COMPRESSIBILITY FACTOR OF NATURAL GASES AT 60 F AND ONE ...

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COMPRESSIBILITY FACTOR OF mATuRAL GASES
AT 6 0 0 ~AKD ONE ATMOSPHERF:

D. McA. Mason and B. E. =in

I n s t i t u t e of Gas Technology
Chicago, Illinois

INTRODUCTION

Modern a n a l y t i c a l techniques Pave markedly improved r;he accuracy
w i t h which t h e composition of a gas can be measured. For many y e a r s
f u e l gases were analyzed by volumetric techniques, such a s the Orsat
o r Podbielniak. Heating values and s p e c i f i c g r a v i t i e s were a l s o
determined a t atmospheric conditions f o r pure components and f u e l gas
mixtures, thus r e s u l t i n g i n r e a l gas values. However, because of low
accuracy i n the analytical methods, particularly w i t h respect t o
determination of t h e heavier hydrocarbons, d i f f e r e n c e s between calcu-
l a t e d and observed h e a t i n g values were g e n e r a l l y a s c r i b e d t o experi-
mental techniques, and not i n t e r p r e t e d on the basis of gas law
deviations.

With t h e advent of the mass spectrometer, more a c c u r a t e and
complete analyses o f gas mixtures may be obtained. These analyses
a r e on a t r u e mole f r a c t i o n , OP i d e a l gas, basis. Also, accurate
s.,i d e a l gas heat of combustton values can now be derived from t h e work
of Rossini et API p r o j e c t 44'. However, f u l l u t i l i z a t i o n o f
t h i s i m p r o v z accuracy can not presently be obtained i n the calcu-
l a t i o n of r e a l gas h e a t i n g values and s p e c i f t c gravit-Les. The l a c k
of s u f f i c i e n t data t o permit a c c u r a t e p r e d i c t i o n of the c o m p r e s s i b i l i t y
f a c t o r f o r r e a l gas mixtures mag r e s u l t i n s i g n i f i c a n t d i f f e r e n c e s
between measured values and those calculated from the gas a n a l y s i s .
Largest devtations occur i n f u e l gases containing appreciable quan-
tities of h e a d e r constituents.

A n experlmental program was t h e r e f o r e planned t o o b t a i n
c o m p r e s s i b i l i t y f a c t o r data a t 60°F and one atmosphere on f u e l gas
components and mixtures. It was a n t i c i p a t e d that these data would
provide the b a s i s f o r development o f generalized procedures f o r
prediction of compressibility f a c t o r s of gas mixtures. This project
was undertaken a s p a r t of the continuing b a s i c r e s e a r c h program of
t h e I n s t i t u t e of Gas Technology, and was sponsored by The Peoples
Natural Gas Company. The r e s u l t s obtained f o r n a t u r a l gas components
and mixtures a r e reported here.

Methods used f o r determination of gas c o m p r e s s i b i l i t y f a c t o r
f a l l i n t o two groups: 1) gas d e n s i t y , and 2 ) p r e s s u r e - r e l a t i v e
volume measurement. I n the ffrst group of methods, gas d e n s i t y i s
measured by gas balance o r by d i r e c t weighing i n a bulb. The
compressibility factor i s calculated from the expression:

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where V i s the volume of one mole c a l c u l a t e d from t h e d e n s i t y and
molecular weight of the gas. The accuracy of t h i s method depends
on a b s o l u t e values of p r e s s u r e , volume and temperature.

I n che second group of methods, measurements of pressure and
r e l a t i v e volume a r e made over a range of p r e s s u r e , and pV o r a
related parameter i s extrapolated t o zero pressure, a s i n che
Wlrnett A method of t h i s kind f o r use a t p r e s s u r e s near
atmospheric was described by J e s s e n and Lightfoot .6 Data are
obtained a t two o r three different p r e s s u r e s by confining t h e
sample of gas i n p r o g r e s s i v e l y l a r g e r volumes over mercury. The
c o m p r e s s i b i l i t y f a c t o r i s calculated by f i t t i n g t h e pV measure-
ments (V is the measured volum h e r e ) with a n equation such a s :

Here t h e accuracy depends primarily on r e l a t i v e volime and pressure
measurements, and constancy of temperature, r a t h e r than on a b s o l u t e
values of these q u a n t i t i e s . Absolute values of pressure and t e m -
p e r a t u r e a f f e c t t h e r e s u l t i n a much less c r i t i c a l way; that i s ,
by t h e dependence of z on pressure and temperature, and not by the
dependence of pV and RT. S i a i l a r l y t h e accuracy of z f o r mixtures
i s not a f f e c t e d by the dependence of gas d e n s i t y on composition,
Hwever, the method i s dependent on the l i n e a r i t y of z w i t h pressure.

According to the s t a t i s t i c a l mechanical expressions f o r vLria1
coefficients i n the equation

p-J = RT ( 1 + Bp + Cp2 + ...) (3)

the second v i r i a l c o e f f i c i e n t , B, represents the deviation from i d e a l

.'behavior involving c o l l i s i o n s between two molecules Frorn t h i s

theory i t a l s o follows that t h e second v i r i a l coefficient f o r mixtures
involves o n l y binary i n t e r a c t i o n terms, and i s of t h e form:

...B, = x12& + xz2B2 + ~ 3 ~ B+ 3

+ 2 ~ 1 ~ 2 B+ ~2 ~2 1 ~ 3 B 1+3 2X2x3B23 + - . a (4)

...where B1, B2, &, a r e second v i r i a l c o e f f i c i e n t s of the pure
...componenT;s, and B12, B13, B23, are interaction coefficiencs.
The l a t t e r can be evaluated by experimental determination of
on b i n a r y mixtures, t o g e t h e r w t t h knowledge of t h e B f s f o r pu%r Is
components. The e f f e c t of t h e t h i r d a r i a 1 c o e f f i c i e n t a t one
atmosphere i s very small and i s consldered l a c e r .

Five components u s u a l l y occur i n n a t u r a l gases i n concentrations
of 5% o r more: methane, ethane, propane, carbon dioxide and nitrogen.
Minor amoimts of isobutane, _n-butane, isopentane and 2-pentane a l s o
occur, t o g e t h e r v i t h t r a c e s of heavier hydrocarbons. A review of
the l i t e r a t u r e indicated that s u f f l c i e n t l y r e l i a b l e values of
atmospheric pre ssum compress i b i l i t y f a c t ors a r e available f o r the
major components, 1'4 but not f o r Ghe minor components. Also, although
o n l y very limited d a t a were a v a i l a b l e on mixtures of t h e s e components,
the data Indicated that composition had a very sizeable effect O L ~
compressibility factor.

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Therefore, c o m p r e s s i b i l i t y f a c t o r measurements were made on the
pure butanes and pentanes, and on a series of s e l e c t e d mixtures. The
InteracLion c o e f f i c i e n t s among the f i v e major components, and of
methane v i t h the four minor components, were determined. I n addition,
the i n t e r a c t i o n c o e f f i c i e n t s of n-butane v i z h a l l tm j o r couponents
vas measured t o provide a b a s i s Tor p r e d i c t i o n of those not determined
experinentally.

APPARATUS
The apparatus i s shorrn i n Fig. 1. The mixture p r e p a r a t i o n system,
gas e:qxnsion system, mercury r e s e r v o i r and the lower p a r t of the
mercury mnorilezer a r e mounted i n a v a t e r b a t h w i t h a p l a t e - g l a s s
f r o n t . The temperature of the bath vas maintained w i t h i n 0.05' of
15.55OC ( G O O F ) , and d i d not vary more than O.0loC d u r i n g any one run.
A and B a r e Hoke s t a i n l e s s s t e e l cam-closing valves. Connection
t o g l a s s parts i s made through s t a i n l e s s s t e e l Swagelok fittings and
1/8-in. stainless steel tubing sealed t o the glass w i t h Cekhotinsky
cement. Other valves a r e 3oke toggle o r needle valves. The upper
part o f t h e long manometer arm extends out of the b a t h and i s equipped
w i t h a jacket for water c i r c u l a t i o n . F!anometer arms and jacket a r e
constructed of p r e c i s i o n bore tubing; arms 1 and 2 , 1 2 m ID; arm
3 , 13.8 mm I D . Vacuum i s maintained a t 0.02 mm o r l e s s w i t h a
mechanical vacuun pump and dry i c e cold t r a p , and i s read w i t h a
t l l t i n g IkLeod gage s e n s i t i v e t o 0.01 mm Hg. Mercury l e v e l s a r e
read t o 0.05 mm w i t h a Gaertner 100 em cathetometer.
The mixture p r e p a r a t i o n bulbs, gas expansion b u l b s , and manometer
arm 1were caltbraced i;j 1-eight o f mercury d e l i v e r e d . The volume
of valve 3 and t h e c a p i l l a r y manifold i n t h e expansion system, and t h e
volume betireen valves A and ,B and t h e t o p of t n e upper bulb i n t h e
mixture p r e p r a t i o n system, were determlned w i t h a 1 0 cc gas b u r e t t e .
Tine volume betveen the t o p and bottom menisci of t h e manometer arm 1,
and c o r r e c t i o n s f o r c a p i l l a r y depression, were t a k e n from K i ~ t e m a k e r ~ ' ~ .

PNrnALS
T'ne hydrocarbons used were P h i l l i p s r e s e a r c h grade. According
t o the s u p p l i e r , p u r i t y of t h e methane vas 99.68 mole $, with i m -
p u r i t i e s of ethane and n i t r o g e n . P u r i t y of t h e o t h e r hydrocarbons
was 99.9s o r better. The carbon dioxide was PIatheson, bone d r y grade;
t h e n i t r o g e n vas Matheson p r e p u r i f i e d . Components o t h e r than n i t r o g e n
and methane were condensed i n the freezeout t r a p t o remove a i r .

.The p a r t of t h e apparatus being f i l l e d was f l u s h e d w i t h t h e component

before filling
PROCEDURZ

I n making binary mixtures the f i r s t component was measured i n
t h e gas expansion system. The second component was measured i n t h e
mixture p r e p a r a t i o n system by means of manometer arms 2 and 3 .
Thsn valve 4 was closed and t h e gases were mixed by being forced
from one b u r e t t o t h e o t h e r s e v e r a l t i m e s . A s i m i l a r procedure was
folloved i n maldng multicomponent mixtures, except t h a t t h e i n t e r -
mediate components vere,measured one a t a t i m e i n the mixture prep-
a r a t i o n buret and t r a n s f e r r e d t o the gas expansion b u r e t . Cas

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WERCURY

/

F i g . l.-SCHEMATICDIAGRAM O F LOW PRESSURE P-V APPARATUS

WITH INTEGRAL VOLUMETRIC MIXTURE PREPARATION SYSTEiM

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r e m i n i n g i n the s e c t i o n betiieen valves A and B a f c e r t h e t r a n s f e r
vas discarded.

I n mos;'cases, t h e two upper b u l b s of the %as expansion buret
riere f i l l e d t o a pressure of about 950 mn Hg. Three t o f i v e readings
of t h e t o p aDd bottom of t h e mercury menisci i n t h e t u o z m s of t h e
m n o u e t e r , and of a reference p o i n t on manometer a m 1 riere caken.
The p o s i t i o n of the mercury vas changed a m i l l i m e t e r o r s o f o r each
readins, to minimize the e f f e c t of l o c a l imperfections of the

.nanomezer cubing and place g l a s s irindow S i m i l a r readings were

taken a f z e r :he gas rras expanded t o f i l l a l l ?;hree bulbs. Heasure-
mencs on C5 hyd~ocarbonshad GO be taken a t lower pressures; a
maximum of about 80% of s a t u r a t i o n pressure was used.

Tke a p p r a t u s vas conszructed v i t h t h r e e bulbs t o alloir measure-

pments a t shree d i f f e r e n t p r e s s u r e s on t h e same c'harge of gas i n order

t o d e t e c t vayiation of t h e slope i n equation ( 2 ) v i t h pi-essure.
EoTiever, from variances o f t h e pV measurements a t d i f f e r e n t p r e s s u r e s ,
i t vas concluded thas betcer accuracy might be obtained i f o n l y t h e
l a r g e r volumes rrere used s i n c e volume measurement a s well a s pressure

measurement contributed to ?;he e r r o r . Accordingly , data a t loiier

p r e s s u r e s Tvas obtained by relnoval of p a r t of che sample and r e p e a t i n g
tzeasurements on t h e same two combinations of b u l b s .

B t a riere f i t t e d t o equation ( 2 ) by l e a s t squares computation
on a n Alrrac I11 d i g i t a l compucer.

COtFiESSII3IUTf FACTORS OF PUR3 COEIPONENTS

/Fiesulzs f o r the c o e f f i c i e n t equation ( 2 ) obtained cn C 4
and C5 p a r a f f i n hydrocarbons a r e presented i n Table 1. Determina-
-nt i-obuncsanaet, lover pressures were made a s a p a r t of Runs 2 and 6 on
and of Runs 3 and 4 on isobutane, t o detect the change
of c o e f f i c i e n t \ r i c h pressure. A trend i n the d i r e c t i o n of smaller
numerical values of the coefficient a t lover pressures i s evident.
Such t e s 5 s Irere conf'ined t o the C4 hydrocarbons, since the e f f e c t

i s b a r e l y d e t e c t a b l e with them and i s l i k e l y t o be less w i t h l i g h t e r
hydrocarbons and w i t h mixtures. Determination on t h e C5 hydro-
carbons a t n e c e s s a r i l y lower p r e s s u r e s would not y i e l d s i g n i f i c a n t

r e s u l t s on t n i s question.

I n order t o o b t a i n the best value of 1 - 2 (1a t m , GOOF), the t h i r d
v i r i a l c o e f f i c i e n t was introduced t o account f o ne observed
trend of 2 w i t h pressure:

m,=KIv z = 1 + B p + c p 2 (5)

An approximation formula vas derived from which values of B and C/B

pcould be evaluated from tiT0 measured values of and the average

5 n i t i a l and f i n a l p r e s s u r e s , p1 and p2 of each run

c1

(6)

Tae ratio C/B was evaluated by a p p l i c a t i o n of formula ( 6 ) t o the eats

from p a i r s of runs on t h e same sample, one made a t normal pressures

1

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Table 1.

Rcn No. -0.03413 1.901 0.0336l
-n- Butane -0.03466
-0.03433 1.826 0.03417 Y
1 -0 -03422 1.437
2a -0 .03423 0.03410 iI
-0 .034 53
2b -0.03411 1.044 0.03419 A
-0.03437 1.849 0.07374
2c -0 03397 1.830
3 -0.03416 1.845 0. o j 4 o h
4 1.852
5' -0.03022 o .03362
6a 0.03387
6b LO. 02993 1.458 0 A3370
5c -0.03013
-0.02973 1.027 0.03414
Isobut ane Yeighted Avc 0.03394
-0.03022
1 u20.00008 ( a )
2 -0.02942
3a 1.894 o .02982
3'0 -0.05504
4a -0 .05632 1.a91 0.02953
4b 1.a89 0.02973
-0.05199 0 -937 0.02970
-n-Pentane -0.05076 1.881 0.02982
0-895 0.02947
1 ifelghted Av&O ,02971

2 -0.00010 ( a )

Isopentane 0 578 0.05570
0 525 0.05703
Veighted Avg+O .0565

-0.0007 ( b )

0-723 0.05236
0.766
0.05107
ireighted Ave: 0.0518
20.0005 (b)

( a ) From pooled variance or' b values f o r both butanes.
(b) From variance of t h e p'r measurements.

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and the o t h e r a t about h a l f the normal p r e s s u r e s . Runs 2a with

2c, and 6a with 6c, on q-butane, and Runs 3a w i t h 70,and 4a rrith

4b, on isobutane, g i v e C/B r a t i o s of 0.016, 0.007, 0.014 and 0.026,
r e s p e c t i v e l y . The average of the f o u r , weighted according t o the
variance of t h e pV measurements, was 0.016; i t s standard deviation,
a l s o c a l c u l a t e d from t h e variance of t h e pV measurements, vas 0.006.

It was d e s i r a b l e t o apply s i m i l a r c o r r e c t i o n s t o our r e s u l t s
for C5 hydrocarbon and mixtures. However, the C/B r a t i o f o r
l i g n t e r gases a t 60°F i s smaller than those observed f o r the
butanes, e . g . , et'hane, 0.0061'3, v h i l e higher r a t i o s a r e t h e o r e t i c a l l y
expected f o r heavier hydrocarbons3. The observed r a t i o of C/B f o r
t h e butanes iras approximately equal t o B/2. I n o r d e r t o c o r r e c t
o u r r e s u l t s on a consistent b a s i s over the complete range of
d e v i a t i o n s , we adopted a C/B r a t i o of B/2 for a l l the hydrocarbons
and mixtures.

Gas law d e v i a t i o n s of the C4 and Cs hydrocarbons, i n t h e form
pof 1-2
f o r easy comparison v i t h the uncorrected values,
a r e shorrn i n Table 1. blaximum e r r o r of t h e c o m p r e s s i b i l i t y f a c t o r s
of t h e butanes i s estimated t o be 0.03$, and f o r t h e pentanes, 0.1s.

Vzlues of gas law deviations f o r pure components used i n the
c a l c u l a t i o n of data on mixtux%s a r e c o l l e c t e d i n T a b l e 3 .

BINARY IDTI?ERACTIOl'?COE?FICIETlTS

I n t e r a c t i o n c o e f f i c i e n t s determined by neasurernent of r;he
compressibility f a c t o r s of binary mixtures of n a t u r a l gas components
a r e presented i n Table 2. These have been c a l c u l a t e d by means of
the mixture rule of equation ( 4 ) but using the gas lair deviation
(1-z) r a t h e r than a c t u a l second v i r i a l c o e f f i c i e n t s . Since
t h e t h i r d and higher v i r i a l c o e f f i c i e n t s m&e o n l y a small con-
t r i b u t i o n (mxinilm about l$) t o t h e d e v i a t i o n , e r r o r from t h i s
approximation should not be s i g n i f i c a n t .

-methaEnesti-mnat-epde
stan dard r devi at ions of th e methane n-butane t and l y .
ntane ave ages a r e 0.00034 ive
and 0.000455 respec
The estimated standard d e v i a t i o n of o t h e r average i n t e r a c t i o n
coefficients i s 0.00022.

Agreement of experimental gas law d e v i a t i o n s with t h e mixture
r u l e v a s tested by determinations of i n t e r a c t i o n c o e f f i c i e n t s on
1:3 and 3:1 mixtures o f methane w i t h ;-butane, and of carbon
dioxide Tritn ;-butane. On t h e carbon diox'ide - n-butane system,
agreement i s w i-t hni n-buetxapneer i mental e r r or and no wend is a pparent.
On t h e nethane system, a tre nd of i ncrea s ing interact i o n
coefficien; i r i t h Tncreasing mer;hane content i s noGiceable; hovever ,
t h e difference between j:1 and 1 : 3 c o e f f i c i e n t s i s not s i g n i f i c a n t
-an-tb the 9 05 confid eme le vel. Compressibility I'acLors of t h e nethane -
utane mixtures calc ul ated from t h e average interaction coefficLenz
reporGed i n Table 2 agree v i t h t h e detemnlned values v i t h i n O.OE:',
maximum d e v i a t i o n , and agree t o 0.035 on the average.

4 YY

00 dd

CuOCurlN mo

C\DOwln\ooouc)-uN) nt-

. . . . .0 0 0 0 0 00

00000 . .r l d

Y 00

4

s%%mOCoOtJ- r l w m nuJrllnn;ft-=tt-

lnlnln
000 000 0 0 0 0 0 0 0 0 0

999 9 9 9 ?????????

YY

44

- 57 -

I n she %nedict-i*lebb-Rilbin equation the l i n e a r square root
coa'Jination i s u s e d t o p r e d i c t c o n s t a n t s f o r mixtures from con-
s t a n t s for t h e pure components f o r the second v i r i a l terms. Applied
t o c u r parameter t h i s yields:

7..Comparison i r i t h eqiacion 4 i n d i c a t e s that i n t e r a c t i o n C o e f f i c i e n t s

f o r a given component s h o u l d l i e on a s t r a i g h t line of slope b"l
i f p l o t t e d a g a i n s t bZ1 T h i s p l o t i s s h o 1 m i n F i g . 2. Ekperi-
mencal points f a l l TeasoEably close t o t h e predicr,ed value, except
f o r carbon dioxide mixtures. I n o r d e r t o r e t a i n t h e advantages of
the linear square rooc combination f o r calculating the comgressi-
b i l i t y f a c t o r of complex mixtures, a gseudo b value f o r carbon
dioxide :has Seen calcula2ed (Table 3 ) .

?,TO.four-component mixtures were p?epared and t h e i r gas law
deviations measured t o test Eethods of p r e d i c t i n g the c o z p e s s i -
b i l l c g f a c t o r s of complex mixtures. The measured values a r e
presented i n Table 4 , toger;her v i t h values c a l c u l a t e d ,by:

Linear combination of gas lair deviations of compone~%s, acd

idixture r u l e with binary i n t e r a c t i o n c o e f f i c i e n t s , e q g a t i o n !",

Linear sauare r o o t combination of gas law d e v i a t i o n s of p"-lre
components equation, ( 7 ) , w i t h t h e pseudo d e v i a t i o n for carbon
dioxide.

T'nese results i n d i c a t e that t h e c o n p r e s s i b i l j t f a c t o r of complex
mixtures can be c a l c u l a t e d by e i t h e r 2 ) o r 3 7 w i t h a n accuracy
of 0.035 o r better. This represents a d i s t i n c t improvement over
the linear combination.

Heating value s of ga s mid xtur e%s ai,r e sometimes i c alc u l a ted by
a sumling of terms of he f orm x where s th e r eal gas
fik?lheating value of the pure componknt. T h i st l a y i e l d s values

f o r t h e tvo mixtures i n Table 4 which are about 3 Btu p e r SCF, o r
almost 0.3$, too high. As a result of this study it is recomended
t'hat the heating value be calculated on the basis of the i d e a l gas
heating values, and be corrected f o r n o n i d e a l i t y by t h e mixture
compressibility f a c t o r , calculated by e i t h e r of t h e recomended
combining rules, methods 2 ) o r 3 ) . E5y t h i s procedure t h e e r r o r
of c a l c u l a t i o n f o r t h e mixtures i n Table 4 i s reduced t o only
0.3 Btu/SCF, o r 0.03%.

ACIINO?kmm

This papar summarizes the results of a continuing study of
"14easurement of Physical and Thermochemical P r o p e r t i e s of Fuel
Gases," sponsored by The Peoples Natural Gas Company s i n c e January,
1959. The authors r a s h t o e x p e s s t h e i r a p p r e c i a t i o n t o t h e sponsor
f o r permission t o p u b l i s h t h i s d a t a . IQ. B. X. Tike performed
most of the e x p r i m e n t a l measurements agd c a l c u l a t i o n s .

- 58 -

-1 EXPERIMENTAL ;LINEAR SQUAREROOT COMBINATION

0.015

0.0I O

0.005
0

- 59 -

Table 3 . GAS L A W DEVIATIONS OF NATURAL GAS COMPONENTS AT 6 0 O ~
AND ONE ATMOSPKECRE

Hydrocarbon Gas Law Source
Methane Dedation MI1
Ethane
Propane bZl-2 API=
Isobutane
0.0019 MI1
-n-Putane
0.0084 - IGT
Isopentane
0.0180 IGT
-n-Pentane IGT
0.0297
Nitrogen 0.0339 IGT
Carbon Dioxide 0.0518
0.0565 NBS4
o .0003
NEB4
0.0057*

* For c a l c u l a t i o n by l i n e a r square r o o t combination, u s e the

pseudo value, b = 0.0041.

Table 4. GAS LAW DEVIATIONS OF FOUR-COMPONENT NIXTURES AT
6 0 O ~AND ONE ATMOSPIIERE

Methane , mole % Nixture 1 Mixture 2
67.44 64.14
Ethane, mole $ 16.37 20.68
Propane, mole %
Mitrogen, mole $ 5.30 5.02
Carbon Dioxide, mole $ 1 0 .a9 0 -00

bm measured 0 .oo 10.16

0.00274 0.00331

bm calculated* (1) 0.00364 -0.00444

(2) 0.00292 0 00357
(3) 0.00285
o .00362

* bm c a l c u l a t e d from (1) bm = zxibi

(2) bm = Cxi2bi+C2xixjblj
( 3 ) bm = ( XibiJ 2 ) 2

- 60 - 1

LJTERATCiRE CITED r

Amertcan Petroleum Institute, "Selected Values of Properties i
of Hydrocarbons and Related Compounds," M I Research Project
\j
-44. Pittsburgh: Carnegie Institute of Technology, October 31,
1
I358
Burnett. E. S., J. Applied Mechanics 3 , A-136-40 (1936). 1
,
DavLd, 9. GG and Hamann, S. D., in Institution of Mechanical
Engineers, Themoapmics and' Transport Properties of Fluids,'I i

74-78. London: The Institution, 1958. 1

-Hilsemath, J. et al., "Tables of Thermal Proprties of Gases."

mBs Circular 5 c Tashington, D. C.: Govt. Print. Office, 1955.
Hirschfelder, J. O., CurtisstlC. F. and Bird, R. B., "Molecular
Theory of Gases and Liqulds, 132. New York: John Wley
and Sons, Inc., 1954.

.Jes'sen,F. W. and Lightfoot, J. H., Ind. Eng Chem. 28,

870-71 (1936)J u l y .
Kistemaker, J., Phgsica 11, 270-76 (1945) December.
Kistemaker, J., Physica 11, 277-86 (1945) Ikcember.

Rossin€, F. D. e t al., "Selected Values of Physical and Tkermo-

AdmynearmiiccanPPreotpreorletuEmsTInfsHtyidtruotceaRrebsoenasracnhdPRreoljaetcetd-44C.ompPoiutntdssb,urgh:

Carnegie Press, 1953.

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