R A L P H A . M O R G E N AND J . H O W A R D C H I L D S
1
U n iv e r s ity o f F lo rid a , G a in e sv ille, F la.
O
N E of th e handiest tools in chemical engineering calculations is th e use of an approxim ation m ethod when em pirical d a ta are lacking.M any of these approxim ation m ethods (3, 4, 5, 14, 15, 16, 18) depend on th e law of cor
responding states which, in essence, says th a t gases and vapors the same relative dis
tances from th eir critical states behave alike.
These m ethods have led to th e use of reduced tem pera
ture, pressure, or volume relations plotted against some factor whose value is to be d e te rm in e d . T h e red u ce d v a lu e is d e f in e d a s th e actual value divided by the critical value; i.e., reduced
tem perature, Tr, is actual T divided by critical T in absolute units.
In using th e general reduced sta te curves for determ ining z, the compressibility factor in th e equation, P V = zR T , it has been noted (3, 4, 5) th a t there is a d rift in th e error, depending on which gases are used to m ake th e curves. T his is also tru e in determ ining f / p , th e activ ity coefficient (15). T he purpose of th is paper is to analyze th is d rift an d to find a general factor to correct for it. N ew ton (15) recognized th a t hydrogen, helium, and neon needed a correction factor an d suggested th a t an em
pirical constant of + 8 be added to th e critical tem perature and pressure of each of these gases. T his is shown to be tru e for only a lim ited range.
E arly in th is analysis it was recognized th a t bo th z and f / p re
quired three correction factors dependent on T, P, and V, respec
tively. F o r z th e reason is obvious from its definition, z = P V / R T . A ctivity coefficient f / p is determ ined by evaluating E quation 1 from Lewis and R andall (13) by graphical in te
gration :
In d e te r m in in g c o m p r essib ility fa cto r a an d a c tiv ity c o ef
ficien t f / p , co rrectio n fa cto rs for th e law o f corresp on d in g s ta te s are in d ic a ted . E th y le n e a n d n itr o g en are u sed as th e referen ce, a n d th e cu rves o f ; vs. Pu a n d f / p v s. Pit are given for th e se g a ses. A c h a r t (F ig u re 3) gives a p o sitiv e correctio n fa cto r for th o s e g a ses (T j > 1) for w h ic h th e re
cip rocal o f th e c o m p r essib ility fa cto r a t th e c ritica l p o in t (l/z c ) is 3.35 or less; th e co rrectio n fa cto r m a y b e n eg lected for th o s e vapors (T r < 1) for w h ic h th e reciprocal o f th e c o m p ressib ility fa cto r a t 1/*« is 3.35 or le s s. A ch a rt (F igu re 4) gives a n e g a tiv e correctio n fa cto r for th o s e gases (T r > 1) for w h ic h th e reciprocal o f th e co m p ressib ility fa cto r a t l / z c is 4.0 or greater; F igu re 4 gives a p o sitiv e co rrectio n fa cto r for th o s e vapors ( Tr < 1) for w h ic h th e reciprocal o f th e c o m p r essib ility fa cto r a t 1/s« is 4.0 or greater. For th o s e co m p o u n d s w h o se l / z c valu es are b e tw e en 3.35 a n d 4.0, th e curve for z a n d / / p for e th y le n e and n itr o g en sh o u ld b e u sed w ith th e k n ow led ge t h a t valu es d rift in th e d irectio n in d ic a ted b y th e correctio n fa cto rs.
R T In
where
àP a t co n stan t T (1)
W ithin th e lim its of accuracy of these correction factors, the above equation is w ithin th e modification of T unnell’s E quatio n 1 (17). I f reduced values for P are plotted a g a in st//;» a t constant reduced T, generalized a ctiv ity coefficient curves are obtained.
T he error and d rift of values in these curves were found to be of th e same order of m agnitude and direction as those for th e z curves.
1 P resen t address, N atio n al A dvisory C o m m ittee for A eronautics L ab o ra
tories, Cleveland, Ohio.
P relim inary c alc u la tio n s showed th a t a correction to T r m ade th e largest contribu
tion to bringing the individual values onto th e general curve.
A correction to P r was found to be second in im portance, and a correction to V r was, in m ost cases, of m inor impor
tance. T he only exception is in th e vicinity of th e actual critical region, T r — 1 and P r = 1, where th e correction factor for V r becomes ap
preciable.
I t was next observed th a t those substances whose criti
cal constants were much lower th a n th e substances for which th e general curves were draw n, had deviations above th e reduced tem pera
tu re isotherm s when T r was greater th an 1 an d below th e iso
therm s when T r was less th a n 1. On th e other hand, those sub
stances whose critical constants were greater th a n th e reference substances deviated in exactly th e opposite direction.
T his suggested th a t if a curve were p lo tted using th e m aterial w ith th e lowest critical constants, helium , all th e deviations would be in th e sam e direction. However, there were insufficient reliable d a ta to m ake this curve, and th e correction factor would be too cumbersome for substances now satisfactorily handled w ith th e present curves.
I f th e law of corresponding sta te s were absolute, then a t the critical point, zc = P CV C/R T C would be th e sam e for all sub
stances. B y plotting l/z c against T c for various substances, there is a steady d rift from l / z c for helium = 3.28 and hydrogen = 3.27 w ith Tc a t 5.26° and 33.3° K ., respectively, to \ / z c for w ater
= 4.30 and am m onia = 4.12 w ith T c a t 647.3° and 405.56° K., respectively. I t was decided, therefore, to m ake a curve for a substance w ith an interm ediate value of l/z„. E thylene (9, 20), w ith a l/zc value = 3.58, was chosen; an d in th e range where d a ta could n o t be obtained for ethylene, values (10) for nitrogen, 1/ze = 3.43, were used. In th e overlapping range th e two check as closely as th e d a ta justify. T he curves of Figures 1 and 2 were obtained.
These curves agree closely w ith those published by Dodge (6), W eber (19), H ougen and W atson (7), and N ew ton (15). How
ever, values for th e individual hydrocarbons on the family from m ethane (CH<), 1 /ze = 3.46, to decane (C10H22), 1 / z c — 3.91, shift progressively from this curve w ith a regular deviation.
T he error is usually to indicate too high a value of z or f / p when T r is 1 or more, and too low a value when T r is less th an 1. For m ost purposes th e error is no t large enough to w arran t a calcu-667
REDUCEDPRESSURE 3.5 4-.0 4-.S 5.0 5.5 «.0
I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y
^ A l l l S O d
i y N / \ d = z
July, 1945 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 669
l a t e d c o r r e c t i o n f a c t o r e x c e p t t o have a knowledge of th e direction of th e error. A t its m a x i m u m , t h e d e v ia tio n u s u a lly does n o t exceed 2 or 3% .
W ith w ater, am m onia, an d o th er s u b s t a n c e s f o r which 1/2, > 4.0, th e error a t several places in th e curves becomes apprecia- ble. Two correo- Z tio n factors were
£ therefore m ade, one for substances for
^ ,• which 1/ze < 3.35,
; I an d th e o th er for
~ ® s u b s t a n c e s f o r s 1 which 1 Jzc > 4.0.
£ *
“ g Since th e equations
£ ■£ of these factors are
™ 1 c o m p lica ted , th e .s ; c o rre c tio n s w ere a | p lo tted and appear a j a s F i g u r e s 3
— : HTid 4.
I Î
1 2I J
C O R R E C T IO N F A C T O R S
O £ Te m f e e a t u b e T/
» | Wh e n- 1/z= < 3.35.
- .z Figure 3 w as
ob-~ J tained by tak in g a s e r i e s o f P Y T values for hydrogen S » (8) and adding a
© e;
c o rre c tio n fa cto r,
^ ® T/, oi such size
“ j t h a t a p s e u d o '■g i v a lu e o f t h e r e
-< J ’ duced tem perature,
^ : 1% w ould fit th e
£ z — Pr curve for N ethylene an d
nitro-“ gen. W hen p lotted i f o n l o g a r i t h m i c
“ paper, these values (Table I ) give ap- p r o x i m a t e l y a straig h t line whose slope is 2.03 and intercepts th e line for pseudo reduced tem perature T% = 1 a t T , = 1-05.
T h e e q u a tio n fo r th is line is, there
fore,
log Tj = log 1.05 + 2.03 log T% (2) or T/ =
1.05 T V - " (3)
670 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y for these values except th a t th eir use m akes th e values for hydro
gen an d th e o th er gases w ith low critical constants fit th e general positive direction give a V-shaped curve m ade up of two straight- line segm ents. T he num erical value of th e correction factor is approxim ation w hen em pirical d a ta are lacking.
NOMENCLATURE
July, 1945 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 671
Pr — pseudo reduced pressure, atm ospheres P/ = correction factor for critical pressure to get P i z = com pressibility factor, P V / K T
<= com pressibility factor a t th e critical sta te . P eV e/ R T „ a — (R T / P ) — v (deviation from perfect gas law)
L I T E R A T U R E C I T E D
(1) B a rtle tt, E . P., / . Am. Chem. Soc.. 49, 687,1955 (1927).
(2) B e attie , J. A., an d Lawrence, C. K., Ibid., 52, 6 (1930).
(3) Brown, G . G ., Souders, M ., and Sm ith, R . L., In d. En q Chem., 24, 513(1932).
(4) Cope, J . Q., Lewis, W . K., and W eber, H. C., Ibid., 23, 887 (1931).
(6) D odge, B .F .,/b t< i., 24,1353(1932).
(6) D odge, B. F „ “ Chemical E ngineering Therm odynam ics", pp.
161,238, New Y ork, M cG raw -H ill Book Co., 1944.
(7) H ougen, 0 . A., and W atson, K., “ In d u strial Chemical Calcula
tions” , 2nd ed., p. 398, 427, New Y ork, John W iley & Sons, 1936.
(8) In tern atio n al C ritical T ables, Vol. I l l , p . 4, New Y ork, M c
G raw -H ill Book Co., 1929.
(9) Ibid., Vol. I l l , p. 14.
(10) Ib id ., Vol. I l l , p. 18.
(11) K ay , W . B., In d. En g. Chem ., 28, 1014 (1936).
(12) K eenan, I. H „ an d K eyes, F . G ., “ Therm odynam ic Properties of S team ", N ew York, Jo h n W iley & Sons, 1936.
(13) Lewis, G. N ., and R andall, M ., “ T herm odynam ics” , p. 194, N ew Y ork, M cG raw -H ill Book Co., 1923.
(14) Lewis, W. K ., In d. En g. Chem ., 28, 259 (1936) (15) N ew ton, R . H „ Ibid., 27, 302 (1935).
(16) O thm er, D. F., Ibid., 32, 841 (1940).
(17) T unnell, G ., J . Phy». Chem., 35, 2885 (1931).
(18) W atson, K . M ., In d. En q. Chem ., 23, 360 (1931); 35, 398 (1943).
(19) W eber, H . C., "T herm odynam ics for Chemical E ngineers” , p. 198, New Y ork, Jo h n W iley <fc Sons, 1939.
(20) Y ork, R., and W hite, E . F., Tran». Am. Inet. Chem. Engrs., 40, 227 (1944).