s su m p t i o n i s , o f co u rse , n o t s t r i c t l y

co rrec t s i n ce t h e v i s co s i t y rep resen t s a w e i g h t ed a v era g e o f t h e p o t en t i a l , ' b u t

ev ident ly on ly a l imi ted part o f the potent ia l on each s ide o f the po int in

q u es t i o n i s i m p o r t a n t f o r t h e d e t erm i n a t i o n o f t h e v i s co s i t y a t t h e co rre

s p o n d i n g t e m p e r a t u r e .

K a l e l k a r a n d K es t i n e s t i m a t e t h e ra n g e o f s ep a ra t i o n d i s t a n ce s , r , s i g

ni f icant for the c a l c u l a t i o n o f v i s co s i t i e s i n a g i v en t em p era t u re ra n g e b yco m p u ti ng t h e a v era g e sep a ra t i n g d i s t a n ce a t a g i v en t em p era t u re . T h e r e

s u l t ia v e r y m u c h t h e s a m e a s e s t i m a t e d b y t h e p r o c e d u r e of A m d u r and

U o i b . T h e i i r r . p c r - t u r c r a n g e 3 00 t " 1! 10 K g i v e s a rsmse of a v e r a g e » e p a r -"-12 -12

a t io n d i s t a n c e s c o r r e s p o n d i n g to e n e r g i e s f r o m 0 . 0 3 ' I 0 t o 0 . 1 • 1 0-1 2

e r g , w h i c h s h o u l d b e c o m p a r e d w i t h c • 0 . 0 0 1 4 • 10 e rg fo r c / k » TO K.



- 26 -

Kale lkar and Kest in conclude that the physical s igni f icance o f c is dis tortedwhen c /k i s d etermined by fitt ing a potential of the types con sid ered he reto v iscos ity data at higher temperatures .

The difference between the "Exp-6" and the (6-n) potential and their

abi l ity to g ive a pr ec ise f i t to the m ea su rem en ts is i l lu stra ted in f ig s . 6Aand GB. Fi g . 6A is a plot o f the d eviat ions o f the v is cos ity va lu es ca lcula tedby mea ns of a "Exp-6", a = 12 , p otentia l from the adopted v isc os it y v a lu es .

The pa ram eter e / k is var ied from 10 to 50 K. Sim ilar ity f ig . 6B i s a de viation plot for the (6-9 ) potential with va lu es of c /k from 20 to 80. A s wi llbe seen the "Exp-6" potential is superior to the (6-9) potential , but on the

other hand , the "Exp -6" poten tial ca n be fitted to within 0. 5%, ev en if c / kis a rbitra rily sele cte d bexween 20 and 50. Th is ind ica tes that the "E xp- 6"potent ia l i s rather insensit ive to a var iat ion of the parameters .

Fig . 7 is presented as an i l lustrat ion of the features mentioned andalso for comparison of the potent ia ls calculated from gaseous transportpropert ies with potent ia ls der ived from measurements o f the scatter ing of

high-veloc ity hel ium atoms performed by Amdur and Harkness , 1 954, curv est and 2 , and by Amd ur, Jorda n and C olgate , 1961, cu rve 3 . C u rve 4 iscalculated from the "Exp-6" pot . , eq. (6 -2) , with the parameters a • 12 ,

c /k - 20 K and r m = 2 .925 A. C u rve 5 is calcu lated from the (6 - 9) p ot . ,eq. (6 -3) , with the pa ra meters c /k = 20 K a nd c « 2 .530 A. C urve 6 i sthe exponential potential , eq. (6- 4), with the pa ra m eter s r = 2, a - 10

and c /k = 560. C u rve 7 is the inv erse power potent ia l , eq. (5 -12 ) , » (r) =7 7 - 1 0 " / r , f rom which the equa tion for the vi sc os ity adopted fo r thiswork, eq. (6 - 1) , ca n be d er ived by insert ion of W , . . = 1 .0 31 .

Of the potentials shown the exponential potential will give the best f itto the resu lts o f Amdui et a l . when extrapolated to higher ener gie s . How

ever, it should be noted that none of the individual potentials seems to f itat a l l temperatures and for calculat ion of the di f ferent propert ies whichcan be compu ted from the interm olecu lar potent ia l . As an exam ple f ig . 1shows the second v ir ia l coef f icient , and cu rve b by. Ma son and R ice f i ts

wel l over the ent ire temperature range in spite o f lack o f abi l i ty o f thepotent ia l to f i t the new visc os ity mea su rem ent s . C u rve c , an in ver sepower potent ia l by Amdur and Mason, a lso f i ts wel l , but as seen fromf ig . 3 the potent ia l g ives too smal l v iscos ity values at 1000 K.

It ma y be conclud ed that the "Exp-6" and the (6-n) poten tia ls , althoughgiv ing very good interpolat ion formulae for the propert ies and also beingu sefu l , when m ixtu res o f he l iu m and other ga s eg a re treated , do not re

present the intermolecular forces for he l ium above room temperature as



+127 •

f-so40

#

273 400 600 800 1000 1200 MOO 1600* - T K F i g . 6A

P a r a m e t e r s

"E x p- 8

c / k K

za

" potent ia l , a

103 . 1 3 4

202.925

= 1230

2.809

4 02.728

502.663

• 2

-2

^

vAO

^ 2 ,

=80

40

273 400 600 800 1000 1200 VM 1600— — T K F ig - 6B

(6 -9 ) po tent ia l

P a r a m e t e r «

' --—1

c / k K

< r Å

20

2. 530

40

2.372

50

2.330

6 0

2.268

80

2 . 200

D e v i a t i o n « in % f ro m n - 1 . 865 - 1 0 " 5 ( T / T o ) ° ' 7

D e v i a t i o n p l o t« o f c a l c u l a t e d v i* c o # i t y c o e f fi c i e n t «



22 2A 26 2.8

Pig. 7

Infcermoleoular potential functions for helium.



- 29 -

w e l l as other potent ia l s do . It seems most l ike ly that the exponent ia l po ten

t i a l is t he m o s t v e r s a t i l e f unc t io n pa r t i c u l a r ly f o r t e m pe r a t ur e s h ig he r t ha n

those cons id ered in th i s work . The inv ers e pow er potent ia l chose n for th i s

w o r k g i v e s a s u f f i c i e n t ly a c c ur a t e in t e r po la t i o n f o r m ula of a m p l e s i m p l i c i t y

f o r p r a c t i c a l p u r p o s e s .F ina l l y it should be noted that the v i scos i ty is independent of t h e p r e s s

u r e w i th in t he a c c u r a c y of t he m e a s ur e m e nt s h i t he r t o pe r f o r m e d .

7 . THERMAL COND UC TIVITY D ATA

The l i t erature presents much informat ion on the coe f f i c ient of t h e r m a l

conduct iv i ty ofh e l i u m b a s e d o n m e a s u r e m e n t s of the conduct iv i ty in the

s t e a dy - s t a t e c o n f ig ur a t io n in a ce l l w i th e i ther a t h in w ir e s us pe nde d a lo ng

the cen tre l ine or conc entr ic cy l ind ers . The da ta ator be low 1 b a r a r e

pr e s e nt e d in the form of a deviat ion plot , fig. 8, based on the fo l lowing

equat ion adopted for th i s work:

_4

k = 2. 682 - 10"3(1 + 1.1 23 • 1 0"3P) T(0-71 ( 1 " 2 ' ' ° P , )

-4 <7-U

= 0.144(1 + 2.7 • 10"4

P(T/To)(0

-7

'(

'"2

'1 0 P ) )

W/m K.

P is t h e p r e s s u r e i n b a r .

A t 1 b a r t h e p r e s s u r e t e r m is neg l ig ib le , th ere fo re the fo l lowing

equat ion is us e d in f i g . 8:

k = 0.144(T/To)0-7'. (7-2)

T he de v i a t i o ns a r e a l s o g iv e n f o r a s u r v e y of t he da t a pr e s e nt e d by t he '

N . B . S . , USA , 1 9 6 6 , c u r v e 6, and for the express ion for the conduct iv i ty

ba s e d o n t he v i s c o s i t y da t a , eq. ( 5 - 1 0 ) :

T he r a t i o of t he c o r r e c t ing f a c t o r s is a bo ut 1 . 0 0 4 . I ns e r t i o n of th i s

a nd c in t o t he e qua t io n ba s e d o n v i s c o s i t y da t a g iv e s :

k - 2.51 c • >i ' 7 8 2 4 M , (7-»}

r e pr e s e nt e d by c ur v e 9 in the f igur« .



- 30 -

The deviation plot shows that the adopted values of the conductivity arechosen as being between the measured values and the values der ived fromthe vis co sit y da ta, cu rve 9. It is felt that this is the m os t app ropriate

choice , s in ce so m e u ncertainty s t i l l ex i s ts a s to the a ccu ra cy of both a pproa ches to the problem. If this is taken into accou nt , the s tanda rd d evia tion of the conductivity d&t* calculated from the adopted equat ion can beestima ted to be about t% at 273 K and 6% a t 1 8 00 K , i . e . ,

o - 0 .0035 T% (7- 4)

Whi le the conduct iv i ty measurements a t low pressure are compl ica ted

by the inaccu racy of the correct ion for the accom od at ion e f f ects , m ea su re

ments a t h igh pressure are d i f f i cu l t to per form in s teady -s ta te exper imentsbeca u se o f the r isk o f natu ral convect ion in the ga s . How ever , the ag re e

ment o f di f ferent measurements at 100 bar is reasonably good as is ev ident

from the deviat ion plot, f ig . 9. The pr ess u re e f fect i s sm a l l and se em s to

vanish at e levated tempera tu res . Fig . 9 i s based on eq. (7 -1) for P • 100 ,

i. e.

k - 0 .1 4 7 89 ( T / T 0 ) 0 - 6 9 5 8 . (7 -5)

The adopted formula eq. (7 -1) takes the pressure e f fect into account .

8 . PRAND TL NUMBER

The Prandt l number is

P r = c 4 j , or , by insert ion of eq s . (4 -4 ) , (6 -1) and (7 -1 ):

P r = 0.71I I T - (0 .01 - 1 .42 • 10" 4 r )

1 + 1 .123 - 10" 3P

0.6728 ( T / T ,-(0 .0 1 - 1 . 4 2 - 1 0 "4P) ( g _ , ,

1 + 2 .7 - 10"*P °

The standard deviat ion I I I M , 004 T%. (8- 2)

Fina l ly i t should b e mentioned that the condu ct iv ity and the Pra nd t lnum ber c a n be m e a s ur e d ind i r ec t ly by s ho c k - t u be a nd t e m pe r a t ur e - r e c o v e r y

methods . Such me asu reme nts , however , appear to be le s s a ccu rate thanthose considered in this work for which reason a detai led analys is o f theresults by those methods is not incorporated herein.



300273

900

• 9

• 6

" " ^ 1 ? S * - — V . . . _1000 1500 2000T K

Fig. 8

Thermal coiutactivity k of helium at 1 bar

Deviations in % from k ' 0. 1 4 4 ( T / T 0 ) 0 - "

1 - Johnston and G riUy. 19462 . Kannuluik and C arma n, 19523 - Zaitaeva, 19594 - Mann and B la ls, 19595 • VargafUk and Ztinina, 19656 - Pow ell, Bo aad L iley, 1966

7 - Ho and Leidenfrost, 19698 - L eNeind re, . . . and Vodar, 19699 - derived from viscositjr,

I .e. k*2.»1 • e v ' » .Opea circle« ara obsei »alle— bynine other experim enta lists.



- 32 -

+ 1

r&f*& ,,

i ' *

1 — i

— * •

T^ X

I273

400 600 800 1000 1200T K

Thermal conductivity k of helium at 100 bar

Fig. 9Deviation in % from k = 0 .14789 (T/T Q )

1 - Vargaftik and Zim ina, 1 9652 - Ho and Leidenfrost, 1 9693 - L e Neindre, and Voda r, 1 969

0 . 6 9 5 8

ACKNOWLEDGEMENTS AND NOTES

Tes author is indebted to Mr. K. Sull ivan, UKAEA, Rislev, and Mr.Jan Blomstrand , OECD -H. T. R. -pro ject DRAGON, for Interest and in spira tion, and to Mr. N.E. Kaiser and Mrs. Susanne Jensen for the computat ionalhandling of the data.

After the report was written a recent work on the conductivity of heliu-nand other gaBes by D r. J . W. Haarman of the Netherlands was brought to theau thor's attention. Th is work is pa rticu larly interes ting as it giv es high-

precis ion measurements by the transient hot-wire method with an extremelythin wir e. The resu lts a re d ealt with in the la st-m inu te note on p. 37.

A s imilar apparatus is at present being f inal ly tested at the laboratoriesof the Res ea rch Establishment RisS . In this set-u p the da ta logg er is adig ita l computer , and in this the method di f fers from that u sed by D r. Ha arman, Moreover , the apparatus is des igned for larger ranges o f pressureand temperature .



- 33 -

1 0. R E F E R E N C E S

A m d u r , I . an d H a r k n e s s , A . L . , 1 954.

" S c a t t e r i n g of H i g h - V e l o c i t y N e u t r a l P a r t i c l e s . I I. H e l i u m - H e l i u m " .

J . C h e m . P h y s . , 22 , 6 6 4 - 6 9 .

A m d u r , I . , J o r d a n , J . E . a n d C o l g a t e , S . O . , 1 9 6 0 .

" S c a t t e r i n g o f H i g h - V e l o c i t y N e u t r a l P a r t i c l e s . X L F u r t h e r S tu dy o f t h e

H e - H e P o t e n t i a l " .

J . C h e m . P h y s . , 3 4 , 1 5 2 5 - 3 0 .

A m d u r , I . a n d M a s o n , E . A . , 1 9 5 8.

" P r o p e r t i e s of G a s e s a t V e r y H ig h T e m p e r a t u r e s " .P h y s . F l u i d s , 1 , 3 7 0 - 8 3 .

A m d u r , I . a n d R o s s , J . , 1 9 5 8 .

"O n t h e C a l c u l a ti o n of P r o p e r t i e s o f G a s e s a t E l e v a t e d T e m p e r a t u r e s " .

C o m b u s t . F l a m e , 2 , 4 1 2 - 2 0 .

B u c k in g h a m , B . A . , D a v i e s , A . E . a n d D a v i e s , A . R . , 1 9 5 7 .

" T h e D e d uc t i o n o f I n t e r m o l e c u l a r F o r c e s fr o m t h e T r a n s p o r t P r o p e r t i e s

o f H y d r o g e n a n d H e l i u m " .

T h e r m o d y n a m i c a n d T r a n s p o r t P r o p e r t i e s of F l u i d s , P r o c e e d i n g s of t h e

J o i n t C o n f e r e n c e , L o n d o n , 1 0 - 12 J u l y 1 9 5 7 . ( T h e I n s t i t u ti o n o f M e c h a n i c a l

E n g i n e e r s , L o n d o n , 1 958) 111- 19 .

D i P i p p o , R . a n d K e s t i n , J . , 1 9 6 9 .

" T h e V i s c o s i t y o f S e v e n G a s e s r.p t o 5 00 ° C a n d I t s S t a t i s t i c a l I n t e r p r e t a t i o n " .

4 th S y m p o s i u m o n T h e r m o p h y s i c a l P r o p e r t i e s , A p r i l 1 - 4 , 1 9 6 8 . E d i t e d b y

J . R . M o s z y n s k i , A m e r i c a n S o c ie t y of M e c h a n i c a l E n g i n e e r s , N e w Y o r k ,

(1 968) 304-1 3 .

G u e v a r a , F . A . , M c l n t e e r , B . B . a n d W a g e m a n , W . E . , 1 9 6 9 .

" H i g h - T e m p e r a t u r e V i s c o s i t y R a t i o s f o r H y d r o g e n , H e l iu m , A r g o n a nd

N i t r o g e n " .

P h y s . F l u i d s , W, 2 4 9 3 - 3 5 0 5 .

H a n l e y , H . J . M. a n d C h i l d« , G . E . , 196 8 .

" D i s c r e p a n c i e s b e tw e e n V i s c o s i t y D a t a fo r Si m p l e G u e i " .

S c i e n c e , 1 5 9 , 1 1 1 4 - 1 7 .



- 35 -

K e s t i n , J. a n d L e i d e n f r o s t , W., 1 9 5 9 .

" T h e V i s c o s i t y of H e l i u m " .

P h y s i c a , 25, 5 3 7 - 5 5 5 .

L e F e v r e , E . J . , 1 9 5 7 .

" T h e C l a s s i c a l Vi s c o s i t y ofG a s e s at E x t r e m e T e m p e r a t u r e s " .

T h e r m o d y n a m i c a n d T r a n s p o r t P r o p e r t i e s of F l u i d s , P r o c e e d i n g s of th e

J o i n t C o n f e r e n c e , L o n d o n , 1 0-1 2 J u l y 1 9 5 7 .

( T h e I n s t i t u t i o n of M e c h a n i c a l E n g i n e e r s , L o nd on , 1 958) 1 2 4 - 2 7 .

L e N e i n d r e , B. , T u f eu , R. , B u r y , P. , J o h a n n i n , P. a n d V o d a r , B. , 1969 ." T h e T h e r m a l C o n d u c ti v it y C o e f fi c i e n t s of S o m e N o b le G a s e s " .

T h e r m a l C o n d u c t iv i t y . P r o c e e d i n g s of t h e E i g h t h C o n f e r e n c e .P u r d u e U n i v e r s i t y , W e s t L a f a y e t t e , I n d . , O c t o b e r 7 - 1 0, 1 9 6 8 .

E d i t e d b y C . Y . H o a n d R . E . T a y l o r ( P le n u m P r e s s , N ew Y o r k, 1969)7 5 - 1 0 0 .

L i c k , W . J . a n d E m m o n s , H . W . , 1 9 6 5 .

" T r a n s p o r t P r o p e r t i e s ofH e l i u m fr o m 2 00 to 5 0 , 0 0 0 ° K " .

( H a r v a r d U n i v e r s i t y P r e s s , C a m b r i d g e , M a s s . , 1 96 5 ).

M a n n , J. B . , 1 9 6 0 .

" C o l l i s i o n I n t e g r a l s a n d T r a n s p o r t P r o p e r t i e s fo r G a s e s O b e y in g an

E x p o n e n t i a l R e p u l s i v e P o t e n t i a l " ,

L A - 2 3 8 3 .

M a n n , J . B . a n d B l a i s , N . C , 1 9 5 9 .

" T h e r m a l C o n d u c t iv i t y of H e l i i j a a n d H y d r o g e n a t H i g h T em p era t u res" .

L A - 2 3 16 , A b r i dg e d v e r s i o n : J. C h e m . P h y s . , 32, 1 4 5 9 - 6 5 , 1 9 6 0 .

M a s o n , E . A . a n d R i c e , W . E . , 1 9 54 .

" T h e I n t e r m o l e c u l a r P o t e n t i a l s of H e l i u m a n d H y d r o g e n " .

J . C h e m . P h y s . , 22, 5 2 2 - 3 5 .

M a r g e n a u , H. a n d K eatner , N. R . , 1 9 6 9 .

"Theory of I n t e r m o l e c u l a r F o r c e s " .( P e r g a m o n P r e s s , O xfo r d, 1 9 6 9 ) .

i in in ^ ^ ^ É ^ M T : ••niiÉEiii^^^^aha^is^ÉÉfr



- 36 -

Monchick, Louis, 1959.

"Collis ion Integrals for the Exponential Repulsive Potential".

P h y s . Fluids , 2 , 695-700.

Po w e ll , R . W . , H o , C . Y . a nd I j l e y , P . E . , 1 96 6.

"Thermal Conductivity of Selected Materials".(NSRD S-NBS 8). (National Bu reau of Stand ard s, W ashington, D . C . 1966).

Schneider, Vf.G. and Duff i e . J .A .H . . 1949."C ompressibil ity of G a ses at High Tem pera tu res . II. The Second Vir ia lCoefficient of Helium in the Temperature Range 0 °C - 600 °C".J. C hem . P h y s ., 1_7_, 751 - 5 4 .

Stroud, L . , I d l e r , J . E . and Brandt , L . W . , 1 960."Physical Pro per t ies , Evalua t ion of C ompounds and M a teria ls . P t . 2 :C omp ressibil ity of Helium at -1 0° to 1 30 °F and P re ss u re s to 4000 P . S, I . A. ".J. C hem. Eng. D ata , 5 , 51 -5 2.

Vargaftik, N. B. and Zimina, N. K h. , 1 965.

"Thermal Conductivity of Helium at Temperatures of 0 - 200 Atm",

Sov. At. Energy, 1_9, 1221-23.

Wi l son, M.P. Jr . , 1960 .

"Thermodynamic and Transport Propert ies o f Hel ium".GA-1 355. Janu ary 1960.

Yntema, J . L. and Schneider, W. G ., 1 950.

"C ompress ibil ity o f G a ses at High Tem pera tu res , i n . The Second Vir ia l

Coefficient of Helium in the Temperature Range 600°C to 1200°C".J . Chem. Phys . , 18 , 641 -46 .

AiÉi&L^.



- 37 -

L a s t - Minut e N o t e

The work mentioned in sect ion 9 , Acknowledgement and Notes , by Or.

J . W. Haarman "EEN NAUWKEURIGE METHODE VOOR HET BEPALEN

VAN DE WAEMTEGELE1DINGSCOÉFFICIÉNT VAN GASSEN (DE NIET-

STATIONAIRE D RAADMETHODE)", D eu tsch e U i tgevers Ma atschappi j , NV,De l f t , 1969 , presents measurements o f the conduct iv i ty o f he l ium at about

atmospher ic pressure and a t t emperatures f rom 55 to 1 95 ° C . T he m e a s u r e d

va lu es g ive the points ma rk ed 10 in the d eviat ion plot , f ig . Sa. The d evia

t ion a re sm a l l and do not ne ces s i ta te a change in the adopted formu la , eq .

( 7 - 1 ) , for the thermal conduct iv ity o f he l ium.

• 3

•» T KF ig . 8a . Th erm a l condu ct iv i ty k of he lium at 1

De v ia t i o ns in % from k - - - - —10 - Haarman, 1969

0.144 (T/T 0 ) " > • 7 ,

1 1 . TABLES OF RECOMMENDED DATA

The tables on the fo l lowing sheets g ive values o f tho propert ies at

s e l e c t e d p r e s s u r e s a n d t e m p e r a t u r e s .

T a b le s w i th s m a l l e r in t e r v a l s of pr e s s u r e a nd t e m pe r a t ur e , * j » l ta | le .

for l inea r interpo la t ion , a re presented in Dragon. Pr o j ec t B «p ert Mo. M 4:"T a b le s of T b* nno pt y » i c * l Pr o pe r t i e s o f H e l iw s " , * » a # d by © , J & C f r , ;

High Temp eratu r« K ew to r Pro jec t , W tAOON, A. E . E . W iabt tb , Bpt t$w» t»r ,

Darfti , E u f b w d , , s • ...' . - - . ., :

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1(0 £0

1.00053

1.000*7"1.000U31.000J9

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1.02120

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100.00126.8*

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LOOSW•°°513

1.01733

1.01575

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1.(112*1.011651.010971.01026

1.025991.023621.021871.02012

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1.01539

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1.00359

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1.00972

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LOO7I191.00718

1.00685

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1.01077

1.01028

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723.16750.00773.16800.00

1.000161.00016

1.00015

1.00015

1.0056*

1.005*3

1.00526

1.00507650.00676.8*7OO.OO7 2 6 . 8 *

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1000.00

1.0001*1.0001*1.00013

1.000131.00*92

1.00*75

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1.00012

1.00012

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1023.161050.001073.161100.00

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1223.161250.001273.161300.00

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1.00009

1.00008

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1.00160

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1.00007

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1000.001026.8b1050.001076.8b1100.001 i 2 6 . 8 b1150.001176.8b1200.001226.8bI230.OO1276.8b

1300.001326.841550.001376.841400.001*?6.9>1450.001476.841500.001526.84

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1273.161300.001323.161350.001373.161VO0.0O1b23.16* 5 0 . 0 01*73.161500.001 5 2 3 , «1550.00

1573.161500.001 6 2 5 . «1650.001*73.161700.001723.161750.001773.161600.00

Preenure tar abs1 20

0.1761b 3.b8770.16039 3 .1792O.1b890 2.95370.137b9 2.729b0.1289« 2.56150.12031 2 .39100 . i i J 7 3 2 . 2 6 1 10.10695 2 .1*720.10171 2 .02570.09626 1.9158O.09200 1.6315O.08751 1.7b260.08397 1.6725O.O8022 1.59810.0772b 1.53900.07b05 1.b757O.07150 1.b252

0.06876 1.37070.06656 1.32700.06b18 1.27970.06225 1.2b150.06017 1.20000.058b7 1.16630.05665 1.1296O.05513 1.09980.053b8 1.06710.0521b 1.0*040.05067 1.0111O.ob9b6 0.9871

0 .0b8 l3 0 .9607O.ob705 O.9390O.ob58b O.9150O.Obb85 0.8953O.Ob376 0.8735O.Ob286 0.85560 .0b l86 0 .83560.0b103 0 .81920.0b011 0 .8009O.03935 0.7858O.03851 0.7689

0.03781 0 .75500.0370J 0 .739b0.03638 0 .72650.03566 0 .71210.03506 0 .70010.03b38 0.6867O.03362 0.67550.03320 0 .66310.03268 0 .65260 .03209 _J» .6bW

t) .03160 0 .63130.03106 0 .6203

0.03060 0 .61120.03009 0 .60100.02966 0.592b0.02917 0.5826O.02877 0.57*70.02832 0.56J70 .0279* 0 .5501O.02751 0.5*950.02715 0 .5*240 .0267* 0 .53*3

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2.73212.6655"2.551b2.b7552.39302.32622.25322 .19382.12882.07572.017b1.9696

1.91701.87391.82621.78701.7*361.70781.66811.6353

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1.507b1.b76b1.*5071.b2191.39801.37131 . 3 * 9 *1.32*21.3035t . 2 8 0 21.26081.2391

1.22091.200b1 . * 3 *1.16*21.1*611.15001.11*91.09761.08351 . 0 * 7 *

60

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100.00126.84

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450.00476.8450O.OO526.84550.00576.84600.00626.84650.00676.84700.00726.84

750.00776.84800.no826.84850.00876.84900.00926.84950.00976.84

1000.001026.841050.001076.841100.001126.841150.001176.841200.001226.841250.001276.841300.001326.641350.OO1376.841400.001*26.841450.00* 7 6 . t t1500.00

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273.16300.00373-16350.00373.16400.00

423.16450.00473.16500.00523.16550.00573-16600.00623.16650.00673.16700.00

723.16750.00773.16800.OO823.1685B.no873.16900.009 2 3 . , 6950.00973.16

1000.00

1023.161050.001073.161100.001123.161150.001173.161200.001223.161250.001273.161300.001323.16I350.OO1373.16i4oo.no1423.1é1450.00* 7 3 . 1 61500.001523.161550.no 1573.161600.001623.161650.001673.161700.00r-23.16I750.OO1773.1«

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1.86481.99122.09762.2181

2.31992.4355

2.53332.64462.73932.84722.93893.04363.13283.2348

3.32173.42123.50613.6034

3.68643.78173.86303.95644.03634.12794.20634.29644.37354.46224.53804.6253

4.7OOO4.786O4.85964.94445.OI7O5.IOO75 . 1 7 *5.25495.J2575.40725.47725.55785.62695.70655 . n ^ 95.85375.92155.99926.06626.14336.Z0966.28606.35166.42726.49226.56726.6316

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TBapuf t ture

O.no 273.1626.84 300.0050.00 323-1676 .84 350 .00

100.00 373.16126.84 400.00

150.OO 423.16176.84 450.OO200.00 473.16226.84 500.00250.00 523.16276.84 550.00300.00 573.16326.84 600.00350.00 623.16376.84 650.00400.00 673.16

426.84 700.00450.00 723.16476.34 750.00500.00 773.16526.84 800.00550.00 823.16576.84 850.00600.00 873.16626.84 900.00650.00 923-16676.84 950.00700 .00 973 .16

726.84 1000.00750.00 1023.16776.84 1050.00800.00 1073.16826.84 1100.00850.00 1123.16876.84 1150.00900.00 1173.16926.84 1200.00950.00 1223.16976.84 1250.00

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