« Riso Report No. 224 5 o, « Danish Atomic Energy Commission .1 * Research Establishment Ris6 The Properties of Helium: Density, Specific Heats, Viscosity, and Thermal Conductivity at Pressures from 1 to 100 bar and from Room Temperature to about 1800 K b y Helge Petersen September, 1970 Sain dUiributort: JUL GjiUtrap, 17, » W pd t, D K-U07 Capubaftn K, Dnaurk
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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.
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
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 )
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 «
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
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 . 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
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
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,
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 - 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 ,
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
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
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 .
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 :
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 .
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
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.
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
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.
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.
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 .
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 ) .
"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 .
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 .