A non-cubic equation of state for PVT and phase equilibrium calculation (pure compounds)

8/12/2019 A non-cubic equation of state for PVT and phase equilibrium calculation (pure compounds)

1/8

Fluid Phase Equilibria, 56 (1990) 39-57Science Publishers B.., !"s#erda" - Prin#ed in $e %e#herlands

A non-cubic equation of state for and phase equilibriumcalculations (pure compounds)

&i'ri' S'ae, Eniricerche *-+0 097 San 'na#' ilanese, *#al/

Abstract: A new quartic van der Waals-type equation of state is proposed. The repulsive part is an extension ofthe original form, with a behaviour similar to that of the rigorous expression by but retaining its original simpli-city. The aractive part has the same form as in the classical cubic !"#.

$orrelation of and vapour-pressure data proves that the new !"# is superior to the classical cubic equations.The laws of dependence of the parameters on temperature are simple and some general constraints can be so thatthe number of ad%ustable parameters can be reduced down to two only& they can be determined from one vapourpressure and one liquid density value and applied with confidence in a temperature range from the boiling pointup to supercritical temperatures.

With one more ad%ustable parameter, determined from an additional vapour pressure value, the applicationrange can be extended down to the triple point.

$ubic equations of state have found wide application for process design, especially calculations, as they o'er agood balance of simplicity and accuracy.

Their wea( point is in general their low accuracy in calculations )which does not a'ect their good accuracy incalculations, due to an error between the two phases*.

The need of a more accurate description of the behaviour of pure compounds and their mixtures becomes im-portant in particular conditions, such as those found in supercritical f luid extraction.

ForewordSi"le cubic hae been rendered s'his#ica#ed b/

"an/ au#h'rs in 'rder #' 'erc'"e #ha# ea2 'in#, bu# i#has bec'"e eiden# #ha# #heir accurac/ cann'# ' be/'nd a

cer#ain i# is en'uh #' c'"are #he alues ' #he cri#icalc'"ressibili#/ ien b/ an/ cubic i#h #he e4eri"en#al'nes #/ical calcula#ed alues are #'' hih b/ "'re #han10 and such an err'r is "ain#ained 'er #he cri#ical area.

#' n' #he need 'r "'re accura#e da#a has 'und as'lu#i'n in #he deel'"en# ' accura#e bu# c'"lica#edequa#i'ns, i#h "an/ ad8us#able ara"e#ers, h'se de-#er"ina#i'n requires #he aailabili#/ ' len#iul accura#ee4eri"en#al da#a, n'# aailable usuall/ #heir alica#i'nis #hus res#ric#ed #' a handul ' ell s#udied subs#ances.

*# is s#ill desirable #' hae an equa#i'n ' s#a#e i#h ali"i#ed nu"ber ' ara"e#ers and a be:er accurac/ #han i#can be a:ained b/ a cubic E;S.


2/8

3G G. Soave

Zre1+ b

vb/1.6(C)

Eqs. (1) #' (C) are c'"ared in Fi. 1 'r ari'us alues' b v . *# is seen #ha# Eq.(C)is er/ near #' Eq.( +) (-S) a##he l'er densi#ies, hile Eq.(3), less accura#e a# l' dens-i#ies, is be:er a# #he hiher 'nes. Eq.(1) (#he d> e4res-si'n) dieres raidl/ and is ruled 'u#.

Eqs. (3) and (C) can be c'nsidered as ar#icular cases '

#he "'re eneral 'r"

Zre=1+ c b

vb(5)

here c is a ac#'r hich, in rincile, sh'uld increasei#h #he "'lecular di"ensi'ns, as 'in#ed 'u# b/ >ils'nH1CI.

Eq. (5)as selec#ed #' e4ress #he reulsie ar# ' #hene JKL !n/ decisi'n ab'u# #he alue ' c is dela/ed un#il

a 'r" ' #he a:rac#ie ar# has been selec#ed and #he er-'r"ance ' #he ne JKL has been chec2ed.

Aractive part*n ar# #he nu"er'us an der >aals-#/e equa#i'ns '

s#a#e 's'ed in #he li#era#ure, #he a:ac#ie e@ec#s ' #he"'lecular in#erac#i'ns are e4ressed n ari'us 'r"s,hich can be c'llec#ed in#' a eneral e4ressi'n

Za:= a v

v++u b v+w b+

(6)

here #he adi"en#i'nal ac#'rs u , w are ei#her as-sined, 'r c'ns#rained, 'r reel/ ad8us#ed.

$e ?rs# case is #ha# ' #he s'-called #'-ara"e#er cu-bic equa#i'ns, includin e.. #he 'rininal d> JKL (u D wD 0), #he Medlich-N'n (O ) JKL and i#s "an/ aria#i'ns(u D 1, v D 0) and #he Pen-M'bins'n (QO) JKL (u D +, w DR1). !ll #h'se equa#i'ns c'n#ain #' ara"e#ers 'nl/,h'se alues (a# leas#, #heir alues a# #he cri#ical #e"er-a#ue) can be (and are usuall/) de#er"ined b/ i"'sin #hecri#ical c'ns#rain#s. $e/ are s' ?4ed de?ni#el/ and hae#he sa"e alues 'r all c'"'unds, hich is a seri'us li"-i#a#i'n, in ar#icular 'r Q T calcula#i'ns, since i# can besh'n easil/ #ha# #he cri#ical is'#her"s hae a di@eren#'r" r'" subs#ance #' subs#ance.

! sec'nd #/e ' JKL r'" #he li#era#ure (Sch"id#,

>en=el H15I Uar"enss V Nna H16I Pa#el V We8a H17I),al#h'uh ri:en in di@eren# a/s, can be c'llec#ed in#' asinle 'r", i#h 'ne "'re ara"e#er hich is leX ree b/#he cri#ical c'ns#rain#s, and 'be/in #he c'ndi#i'n u Y w Dl. Such c'ns#rain# has n' #he're#ical 8us#i?ca#i'n, bu# inac# a 3-ara"e#er JKL ' #ha# #/e is al"'s# as ''d as aC-ara"e#er 'ne, as i# ill be sh'n ur#her 'n.

$e c'rrela#i'n ' ure-c'"'nen# PVT and a-'r-ressure da#a /ield ala/s alues ' u ,w such as #'ie a 'si#ie discri"inan# (u +Z C w ) b + #he quadra#ic e4-ressi'n a# #he den'"ina#'r ' #he a:rac#ie ar# ' Eq.(6)has #hus ala/s real r''#s and can be relaced, i#h'u#an/ real li"i#a#i'ns, b/ a r'duc# ' #' bin'"ials, hichhas #he adan#ae ' enablin an easier anal/#ical #rea#-

"en#

Za:= a v

(v+d)(v+e)(6A)

$e ne equa#i'n ' s#a#e has #hen #he 'll'in 'r"

Z=1+c b

vb

a v

(v+d)(v+e)(7)

Eq. (7) is ' 'ur#h deree in eneral (e4ce#in #hecases i#h c D 1 'r e D 0) and has + 'r C real r''#s (a# su-er- 'r subcri#ical #e"era#ures, resec#iel/), 'ne 'hich 'u#side #he all'able rane r'" =er' densi#/ #' 1 b .

Fig. 1.Meulsie Z(Eqns. 1 #' C )

0 0.05 0.1 0.15 0.+ 0.+5

1

+

3

C

Table I.E4eri"en#al da#a used 'r c'rrela#i'n

'"'undT [\] T [^_ P[\] P[^_

a#as'urce

N N bar bar (T ` T )

ar'n G6 300 0.GG 1000 H1I

ni#r'en 6C C00 0.15 1000 H9I"e#hane 91 360 0.1+ 1000 H10I

e#h/lene 113 600 0.005 3000 H11I

e#hane 1+0 700 0.003 G00 H1+I

r'ane 1C5 600 0.0015 +00 H1I

n-bu#ane 170 511 0.0011 6G0 H1I

ben=ene +GG 673 0.07G0 +G95 H13I

carb'n di'4ide +1G 6+0 5.5 600 H1I

h/dr'en sul?de +G+ C93 13.3 1706 H+I

"e#h/l chl'ride 175 C9G 0.00G 30C H3I

#riu'r'"e#hane 1+5 C53 0.00+ +00 HCI

a""'nia +03 760 0.11 5000 H5I

ace#'ne +59 53G 0.0C+ 300 H6I

"e#han'l +GG 573 0.09G 690 H7I

e#han'l +93 6+3 0.05G 690 HGI


3/8

G. Soave 39

nli2e cubic JKL As, n' direc# s'lu#i'n r'cedure isaailable #his is a drabac2, bu# 'nl/ i#h a n'n-cubicJKL a real i"r'e"en# can be 'b#ained, as i# ill besh'n ur#her 'n. *n ar#icular, a s#r'nl/ nea#ie (andeen, near #' Zb ) alue ' #he ara"e#er e is essen#ial 'r a''d behai'r a# #he hiher densi#ies. !s an indirec# r'',i #he a:rac#ie ar# ere ("'re c'""'nl/ bu# n' less e"-iricall/) e4ressed b/ a series ' densi#/ 'ers, hih

'ers 'uld be required #' e4ress #he s#r'n deend-ence ' ressure 'n densi#/, a# hih alues ' #he la:er.'re si"l/, #his is e4ressed b/ Eq.(6A) i#h s#r'nl/ ne-a#ie alues ' e .

$e Pen-M'bins'n JKL , hich is clai"ed #' be ar#ic-ularl/ sui#ed 'r QT calcula#i'ns, has in ac# a nea#ie-alue ' e un'r#una#el/, #' ara"e#ers 'nl/ (b'#hde?ned b/ #he cri#ical c'ns#rain#s) are #'' e 'r a reall/sa#isac#'r/ JKL.

Experimental dataW' 2inds ' da#a "us# be c'nsidered in 'rder #' de-

el' an JKL i#h a eneral alica#i'n a'ur-ressure da#a an accura#e descri#i'n ' ure-c'"'und a'ur ressures is a er/ s#ric# cri#eri'n and isessen#ial 'r a ur#her alica#i'n #' "ul#i-c'"'nen#hase equilibria. E4eri"en#al a'ur ressure da#a can berelaced b/ alues enera#ed b/ sui#able e4ressi'ns, suchas #he C-ara"e#er >aner equa#i'n

ln P

Pc

=a1 (1Tr )+a+(1Tr )1.5+a3(1Tr)

3+aC (1Tr)6

'r hich an e4haus#ie c'"ila#i'n ' c'ns#an#s has beenien b/ c&arr/ H1GI and la#er resu"ed and enhanced b/

Meid et al . H19I. PVT da#a. Par#icularl/ cri#ical are #he liquid densi#/alues, hich a@ec# "'s# #he ara"e#er alues e4eri-"en#al da#a can be relaced b/ alues enera#ed b/ #he ac-cura#e Mac2e: equa#i'n

vsa#d =

R Tc

PcZMa

1+(1Tr )+/ 7

(G)

h'se ara"e#ers can be 'und in a c'"ila#i'n b/ Sen-cer and !dler H+0I.PVT da#a a# suercri#ical #e"era#ures are n'# len#iul in#he li#era#ure in s'"e cases #heir alues can be enera#edb/ er/ accura#e (and c'"le4) equa#i'ns ' s#a#e, h'seara"e#ers are aailable 'r a e c'"'unds.

Parameter dependence on temperatureB/ unc'ns#rained c'rrela#i'n ' sinle-#e"era#ure

da#a, #he 'b#ained ara"e#er alues e4hibi#ed an erra#ic#rend, due #' b'#h #he inaccurac/ ' #he da#a and #he inad-equac/ ' #he "'del such erra#ic behai'ur is ar#icularl/eiden# near #he cri#ical #e"era#ure and 'r #h'se ara-

"e#ers (li2e c ) hain a s"aller inuence 'n resul#s.%eer#heless s'"e c'ns#an# behai'urs can be iden#i-

?ed, n'# deendin 'n #he ar#icular c'"'und e4a"ineda) D b P cR T c is al"'s# c'ns#an# i#h #he #e"era#-

ure i leX #' ar/, #he resul#s 'uld be li:le i"r'ed and'ne "'re ad8us#able ara"e#er 'uld hae #' be de#er"-ined, i# as s' assu"ed as c'ns#an#

=c (9)b) f D a P cR T cand g D d P cR T cdecrease i#h increas-

in #he #e"era#ure and a #endenc/ is qui#e eiden# 'rb'#h #' ' #' =er' 'r T #endin #' in?ni#/ (hich, 'r a, ish/sicall/ "eaninul).

F'r g, a ara"e#er i#h s"all inuence 'n resul#s, a

si"le inerse la can be assu"ed

g=gc

Tr(10)

hile 'r f a "'re e4ible e4ressi'n as used, i#h#' ad8us#able c'ecien#s

f=fc h (Tr ) (11)

(Tr )=1+(1Tr) (m+ nTr )

Tr

(1+)

c) D e P cR T c is al"'s# c'ns#an# and ar'aches #healue R al#h'uh i# a@ec#s resul#s c'nsiderabl/, "ainl/ a#hih densi#ies, i# is dicul# #' de?ne an e4ressi'n 'r i#,#hus 'r #he sa2e ' si"lici#/ i# as assu"ed as c'ns#an#

i=ic (13)Basin 'n #he ab'e c'nclusi'ns and assu"#i'ns, #he

?nal 'r" ' #he ne equa#i'n ' s#a#e 'r a ure c'"-'und is

Z=P v

R T=1+c

b

vb

ach( Tr )v

(v+dc /Tr )(v+e)(1C)

'r

Z=x Pr

Tr=1+c

x

fc (Tr )x

(x+gc/Tr)(v+)(15)

here

Table II.c alues r'" unc'ns#rained c'rrela#i'n'"'und %'.P#s. c j Pk , j

ar'n 90 1.69C 1.1

ni#r'en GC 1.6G+ 1.0

"e#hane GC 1.700 0.9

e#h/lene G+ 1.G1C 1.3

e#hane G0 1.736 1.3

r'ane 91 1.630 1.6

n-bu#ane 75 1.55G +.0

ben=ene 51 1.5+1 3.+

carb'n di'4ide 100 1.716 1.7

h/dr'en sul?de 7G 1.C50 1.9

"e#h/lchl'ride GG 1.109 1.9

u'r''r" G0 0.96C 1.9

a""'nia 100 1.+G5 1.7

ace#'ne ++ 1.33C 1.C

"e#han'l1 C7 1.C66 +.9

e#han'l 63 1.6C3 1.6

O.m.L. 1.G1n' liquid densi#/ da#a used.


4/8

C0 G. Soave

x=vPc

R Tc (16)

ritical constraints*"'sin #he cri#ical c'ns#rain#s, i.e. 'rcin #he ?rs#

and sec'nd deria#ies ' ressure i#h 'lu"e #' be =er'a# #he cri#ical 'in#, besides bein #he're#icall/ c'rrec#, hasa d'uble adan#ae ?rs#, i# decreases #he nu"ber ' ad-8us#able ara"e#ers b/ relacin #' ' #he" i#h #he cri#-ical c'ns#an#s T c, P c, hich are 'Xen 2n'n sec'nd, i# en-sures a c'rrec# behai'ur in #he icini#ies ' #he cri#ical'in# (nelec#in, 'bi'usl/, #he n'n-anal/#ic e@ec#s in #hei""edia#e surr'undins ' i#).

Since #he ara"e#er alues ere ri:en as rac#i'ns '#he ac#'r (R T cP c), i# is c'nenien# #' relace #he cri#ical'lu"e v b/ Z (R T P ) #he e4#ra ara"e#er Z cis de?nedb/ Eq. (7) as a unc#i'n ' #he '#her 'nes. >e hae #hus (a##he cri#ical #e"era#ure) 6 ara"e#ers fc, , c, gc, , Z c, 3 'hich (sa/ fc, gc, ) are de?ned #hr'uh Eq. (7) and #hecri#ical c'ns#rain#s.

!nal/#ical de#ails are ien in #he!endi4 .

!ata correlation " first results$e s#ra#e/ 'll'ed here as #' add aailable hih

#e"era#ure da#a #' an equal nu"ber ' a'ur-ressureand sa#ura#ed-liquid densi#/ da#a, hich ere enera#ed b/#he >aner and Mac2e: equa#i'ns resec#iel/, i#h ara-"e#ers r'" #he li#era#ure (Meid et al . H19I).

'"'unds e4a"ined, #e"era#ure and ressureranes and da#a s'urces are re'r#ed in Wab. *.

a#a ere aailable 'r a #'#al ' 16 c'"'unds, in-

cludin C in'ranic ases (!r, %, ;, US), 6 h/dr'car-b'ns r'" "e#hane #' ben=ene, C 'lar, n'n-ass'cia#inc'"'unds (Uol, UFo, %Uo, ace#'ne) and + alc'h'ls("e#han'l and e#han'l). $e #/es ' c'"'unds and #hec'ndi#i'n ranes are ell reresen#a#ie.

F'r #he alc'h'ls n' liquid hase densi#/ da#a ere used,since #he 'ccurrin ' s#r'n ass'cia#i'n hen'"ena"a2es #he" "isleadin.

$e ab'e da#a ere c'rrela#ed #' de#er"ine #he E;Sara"e#ers ' each c'"'und, i#h 'r i#h'u# alica#i'n' ari'us c'ns#rain#s. $e 'b8ec#ie unc#i'n #' be "ini"-i=ed as

( ZcalcZe4 1)+

+100 (ln hd ln h )+

here b'#h Z ^p and Z _ and #he uaci#/ c'ecien#s ere ealua#ed a# #he e4eri"en#al #e"era#ures and res-sures.

$e di@erence (ln R ln ) as assu"ed 'r #he sa2e' si"lici#/ as an ar'4i"a#i'n ' #he "'re ri'r'us rel-a#ie err'r 'n a'r ressure #he di@erence is s"all and#he i#era#ie calcula#i'n ' a'ur ressure c'uld #hus be

a'ided.$e resul#s ' #he ?rs# unc'ns#rained c'rrela#i'ns (bu#

al/in #he cri#ical c'ns#rain#s) are re'r#ed in Wab. **,here #he jP k, j #er" is #he ?nal 'erall OmL err'r

100 [ (jPkPk)++(j) +]/(+ n).er/ l' residual err'rs ere 'b#ained in all cases, re-

ardless ' #he na#ure ' #he c'"'und ('lar 'r n'#) #hisas a ?rs# c'n?r"a#i'n ' #he s'undness ' #he ne equa-#i'n.

$e alues 'b#ained ' #he ara"e#er c ere qui#e er-ra#ic, i#h n' eiden# c'nnec#i'n #' #he na#ure 'r "'lecu-

Table III. Mesul#s r'" c'rrela#i'ns i#h c D 1.6 (ad8us#able ara"e#ers , Z , m , n )

'"'und Z m n g uY w j,

ar'n 0.0950 0.3069 0.39GG 0.0+5 +.33 R0.GG R0.59 1.1

ni#r'en 0.093G 0.3073 0.CC97 0.0+G +.33 R0.G7 R0.56 1.1

"e#hane 0.09+0 0.3065 0.CC70 0.0119 +.3G R0.G6 R0.53 1.0

e#h/lene 0.0GC+ 0.310+ 0.5G11 0.00G5 +.3C R0.77 R0.+C 1.5

e#hane 0.0GC9 0.30GC 0.599C 0.0013 +.39 R0.G0 R0.30 1.C

r'ane 0.0G71 0.3010 0.6055 0.01C3 +.60 R0.G6 R0.50 1.6

n-bu#ane 0.0G65 0.+9G3 0.6390 0.0+0+ +.69 R0.G7 R0.5+ +.0

ben=ene 0.0GC7 0.+966 0.63+6 0.037+ +.77 R0.G7 R0.50 3.+

carb'n di'4ide 0.0G7C 0.+99+ 0.65CC 0.0++6 +.65 R0.G7 R0.53 1.7

h/dr'en sul?de 0.0966 0.+97G 0.CC+5 0.036+ +.53 R0.9+ R0.7+ +.0

"e#h/l chl'ride 0.0GC3 0.+G91 0.57G5 0.0+0+ 3.00 R0.90 R0.60 +.0

#riu'r'"e#hane 0.0765 0.+G77 0.7659 0.037+ 3.+C R0.G6 R0.C1 +.1

a""'nia 0.06GC 0.+797 0.79G7 0.0++6 3.79 R0.GC R0.+C +.1

ace#'ne 0.0717 0.+753 0.7G51 0.036+ 3.G+ R0.90 R0.50 1.6

"e#han'l1 0.05G9 0.+515 1.0950 R0.051 5.CC R0.93 R0.57 3.C

e#han'l 0.0G10 0.+600 0.77C6 0.0359 3.G5 R0.9G R0.9+ 1.3

O.m.L. 1.91%' liquid densi#/ da#a used.


5/8

G. Soave C1

lar di"ensi'ns ' #he c'"'unds all hiher #han uni#/and r'ued aaren#l/ ar'und #ha# alue 1.6 hich assh'n #' 'rce #he aree"en# ' #he reulsie ar# ' #heJKL i#h #he -S e4ressi'n a# l' densi#ies. $is is inc'n#ras# i#h #he c'nclusi'ns ' >ils'n, acc'rdin #'h'" i# sh'uld increase i#h #he "'lecular di"ensi'ns,s#ar#in r'" a alue + such behai'ur as n'# 'und andc'uld be due #' an inedaqua#e a:rac#ie ar# ' #he JKL.

Reducin# the number of ad$ustable parametersWab. ***sh's #he resul#s ' a ne c'rrela#i'n i#h c

?4ed a# a alue 1.6 (C ad8us#able ara"e#ers , Z , m , n )#he err'rs ere rac#icall/ #he sa"e as #h'se i#h c ad8us-#ed #' #he bes# alue, a r'' ' i#s "in'r i"'r#ance.

! ur#her reduc#i'n in #he nu"ber ' ree ara"e#ersc'uld be necessar/ hen #he aailable da#a are er/ li"-i#ed 'r hen #he da#a #he"seles are ' ''r quali#/, #'reen# #he ara"e#ers r'" assu"in unrealis#ic alues.

Fr'" #he alues ' #he ara"e#ers and ' s'"e deriedac#'rs re'r#ed in Wab. ***, #he 'll'in c'nclusi'ns can

be dran $e resul#s are er/ sensi#ie #' #he alue ' , #ha#cann'# be es#i"a#ed a priori and "us# be #rea#ed as ad-8us#able. $e ra#i' g aries "ar2edl/, assu"in hih alues'r 'lar c'"'unds. Een i i# a@ec# li:le #he resul#s, i#ssread n'# all' #' assu"e a c'ns#an# alue 'r all c'"-'unds.

$e ra#i' aries i#hin a narr' rane /e# #hisara"e#er has a s#r'n inuence 'n resul#s and cann'# be#rea#ed as a uniersal c'ns#an#. $e alues ' u D (g Y ) and w D g ar/ sear-a#el/ 'er ide ranes, bu# #heir su" re"ains rela#iel/c'ns#an# since i# has a s"all inuence 'n resul#s, i# can be?4ed a# a c'ns#an# alue

u+w=(gc+i)

+

gc

+=0.5 (17)

$e !endi4 ies #he (i#era#ie) calcula#i'n r'cedure #'de#er"ine Z , in resence ' #ha# c'ns#rain#.>i#h #he ab'e assu"#i'n, #he ad8us#able ara"e#ers leX

Table IV.M''# "ean square jPsa# , ji#h ari'us assu"#i'ns

case a c D 1, g D , D 0 (Medlich-N'n)

b c D 1, g D +.C1C , D R0.C1C (Pen-M'bins'n)

c c D 1, D 0

d c D 1, u Y w D 1 (3-ara"e#er e'sAs r'" li#era#ure)

e c D 1, g D g D (an der >aals Y Penel'u4 'lu"e shiX)

c D 1, n' c'ns#rain#

c D 1.6, n' c'ns#rain# (see Wab. *)

h c D 1.6, u Y w D R0.5

i c D 1.6, D 0

ase a b c d e h i

n'.ara". + + 3.0 3.0 3.0 C.0 C.0 3.0 3.0

ar'n 3.G 5.C 3.0 +.7 +.9 +.5 1.1 1.1 +.9

ni#r'en C.1 5.G 3.C 3.3 3.+ +.6 1.0 1.0 3.3

"e#hane C.1 5.G 3.C 3.3 +.9 +.6 1.0 1.0 3.3

e#h/lene 5.3 5.+ C.+ 3.9 3.7 3.+ 1.5 1.C C.1

e#hane 5.C C.+ 3.G 3.C 3.7 +.9 1.C 1.C C.0

r'ane 6.+ 3.+ 3.7 3.+ 3.5 +.5 1.6 1.6 C.5

n-bu#ane 7.7 3.6 C.3 3.6 C.+ +.9 +.0 +.0 5.+

ben=ene 9.5 C.6 5.5 C.6 5.3 3.9 3.+ 3.+ 6.1

h/dr'en sul?de 7.5 +.7 3.0 +7.0 +.G +.6 1.7 1.7 3.0

"e#h/l chl'ride 6.G C.9 C.6 C.1 C.7 +.5 +.0 +.3 C.9

#riu'r'"e#hane 10.9 3.+ +.9 +.0 C.G 1.9 +.0 1.G C.3

#riu'r'"e#hane 1C.1 6.0 3.C 1.9 C.1 1.9 +.1 +.+ 5.1

a""'nia 1G.6 10.0 3.6 +.0 C.+ +.0 +.1 +.3 C.C

ace#'ne +5.0 13.6 3.C 1.9 5.0 1.9 1.6 1.6 C.5

"e#han'l1 3+.C ++.3 5.3 3.C 6.7 C.9 3.C 3.C 5.G

e#han'l +1.6 1+.0 5.0 3.5 5.G t 3.0 +.G 5.6

O.m.L. 1C.1 G.6 C.0 3.+ C.3 (+.9)+ 1.9 +.1 C.51%' liquid densi#/ da#a used+usin 'r e#han'l #he err'r in c'l. d.


6/8

C+ G. Soave

are 3 'nl/, i.e. and #he c'ecien#s m , n ' (T v) (Eq. 1+). $e alues ' #he c'ecien# n ' (T v) are ala/ss"all, b'#h 'r 'lar and n'n-'lar c'"'unds.e:in n D 0 all's a ''d rer'duc#i'n ' e4eri"en#ala'ur ressures in a "'re li"i#ed rane (sa/, r'" 0.1bars #' P ) and 'uld reduce #he nu"ber ' ad8us#ableara"e#ers #' + 'nl/. Such assu"#i'n c'uld be ad'#ed i#h a er/ li"i#ednu"ber ' e4eri"en#al da#a, as i# ill be sh'n ur#her 'n.

omparin# the di%erent assumptionsWab. * re'r#s #he resul#s 'b#ained i#h ari'us as-

su"#i'ns.S'"e seci?c c'nclusi'ns can be dran

>i#h #'-ara"e#er equa#i'ns ' s#a#e (M-N and P-M,cases a , b resec#iel/), #he resul#s ere acce#able 'nl/'r lih#, n'n 'lar subs#ances. *n #he '#her cases, be:er resul#s ere 'b#ained in en-eral i#h c D 1.6 #han i#h c D 1. >i#h c D 1 (cubic JKL As) #he resul#s 'b#ained i#h #he

c'ns#rain# u Y w D 1 (case d) ere 'nl/ slih#l/ 'rse #hani#h'u# c'ns#rain#s (case ). $e case ' #he an der >aals JKL i#h a 'lu"e#ransla#i'n acc'rdin #' Penel'u4 (case e) ae 'rse res-ul#s #han i#h #he '#her 3-ara"e#er 'r"s. >i#h c D 1.6 and #he c'ns#rain# u Y w D R0.5 (case h, 3ara"e#ers) #he resul#s ere al"'s# #he sa"e #han i#h'u#

c'ns#rain#s (case , C ara"e#ers), #he bes# 'erall. $e case i, i#h c D 1.6 and D 0 (a cubic JKL r''sedb/ W'chii et al . H+1I), is aain a r'' #ha# a nea#iealue ' is essen#ial 'r a real i"r'e"en#.

!s a ?nal c'nclusi'n, #he bes# resul#s ere 'b#ainedhen #he ara"e#er c as leX ree #' assu"e nea#ie al-ues and i#h alues ' c di@eren# r'" #he uni#/ (i.e. i#hn'n-cubic JKLAs) in ar#icular a alue c D 1.6 l''2s #' be

#he bes# uniersal alue.

&sin# limited' low-temperature data*# "a/ haen #ha# 'r cer#ain subs#ances er/ li:le in-

'r"a#i'n is aailable. iquid densi#/ da#a are 'Xen aail-able ('r can be "easured easil/ an/a/), eseciall/ bel'#he b'ilin 'in#. $is 'ne is enerall/ 2n'n, hile '#hera'r ressure da#a are n'# ala/s aailable and "a/ n'#be "easurable 'er a cer#ain #e"era#ure, 'r #he 'ccur-rence ' dec'"'si#i'n hen'"ena.

!s #' #he cri#ical c'ns#an#s, in #he case #he/ are n'#aailable 'r cann'# be "easured (ar#icularl/ 'r hea/

subs#ances) #he cri#ical #e"era#ure can be es#i"a#ed i#hsucien# accurac/ b/ seeral r'cedures, hile a c'nsis#-en# alue ' #he cri#ical ressure can be 'b#ained b/ c'rrel-a#in a# leas# #' liquid-densi#/ alues i#h #he Mac2e:equa#i'n, as r''sed b/ e#ere H++I.

$e leas# aailable da#a are PVT da#a a# suercri#ical#e"era#ures. $is ies rise #' #he ques#i'n he#her ara-

Table V.Para"e#er alues r'" #he a#"'sheric b'ilin 'in#, T w, and #he sa#ura#ed liquid densi#/ a# T w. !lica#i'n #'#he calcula#i'n ' a'r ressures and liquid densi#ies a# T w #' 0.9 T and as densi#ies a# T ` T .

case (a) c D 1.6, u Y w D R0.5, n D 0

case (b) c D 1, u Y w D 1, n D 0

(a) (b)

T x T T ` T T x T T ` T

T w, N jP sa# jy jz jP sa# jy jz

ar'n G7.3 0.7 1.3 1.3 +.1 5.0 3.5

ni#r'en 77.C 0.5 1.3 1.1 1.7 C.7 C.G

"e#hane 111.6 0.C 1.3 1.1 +.1 C.G C.C

e#h/lene 169.3 1.0 1.1 1.6 1.0 C.0 5.1

e#hane 1GC.6 0.5 1.1 1.5 1.5 C.1 C.3

r'ane +31.1 0.3 1.0 1.6 1.+ 3.7 3.1

n -bu#ane +7+.7 0.3 1.0 +.C 0.9 3.C C.6

ben=ene 353.+ 0.9 1.0 C.5 0.9 +.G 6.5

carb'n di'4ide +1G.1 0.3 1.+ +.1 0.+ 3.3 3.1

h/dr'en sul?de +G+.0 0.1 1.5 6.+ 0.5 3.3 7.+

"e#h/l chl'ride +C9.1 0.C 1.+ +.C 1.1 3.+ 1.G

#riu'r'"e#hane 191.0 0.C 1.1 +.9 0.C +.C +.6

a""'nia +39.G 0.3 1.7 3.1 0.G 1.6 +.3

ace#'ne 3+9.+ 0.3 1.+ +.5 0.6 1.6 +.C

O.m.L. 0.5 1.+ +.G 1.+ 3.6 C.3

l'er #e"era#ure li"i# 'r liquid densi#/ da#a


7/8

G. Soave C3

"e#er alues deried r'" da#a (a'r ressures and liquiddensi#ies) a# subcri#ical #e"era#ures "a/ be used a# hih#e"era#ures als'.

F'r #ha# reas'n a #rial as d'ne, here 'nl/ #' da#aere used #' de#er"ine #he E;S ara"e#ers #he a#"'-sheric b'ilin 'in# t w and #he sa#ura#ed liquid densi#/ a#tw. *n 'rder #' reduce #he nu"ber ' ad8us#able ara"e#ers#' + 'nl/, #he 'll'in assu"#i'ns ere "ade c D 1.6, n

D 0 and u Y w D R0.5. F'r #he sa2e ' c'"aris'n, als' #hecase i#h c D 1, u Y w D 1 and n D 0 as c'nsidered.

Subsequen#l/, #he alues ' #he ara"e#ers s' de#er"-ined ere alied #' all PVT da#a aailable a# suercri#ical#e"era#ures and #' a'ur ressure and liquid densi#/alues enera#ed in #he #e"era#ure rane r'" #he b'ilin'in# #' 0.9 #i"es #he cri#ical #e"era#ure. $is da#a se# is#/ical ' #he c'ndi#i'ns in'led in a suercri#ical e4#rac-#i'n r'cess.

$e resul#s are sh'n in Wab. .>i#h c D 1.6 and #he c'ns#rain# u Y w D R0.5, a'ur

ressures and liquid densi#ies ere rer'duced ell 'erall #he rane, hile hih-#e"era#ure da#a ere rer'-

duced i#h al"'s# #he sa"e accurac/ #han #ha# b/ direc#c'rrela#i'n ' #he h'le da#a se#s (cr. Wab. *, case h, c'-erin #' a ider rane, bu# i#h 'ne "'re ad8us#able ara-"e#er) #his is a c'n?r"a#i'n ' #he c'nsis#enc/ ' #he r'-'sed JKL .

$e eneral cubic JKL i#h c D 1, u Y w D 1 and n D 0ae hiher err'rs in all cases, in ar#icular 'r #he liquiddensi#/ near #he cri#ical #e"era#ure and, a"a=inl/, n'n-'lar c'"'unds. $'se resul#s rule 'u# de?ni#el/ #ha# al-#erna#e, hich c'uld s#ill l''2 a:rac#ie 'r i#s rea#er si"-lici#/. $e ab'e r'cedure as n'# alied #' #he alc'-h'ls, h'se densi#/ da#a can be "isleadin, due #' #he 'c-currin ' ass'cia#i'n hen'"ena.

$e ara"e#ers de#er"ined b/ #he ab'e r'cedure

cann'# be alied bel' #he a#"'sheric b'ilin #e"era#-ure, since calcula#ed a'ur ressures 'uld diere ra-idl/. *# c'uld be sh'n h'eer #ha#, b/ in#r'duc#i'n ' asec'nd ad8us#able ara"e#er n , a'ur ressure and liquiddensi#/ da#a c'uld be rer'duced accura#el/ d'n #' #he#rile 'in#.

!s a #/ical. e4a"le, #he alues ' , m , n ere de-#er"ined 'r e#hane r'" i#s a#"'sheric b'ilin 'in#, #heliquid densi#/ a# #he sa"e #e"era#ure and #he a'urressure a# #he #rile 'in# (90.C N, P {^| D 1.1C5}10~ bars)b/ al/in #he ara"e#ers s' de#er"ined #' a da#a se#enera#ed 'er a #e"era#ure rane r'" #he #rule 'in##' 0.9 T, #he OmL ere 1.G 'r a'ur ressure and +.C

'r liquid densi#ies.

AppendixApplication of the critical constraints

e# us ri#e Eq.7 as

Z=1+c( vvb1)a

de(v

v+d

v

v+e) (!.1)hence

P

R T=

1

v+c( 1vb 1v)

a

de( 1v+d 1v+e) (!.+)1R T(

P

v)T=1

v++c[ 1(vb)+1v+ ]

a

de[ 1(v+d)+ 1(v+e)+ ](!.3)

1+ R T(

+

Pv

+ )T=1

v3+c[

1(vb)3

1v

3 ]

a

de[ 1(v+d)3 1(v+e)3 ](!.C)

e:in n'

fc=ac Pc

R Tc

=Pc b

R Tc

gc=Pc dc

R Tc

=Pc e

R Tc

Zc=Pc vc

R Tcand

A=ac

dce=

fcgc

(!.5)

B= vc

vcb=

Zc

Zc(!.6)

=vc

vcdc=

Zc

Zc+gc(!.7)

!=vc

vce=

Zc

Zc+ (!.G)Eqs. !.1, !.3 and !.C are ri:en

{1+c(B1)A (!)=Zc1+c(B+1)A (!++)=01+c(B31)A (!33)=0

e:in n', 'r #he sa2e ' si"lici#/1=1+c(B1)Zc (!.9)

+=1+c(B+1) (!.10)

3=1+c(B31) (!.11)

e hae

A=1

!

=+

!

+

+=

3

!

3

3 (!.1+)

hence

(+ !)=+

1(!.13)

( !)=(+1)+

(31) (!.1C);nce #he su" and #he r'duc# ' and ! are 2n'n,

#he/ can be 'b#ained as #he r''#s ' #he equa#i'n

y+y(+!)+(!)=0

i.e.


8/8

CC G. Soave

" !=(+!)(+!)+C ( !)

+ (!.15)

(assu"in #ha# g ` , #he Z sin alies 'r , #he Ysin 'r ! ).

$e s'lu#i'n r'cedure s#ar#s r'" assined ('r #en#a#-ie) alues ' c , ,Z calcula#e B (Eq. !.6)

calcula#e , , o, (Eqs. !.9 #' !.11) calcula#e ( Y ! ), ( ! ) (Eqs. !.13 and !.1C) calcula#e , ! (Eq. !.15) calcula#e g, (Eqs. !.7, !.G) calcula#e A (Eq. !.1+) calcula#e f (Eq. !.5)

* a ur#her c'ns#rain# is i"'sed,i# is sa#is?ed b/ aseci?c alue ' Z , hich is 'und b/ #rial, and err'r.

e# us c'nsider #he ar#icular case hen #he alue ' uY w is assined. $is ies rise #' #he equa#i'n

1+ !( +B1)+ !

B

!

(1 1

B

)

+ =u+w (!.16)

*n #his case #he alues ' and ! need n'# be de#er"-ined 'n each i#era#i'n. $e ab'e r'cedure is s#ar#ed n'i#h a #en#a#ie alue ' Z and s#'ed aXer #he ?rs# 3s#es a chec2 is "ade acc'rdin #' Eq.16 i n'# sa#is?ed, ane alue

' Z is assu"ed and #he r'cedure s#ar#ed aain. ;nl/hen Eq. !.16 is sa#is?ed, #he ur#her s#es are e4ecu#ed.

ReferencesH1I %.B. araXi2, Wables 'n #he #henn'h/sical r'er-

#ies ' liquid and ases, Ue"ishere Publ. '., >ash-in#'n, , 1975.

H+I .. eis, >.. Frederic2s, 'lu"e#ric r'er#ies 'suercri#ical h/dr'en sul?de, . he". En. a#a. 13(196G) CG+ZCG5.

H3I .. Usu, .. cNe:a, Pressure - 'lu"e - We"era#-ure Pr'er#ies ' e#h/l hl'ride., . he". En. a#a.9 (196C) C5Z51.

HCI . . S/che, $er"'d/na"ic r'er#ies ' re'ns (%a-#i'nal S#andard Meerence a#a Serice ' #he SSM aSeries ' Pr'er#/ Wables), 1s# ed., Ue"ishere Publ.'., >ashin#'n, , 19G7.

H5I . Uaar, .S. &allaher, $er"'d/na"ic r'er#ies 'a""'nia, . Ph/s. he". Me. a#a. 7 (197G) 635.

H6I M.. Neller, .*. S#iel, $e P--W behai'r ' ace#'ne in

#he dense ase'us rei'n, . he". En. a#a. ++ (1977)+C1Z+C3.

H7I M.S. Fin2els#ain, .*. S#iel, $e PW behai'ur ' "e#h-

an'l a# elea#ed #e"era#ures and ressures, he".En. Pr'. S/". Ser. 66 (1970) 11Z15.

HGI U.ashin#'n, , 19G7.

H11I. . S/che, $er"'d/na"ic r'er#ies ' e#h/lene(%a#i'nal S#andard Meerence a#a Serice ' #heSSM ! Series ' Pr'er#/ Wables. 'l. 7), 1s# ed.,Ue"ishere Publ. '., >ashin#'n, , 19G7.

H1+I. . S/che, $er"'d/na"ic r'er#ies ' e#hane(%a#i'nal S#andard Meerence a#a Serice ' #he

SSM ! Series ' Pr'er#/ Wables. 'l. C), 1s# ed.,Ue"ishere Publ. '., >ashin#'n, , 19G7.H13I. &ehri, U. en#=, , , W 'r ben=ene in #he rane 5

#' 300 Pa and 3+3 #' 6G3 N, . he". $er"'d/n. 9(1977) CC5ZC50.

H1CI&.. >ils'n, *n#erre#a#i'ns ' Wr'u#'ns a in Mela-#i'n #' Equa#i'n ' S#a#e Pr'er#ies, in N.. ha', M..M'bins'n (Eds.), Equa#i'ns ' S#a#e, 1s# ed., !"ericanhe"ical S'cie#/, >ashin#'n, , 19G6 . 5+0Z536.

H15I&. Sch"id#, U. >en=el, ! "'di?ed an der >aals #/eequa#i'n ' s#a#e, he". En. Sci. 35 (19G0) 1503Z151+.

H16I!. Uar"ens, U. Nna, $ree-ara"e#er cubic equa-#i'n ' s#a#e 'r n'r"al subs#ances, *nd. En. he".Funda". 19 (19G0) +91Z+9C.

H17I%.. Pa#el, !.S. We8a, ! %e ubic Equa#i'n ' S#a#e'r Fluids ans Fluid i4#ures., he". En. Sci. 37(19G+) C63ZC73.

H1GI. c&arr/, 'rrela#i'n and redic#i'n ' #he a'rressures ' ure liquids 'er lare ressure ranes,*nd. En. he". Pr'cess es. e. ++ (19G3) 313Z3++.

H19IM.. Meid, .. Prausni#=, B.E. P'lin, $e r'er#ies 'ases and liquids, C#h ed., c&ra Uill, %e

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A non-cubic equation of state for PVT and phase equilibrium calculation (pure compounds)