R-LIQUID EQUMBRIA FROM

VAPO
-V-T RELATIONS.

GENERAUZED P

ThaisforfluoogMolMS.

MlCH‘GAN STATE UNNERSWY

John Nakon Hester
$9 59

“W. LIBRARY ‘

Michigan State
University

 

VAPOR-LIQUID mu
ERG!
W P-VJI' MORE

JOMMHBSM

ATEISIS

Submitted to the College of Wing
Michigan Statelhiversity at Agricmlture and
Applied Science in partial mitinmt or
the requirements for the dome of

morscmvcz

V Depart-ant of Chemical Engineering
1959

AM

The author wishes to express his sincere apprecietion to
Dr. Carl 1!. Cooper for his valuable assistance and helpful
guidance throughout the course of this work.

11

VAPCR-IIWID man
not
‘ Mm .P-V-T mucus

Joan mos mm

mm
SubuttedtotheCouegeongineerine
Michigan Stete University of Agriculture and
AppliedScienceinpartielfulfillnentot
therequirenents torthedegreeot

morsem

Department at Chemical Engineering
1959

iv

Jonsmsosm

Am

Mimestigationisastudyotthesatistactorinessot
current eethods for calculating the pressm-volxme-teqperature
relations or mama-u, with particular min on the use or
these lethods to predict binary vapor-lipid equilibria.

Pitzer.‘ s generalized correlation, which expresses thereo-
dynalficpropertiesintermotredncedtmrature, refined
pressure and an "acentric f ", accurately-predicts the vapor
pressuredpurecolpounds. Itdoesnotpa'edicttheazeotropic
telperature, pressure andcomosition for any of the systole
studied, vim, ettwl‘alcohol - water, etlul alcohol - benzene
and ethyl acetate - noml hustle. Various combinations of the
critical properties and acentric factors were attelpfted for the
mcmndsbuttheresults obtainedwerenotanilprovenent
onthoseacquiredwithasinplenolaraverageottheproperties
on! the pure coupounds. auteur, Pitzer's correlation, with this
simple Dollar average cabination of the critical properties,
predicted volunetric properties or netlwl alcohol - nornl butane
fixtures with greater acctn‘acy than'Angat's law or critical
compressibility charts developed by ‘hxwell and by Hansen and Watson.

the past success of the Benedict-Webb-Rubin equation or state
in predicting mdrocarbm vapor-liquid equilibria wanted a stw
or the relation between the critical properties or pure coupounds
and the pseudocritical properties obtained with these equations.

For the ethane - nornl heptane system, the critical coupressibility

JOHHIEIBONEESTER

of the 0.8 Iole fraction ethane mixture is 0.21]. while it is 0.228

for the 0:5 mixture. In direct contrast, the critical coupressibility
of pure ethane and pure norml heptane are 0.285 and 0.260 respectively.
In addition, the effective pseudocriticel pressure for the 0.5 mole
fraction ethane mixture mounted by this method is 25.2 etno-

spheres while the critical pressure of ethane is h8.2 atlospheres

and that for normal heptane is 27.0 atmospheres.

Theseanomlous findings castdoubtontheadequacyofthe
usualassimptionthatasinglephasenixturebehaveslikeasingle
compound with properties which are intermediate between the
properties of the pure components of the mixture.

This study indicates that reliable calculation of vapor-liquid
equilibria from the P-V-‘I' relations of mixtures will not be
successful until an enact equation of state relationship is
established between pure compoundsand mixtures.

APPROVED: / Q’M

MorPr essor /

TAR-301W

Acknowledgnent..........
Abstract.............
Listof'i‘ables..........
ListofPigures .........
Introductim...........
Theory..............
History.............
deectives............
ListofCalculations.......
Procedure ............
ResultsofCalculations .....
DiscussionofResults ......
Conclusions ...........
Reco-endatiais .........
lonenclaimre..........

Bibliography...........
SalpleCalculations

Worm

Accuracy and Couplexity of Equations of State. .

PitaerCorrelationvs. VaporPressures . . . . .

mwmwfiehtimhiptofllmm
VaporPressureMne ............
Azeotropes and'i‘heirRelationship to (er/31).],
AlongSaturationmne ...........

Ibesured Boiling Points vs. Predicted Boiling Points

Measured Pressiires vs. Predicted Presetires

Liquid Values Predictions for Methanol - Normal

0 O O 0

Emma O O O O O O O O O O O O O O

Pitser Correlations for Liquid Volune Predictions .

Pseudocritical properties of Ethane - Dorsal
WWeeeeeeeeeeee

O

52

53

57

vii

III

LISIOPFIGIRES

Pseudocritical mm. of Ethane - Norml
HeptaneSysteI...............
Pseudocritical Pressures of Ethane - Normal
Pseudocritical Densities of Ethane - Norml
”WWW...............
Volumetric Properties of mtmrl Alcohol - Kernel
ButaneMixturesat200’Pandll0psia....

'Volmnetric Properties of lbthyl Alcohol - Nor-a1

Butanemxtures at 2h0'1'andh0psia . . . .
Volumetric Properties of mum Alcohol - Norm].
Butanemxturesat280'Pandh0psia. . . .
Volumtric Properties of Methyl Alcohol - IIorml
Butane Mixtures at 2m'Pend125 p615 . . .

62

53

67

69

viii

momma

DEMO!

A prile requisite for the successful design of use transfer
processes such as distillation, absorption and extraction, is
accuratepbase'equilibriuldata.

'rheexperilentaldeterainationofreliablevapor-liquid
equilibriarequinsacoaplexlaboratorytechniqtie. Several
experimtalnthodshvebeupropoeedlmtthereiscmsider-
abledoubtabouttheaccuracyofthedatathathasbeenobtained
witheach.

Difficulties «centered in the arperiaental deteraination
of such data nuke calculatim techniques for vapor-liquid
equilibria iaportant. In addition, such calculations illustrate
the general behavior of vapor-liquid aixtures.

Predictionsofvapor-liquidequilibriaarebasedupon
rigorousthermfinaaicrelationships. Therearetwobasicaethods.
forcalculatingvapor-liquidequilibria. (neoftheselethods,
m1,mewmmm,mmtymm
activity coefficients;whilethesecmdlethodenploys anequatim
ofstatethatisapplicabletothenixtureandtoitspm'ecoaponents.

lbthodIisinusetodarbyaoetpersonsinterestedinvapor-
liquidequilibria. However,ithaseeveraldisadvantagesofwhich
theprincipaloneisitsdependenceonaxperinntalactivity
.coefficientsfortheaixture;theseareseldolavailable.

Indirectcontrast,)bthodndoesnotnquireamaxperi-
aentaldataonpropertiesoftheaixture. However, itdoes
de-ndanaccurateequationofstateandapreciseaethodfor
cowining the arbitrary constants of the equation of state when
appliedtoaixturea.

mselusiveelseentinlbthodnhasbeenadeteraination
oftheexactmationshipbetweenpropertiesofaaixtureand
propertieeotitep‘n-eceepenente. Theestablishlentofthis
clemathasbeentheddnantlotivatimforthisstudy.

A- W

Equations that involve only pressure, voluae, tenperature
and arbitrary constants are called ”equations of state". These
equations attelpttopredictthebehaviorofmrecoapmmds over
a practical range of conditions. Dodge (8) has pointed out that
overlOOeqmtionsofstatehavebeenproposed, althoughonly
afewhave cone intoco-onuse.

neequtimoretetethnteredirectlgreletedtotnie
work are listed below:

Ideal Gas Law (1)
P - m/v
van der Haals' Equation (2a)

P-m/(v-t) -a/Y2
seducedror-ormderwecu'sqmtion (2b)
P,- - err/(3v, - 1) - 3mg
seettie-nridge-en Equation (3)
P-Rl’d+ (soar-i0 -Rc/'r2.)<12+
(4.9m + Aoa - Race/1‘2)d3 + monk/r2
Benedict, went and Rubin Bquatial (h)
Fem-o- (nom- -A° -c°/'12)d2+(bm -a)d2+
dad6 + cd3/'1‘2 Bl + 3d2)e" n12]
Colpressibility Charts (5)
z - f(Pr, err)

Pitui- correlation (6)
Z ' “Pr: Tn.) )

Riedel Correlation (7)
z - “Pr, {Pp-Cc)

Iadersen, Greenkorn and nougen correlation (8)

Z ' fa’r: Tr: 2:)

nose relationships vary in coapleaity, ranging froa no
arbitraryconstantsintheidealgaslawtoeigbtarbitrary
constants in the Benedict, went and Rubin equation.
AWoftbcmlmtyandtheaccmcythtmbe
achinedinpredictingsouproputiesofnmcoqpandsugven
inTa'blel. Ihistableillustratestheapplicationofthelisted
egationeotetetetoelargemneeroroo-poudsthoughthe
couperieoanrnotbeaccurateforawsinglecoqound.
Inthelowpressureregimeachoftherelatialshipsrepresent
thepropertiesofpmeoupoundsaccm-atelr‘butthereare-Jor
inaccuraciesinlquatims(l)and(2)forthehighpressureregion.
mlynquations (It), (6), (7)and(8) give resealable resultswhen
mliedtothepredictionotliq‘nddeneitiu.
Inthecriticalregimeachdtheequatimsaxcepttheideal
gaslawgivesafanflqofisdthea‘leurvesinthepressure-
voluaeplanewhichhaveahorizontalinflectionpointatthe
critical teqeratureandpressure. Theeritical pressure,
temeratureandvolme-ybedeterainedbyutilizingthefact
tuttheilotherlpalsingthroughthecriticalpointhaltero
slopeatthatpointwhichisalaoapointofiuhction. his

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satisfies the following nthentical require-ants:

(£13!)1P - o (9)

321’ -o . . - lo
(76 )r ()

nae prediction of vapor pressures with these relatimships
isbaseduponthetherasflmaicrequireeantstbatatequilibriua:

PL-Pv (u)
rt-Iv ‘12)

v
I v var . o (13)
v1

'i'heactualcalculatimofvaporpressurewithanequationof
stateinvolvesfindingthevaporvolusandliquidvoluasatwhich
thefugacityofthevaporandtheliquidphasesareequal. The
pressm-eofbothphaeesmstbeequalandisthecaleulatedvapor
pressure. I

theidealgaslumthefirstequatialofstate. Itm
bederivedfroeakinetic viewpointwiththefouowing assulptions:
(a)thevoluaeoftheaoleculesofthegasisnegligiblewhen
cmredto'thetotalvoluasoftheasand(b)thereareno
attractiveforcesbetwaentheaoleculesofthegaa. Actualgasu
donctconfmtothebehaviorpredictedbytheidealgaslu
but-erelyapproachthesecmditionsatlowpresaures.

In1873vanderwaals(33)proposedthefirstpractical
equatimocrstatetoimroveupontheidealgaslaw. Beconsidered
that(a)aportionofthsspaceoccupiedbyagasistakenup
bytheaoleculesthe-elvessothatthe"freespace"thrwghwhioh

theaolewlescanmoveisnotequaltothstotalvolunsandh)
thereisaforceofattractionbetweenthemoleculesofthegas.

Despite these refine-ants, van der Waals' equation is only
an approxintion and exhibits large deviations from acttnl
behavior as the pressure increases.

Equation (2a) is another form of van der wans' equatial
ofstateerpressedinter-ofreducedproperties. Thiswas
probablythefirstexpressionofthetheoryofcorresponding
states and m be considered as a forerunner of Equations (5)
through (8).

The Beattie-Bridgemn equatim at state (2) is one of the
nostnotableimprovenentsonvanderWaals' equation. Itisan
maple of a more eagle: equation of state, with five arbitrary
constants. Ithasprovedtobeveryaccurateintherangeof
conditions for which the calstants were determined, but it gives
eresultsatpressuresoftheorderof250ataospheres
(10). w

The Benedict, new and Rubin eqmtion (h) is probably the
lost accurate equation of state in use today. This equation
was developed as a modification of the Beattie-Bl'idge-n equation
orstatetoilprovethepredictionofthepropertiesoffluids
at high densities. It has successfully predicted P-VJP (pressure-
volune-tenperature) properties within 0.3% of the actual values
forlwdrocarbonsandothercoawwndsuptodensities twicethe
critical density.

lo

Equations (5) through (8) are graphical representations of
thecompresaibility factor, Z,intemofredncedprovperties.
Mesa-pressibilityfactormodifiestheidsalgaslawtoaccount
forthedeviatimotthemopertieeofarealpsfroathoseof
snidealgas. Thisfactorisnotaccnstanthltisafunctimaf
(a) thsgas itself, (b) thepreesureand (c) thetamperature.

Thebehaviorofanyachialpscanberepresentedbymeans
ofagraphotcompressibilityfactorvs.pressm-e, withtelper-
atureasaparameter. Ithasbeenfoundthatthegraphsfor
eachgasshosedthesansgsneraltrendsthoughtheyweredifferent
insegznitude. rhesesisdlaritiesledtotheeppucstionorthis
relationshiptotheflleoryofcorrespondingstatesasproposed
byvandufluh(33)-

'ncethsoryorcorrespaadingststesassatsthstthsr-v-r
relationsofallgasesarethssaaswheneachvariableiserpressed
inter-erreduoedpropsrties. seducsdpresan-e,redneed
temperatm‘e,redncedvolunandrednceddensitymybedefined
asfollows:

g-flg
Tr'T/Tc
vr"'V/Vc
dr'd/dc
wherethesubscript, c, referstothecritical state.

'nlistheoryledtotherelationshiperpressedinnquationG)
whichseveralinvestigators(ll,20,2l)haveueedtodevelop
generalized compressibility charts. Mechartsgiveagood
decadencm‘acywitherrorspri-rilydnetotheapproxi-te

nature of the theory of corresponding states.

Inrecentyears, severalmethodsbavebeenpresentedfor
improving the accuracy of the generalized compressibility charts
by the introduction of a third parameter. Equations (6), (7)
and(8)aresomeofthemoreprominentmethods.

Pitser (25) introdmed as his third parameter the acentric
rector,d, whichisaneasm'eofthedeviationofthe "inter-
molecularpotential”frathatofa"siaplefluid”. Theacentric
factor was defined. as

a) e-logPr-l.000 (11¢)
wherePristhereducedvaporpressm'easastn'edatareduced
temperature of 0.7. The compressibility factor was then
expressed as a power series in the acentric factor

zaz°+dZ'+u) 22"+ ...... (15)
where 2°, 2' and z" are different graphical functions of reduced
pressmandreducedtemperature. Itwasdeterminedthatthe
first two terms in Equatim (15) were sufficient in almost every
region. Pitser and his coworkers (26) attempted to evaluate the
quadratictemZ"forthesmllregioninwhichitappearedto
be significant. However, the resulting values were not used
because they showed irregular behavior. This correlation without
theZ" term, basalsobeenappliedwithfair success tothe
prediction of the properties of both saturated liquids and
saturated gases.

Riedel (28) introduced a teniperature derivative of the vapor
pressin'e at the critical point, Kc, as a third parameter for

the satiation of generalized vapor pressure and liquid density
data, where

'13
ece._°.(2£ . (16)
Pc are

Pitser (26)pointsoutthetcccisre1atedtohisacentricfector
bythe following equation:
ace - 5.808+ L93 a) (17)
Iadersen,GreenkornandBouaen(18) heveselectedastheir

choice for a third parameter, the critical coupressibility factor,
definedas

Zc'Pcvclmc (18)
Theresnltsofthisvorkverepresentedinfourmofcoupmme
bytheirvaluesofzc. niecorreletimincludestheliqnid
regimend it cenalsobe relatedtoPitzer's cornJmtim.
Pitzer (26)gi‘vesthefonowin3equet.ionasthere1ationbetueen
anndhisaeentric factor: .

zc - 0.288 - 0.863 a) (19)

3. Was
Thesolutimtotbperpleximgprobleldpredictimsthe
propertiesofeixtureshesbeenelusivetoscientistsandeosineers
vhohavepm'suedthiscourse. mmwtoerpreuther-v-r
behaviorofalixture,1menitscoqpositimishtovn,isinterme
oftheknounbelnviorotthepurecomonents. 'Bmsifthereis
unequetionofeteteforeechcomonentinthelixtm'e,the
amtnytodescribethebeheviorotthelixturembe
bymsdsaecodainaumofthearbitrarycmstanuofthe

equation for the pure W.

13

fiesnccessachievedviththeapplicationofrednced
propertiestopm‘ecoqpoundssmestedtheiruseforlixtures.
Bovever,itvasdeterunedthettheh'uecriticalpointvasof
littlevalueincorreletingtheplvsicalpropertiesotlixtm'es.
whenitisusedforcalculatingredncedproperties,thecomprees-
ibilitycIn-vesdeviateconeiderablyfrouthoseforpm'eeoewomds,
especiallyinthecriticalregim. Mtherinvestiatimshoved
thatafictitiousvalneoi'criticaltapu'atm'eandpres'sure
couldbeemloyedsotbtthecwibilitycurvesforthe
unrecorreapmdviththoseforpurecm. nepoilrt
for these fictitious values was called the pseudocritical point
andthevaluesvereternedpseudocriticalpropertieebynyu3).

mispseudocriticalpointforaaixtm-eisanalogoueto
the critical point of em coupoundvhoee r-v-r relations in
thesuperhatedregionareidenticalviththoseforthemhn'e.
Ithashanvernorealenstenceandthereforecannotbemeasured
experimentally. muforeixhn'esofnarrowbomngranaeis
itapproriutelyequaltothetruecriticalpoint.

Eherehevebeenamnberdmethodspropoeedforcodaining
thecmstantsai’anequatimotstateandforpredictingthe
Daeudocriticalpropertiesofamixtme. Smeofthemore
thethodsarelistedbelow:

host's!” ' (20)

Va ' YaVa + IbV‘b + “"

van der Waals' Combination

(a) b. elab. 4- thb + -..--
(b) (8.01/2 . 15(5)” 2 + 1(eb)1/2 + ----

w <:-:-> “H

01)”— $6.213): 1b {tr-)1)

Lorentz Combination

3
Km ' E3(Kg)l/3 + “(Ital/é] / 8
Kay's Rule ’

(3) Tc- ‘ raTc‘ + IbTCb + m.--

(b) Pc' ‘ I‘Pca + prcb + ----

- Be'ettie-Bridgeman Combinations

KI'YaKa'FYbe‘D- ---- .
wherexsg’lnb,Ao/2 30101.61/2

Benedict, Webb and Rubin Cabinatims '
m K 8 801/2: Bo: c”1/2, 31/3, bl/B’

Prausnitz and Gun Combinations

(8) Pen' “Tc. z 112

 

vc' 1 c1
(1’) Ten - ,0 + (’2 + rvcnx )1/2
five.

(c) p“ f, IiYJWcTc)”

(21)

(22)

(23)

(2h)

(25)

(26)

1h

15

(d) X " E 1123(Vc1fi)”
(9) $51., " (Tcich)l/2 " £013
(5) vcid ’ 0-5(Vc1 + ch) " Avci'j

Angat's (l) lav of additive volumes, which states the volume
occupiedbyanixtureofgases isequaltothesmlofthevolumes
occupied separately by each conponent at the same conditions of
tenperature and pressure, has been applied to both ideal and real
nixtures of gases. It is accurate for ideal gas mixtures and also
gives good results for m Iixtures of real gases and liquids.

Van der Haels' codinatim (33) for the constant a is based
upon the fact the attractive forces between the Iolecules is
equal to the geosetric mean of the forces~that each molecule
possesses when attracted by a like Dolecule. The cabination for
the constant 13 assunes that the volume occupied by the eolecules
is equivalent to their arithmetic mean. Equatims (21c). and
(21d) are extensions of van der Waals' coebinations to the
prediction of pseudocritical temperatures and pressures.

m's rule (13) is an application of the theory of
corresponding states to mixtures of gases by the use of pseudo-
critical constants to evaluate the reduced properties.

The Lorentz combination (1?) takes into account the packing
effectbetveenlargeandsnllmoleculesvherethesnlllolecules
myoccupythespacebetveenthelargemolecules.

Beattie-Bridge-n (2) and Benedict-Webb-Rubin (5) extended
thehypothesis ofvanderwaalstothecalculationofconstants

16

for mixtures for their «nations.
The codinations of Prausnitz and Gunn (27) represent the
first attelpt to account for the probability of differences beMen
thebelnviorofmixturesandofthepurecompmnds composingthe
mixtures. This method incorporates Pitzer's correlation and
introduces two graphical corrections to the characteristic critical
temperature and elmracteristic critical volume , Mom and Av°l2°

c~ mmde
Therearetvonethodsthtmbeemployedforthecalculation

of vapor-liquid equilibria.

MethodI-Thefirstmethodinvolvestheuseofexperimentelvapor
pressures with the basic We relatims. A general relation-
ship for the prediction of vapor-liquid equilibria by this method
ny be written as follovs:

I . Pop’tP; {f e w ' (27)

This relationship is actually Raoult's luv

1 - Pox/r (28)
with additional factors to correct the various deviations from
idealitya This method has several disadvantages, the principal
one being. the dependence on experimental activity coefficients
for the mixture which are M unavailable.

lethodII-i‘hesecondmethodeuodiesonlytheuseofanequation
Ofstateforthepurecompoundsandthemixture. Thismethodisbased

upon the thermodynamic requirements that at equilibrium

17

T1. =- Tv (12)
N ~
(8.) 1L1 =21 (29)

The actual calculation of vapor-liquid equilibria by this
method requires the use of the equation of state to predict the
P-V-‘I' properties of the mixture and then evaluating the following
thermodynamic equations:

A. V j
A BIG KP - KP/VNN x,T - RT 1n RT/V +

XRTlnx+(l-X)Rr1n(l-X) (30)
E1 .. (1.x)(a‘A’L /a x) m + XL (31)
$71 = (1-y)<a’A’v mm M + ”V (32)

The criteria for equilibrium requires that the partial molal
free energies obtained in Equations (31) and (32) Inst be equal.
The fugacity my be then calculated from the work content, A, by
the thermodynamic relation

RT ln r =- (3A/ON)V’T’N (33)
and vapor-liquid equilibria may be predicted from the fugacity.

The principal disadvantage of this method is that it requires
a coupler equation of state which met be accurate in both the
liquid and vapor regions and which must also predict correct
vapor pressures.

A distinct advantage that the use of an equation of state
enJoys over the more cosmonly used first method is tint no
experimental data on the properties of the mixture are necessary.

 

18

19

name

The existence of a relationship between equations of state
and vapor-liquid equilibria has been recognized for some time.
Van der weals demonstrated the relation between his equation of
stateandthevaporpressureofapurecomound. Barthelot (6)
andGalitzine (9) extendedvanderwaals' equationtomixtures
and van Lear (31;) obtained his equation for liquid activity

 

coefficients by a semi-theoretical derivatim frcm these van
der Weals' mixture equatims.

Studies of vapor-liquid equilibria in high pressure mixtures
wereaotivatedbythepetrolemindustry's interest inthe
distillation and absorption of light hydrocarbons . In 1932,
generalized correlations of pure compcment compressibility factors
werepublishedbylewisoflhssaclmsetts InstituteofTechnoloy
and Brown et al of the thiversity of flichigen. These eon-elatims
were used to determine the values of the fugacity coefficients ‘
from which the values of y/x for the individual lurdrocarbons
were then calculated. This calculation was performed in the
unner described under Method I in the theory section assuming
tint xand 8' wereequaltounity.

Amorerigorousapproachwasmadebynenedict, Webband
mahinm19houeingnathodnunaertheory. Theirwork (It)
involved the development of a new equation of state for pure
compoundsandasetofrulesforapplyingthis equationto
letures. mloyingnethodn, theycalculatechhartsfor

pressures up to 3600 psia and correction factor charts for each
of the twelve light hydrocarbons investigated. The correction
factorchartswerenecessarysinceXand“ innethediwere
not actually equal to unity.

Smith and Watson (30) employed a similar approach based upon
thefugacitychartsandxay's rule. Theworkofnenedictand
his coworkers has led to a number of other investigations based
upon their equation of state. Sage and coworkers (16, 22) have
shown its utility over higher ranges or pressm'e than 3600 psia.
Schiller and CanJar (29) and Stotler and Benedict (32) have
applied the ”dict, Webb and Rubin equation and Method II to
the nitrogen-carbon monoxide and nitrogen-methane systems
respectively with success. However, Cullen and Kobe (7) attempted
to apply this equation to the non-ideal system carbon dioxide and
propane but found the correlation to be unsatisfactory.

. .-... :2:

m I“

‘I ‘44-“, _. r'

 

OBJECTIVES

The ultinte obJect or this work was to satisfactorily
predict the vapor-liquid equilibria of mixtures from equations
of state. Benedict, Webb and Rubin and Smith and Watson have
had some success with this approach but their best applications
have been with mixtures of hydrocarbons which form nearly ideal
solutions.

The only limitation in predicting the vapor -1iquid equilibria
forbothidealandnon-ideal systems isthelackofanaccurate
equation of state for pun-e capounds and their mixtures.

It was hoped tint a combination of the correlations recently
developed by Pitzer and others for pure compounds and simple
rules for obtaining the three constants for mixtures might be
useful in working toward this ultinte objective.

in attempt was first ads to apply these equations and
rules to azeotropic mixtures. When this proved unsuccessful,
the same equations and rules were applied to volumetric data on
non-ideal mixtures. Only moderate success was achieved.

It seemed that the principal difficulty encountered was the
inaccuracy of the rules for combining constants when determining
the pseudocritical temperatures and pressures and the acentric
factors . Therefore it was decided to determine what relation
exists between the critical constants for pure compounds and the
Pseudocritical constants for mixtures when the reputedly
Necessful Benedict, Webb and Rubin method is applied. Prom

23

theseresults,aneffortwasndstodevelqprules forcombining
pseudocritical constants. Moreover, an evaluation and a
comparisonwereudeoftheprominentmethods thathavebeen

proposed for estinting pseudocritical constants.

 

 

LIST OF WINS

21+

A.

B.

C.

D.

25

LIST ”W10”

Pitzer Correlations vs. m Pressures
l. Benzene

2. Water

3. Ethyl Alcohol

h. floral hexane

5. Ethyl Acetate

6. lettnrl Alcohol

Aseotrgpes and Their Relationshito the Reduced “12!
Pressure Line

1. Benzene - Ethyl Alcohol

2. Water - Ethyl Alcohol

3. lornl Beans - Ml Acetate

W and Their Relationship“ (82/0th

Saturation Line

1. Bensem - Ethyl Alcohol

2. Water - Ettwl Alcohol

3. [oral Hermann - Btlvl Acetate

we Volume Predictions For nth-mi Jon-.1 Butane Mixtures
l. Angat‘s Law

2. Pseudocritical lbthod

3. W W

1.. Pitser Correlation

P.

G.

26

Pitzer Correlations for Liglid Volume Predictions
1. floral Butane

2. Benzene

3. Nor-l Pentane

Pseudocritical Properties of_ Bthane-llorml W
l. Benedict-Webb-Rubin Equation of State

2. Prausnits-Gunn Correlation

3. Kay's Rule

1... Van der Waals' Combinations

mscellaneous Calculations
1. Empirical Cofiination for Pseudocritical Properties
2. flirtin-Bou Equation of State

28

.A. Pitzer Correlations vs. Eager Pressures

These series ofcalculationswereperfomdtoascertainthe

accuracy that could be achieved with the Pitser correlations. The
following procedure was used:

1.

2.
3.

h.

S.

Calculatethereducedtemeratureandpressurefromths
experimentalvaporpressureandtelperature.
Determinetheco-onlowithmoftheredueedpressure.
Determinethenegativecftheco-onlogaritlnofthe
reducedpressure.

Determine the values of - log Proand - (biog Pr/Ofl)!

corresponding to the reduced tempera‘mre by interpolation

ofTableVIofPitzer'swork. Eaten-103B,°refers
towhentheacentricfactcrisequsltozero.

a. Calctflatethechangeinthelogarithmoftherednced
pressure

-AlogPr= «Hausa/30).,

b. Thsacentricfactorsforseveralcompwndsare
presentedinTableIofPitser'swork. beynyalso
becalculatedbythisrelationship

-logPr- -103Pr°+U(-.103Pr/.0)T
(mi-lentil)

6. Calculate the theoretical value of the cm logarithm
of the reduced pressure.
-logPr- -logPr-logPr°
(Theoretical)

B. Azeotrgpes and Their Relationshiw the Mced Vgpgr Pressure
Ling

Thesecalculationswereperfonedtodetermineifthe

azeotropes lay on the reduced pressure line.

1. These calculations followed the six steps outline in
calculation (A) with the exception of the use of five
different methods for combining the acentric factors ,
critical temperatm'es and critical pressures for the
azeotropic mixtures.

2. Assumption - Key's rule was applied to the pseudocritical
temperatures and pressures while the acentric factors were
combined linearly.

a. P... - X1P¢1 + XJPcJ
b. Tc. - xiPci + xJPcJ
c. d. :- X101 4- X303
3. ”Max 11 - Van der Hast' combinations were applied
to the pseudocritical temperattn'es and pressures while the
acentric factors were again combined linearly.

”(if HG: JG?-

“.1

-: W m)

5.

6.

“Mon III - Key's rule was used for the pseudocritical
temperatures and pressures. The acentric factors were
determined by an empirical relationship.

‘s Pc- . Xipci + XJPcJ

1). Tc. . xiTci + XJTOJ

c. ‘ (0.1 - O.f+ 1):/2 - X1(O.l - 0.1.0" 1) 1/2

. XJ(O.l - o.i‘* 1) 1/2
m

”Mon IV - An empirical relationship has used to
calculate the pseudocritical temperatures and pressures.
This relatiaiship was derived with results obtained in
a later calculation with the Benedict-webb-Rubin equation
ofstate. Theacentricfactorsweredeterminedinthe
same manner as used in Assmtion III.

(5-8).,- (*8) we )
“(z—m) - *1 on) we)

m i J

c- (0.1 - 0.1 W 1) 1/2 =- Xi(O.l - o.1"+ 1) i/Q

U+ 1 l 2
+ X: (0.1 - 0.1 )./

“mun V - Van der Haals' coininations were used

for the pseudocritical temperatures and pressures. The
acentric factors were combined with an emiricel
relationship.

m °i J
b. Tc) _ x1(_r£ + 13(5)
4%: r.) n:
m i J
c. l 3 xi + X
a“: a; ..

c. Azeotropes and Their Relationship to (a P/a xh- Along
Saturation Line
These calculations were performed to determine if the
(3 P/ ax)T along the saturation line was zero at the azeotropic
composition. Three methods of combining the acentric factors,
critical temperatures and critical pressures were employed.
1. Assumption I was used to combine the parameters.
a. A - so + D + B
b. A - (31oz 15/3101.
c. s - (slog Pr/O Tr)x
d. c - (err/ax» . -(Tr/Tc)(ch/dx)
- ‘Tr/Tc('1'ci - Tea)
e. D :- (Clog 5/0th - (0 log Pr/aflhzxdd/dx)
- (3103 madam, 403)
f. E - (d log Pc/dx) - (6. ln Pc/dX)/2.303
= (Pei " PcJ)/2-3°3 Pc
The values for 'B' were determined by taking the difference
in two valuesfor Pitzer's value of - log Pr and dividing
it by the difference between the corresponding reduced
temperatures. The retaining values were obtained from

simple substitutions .

. AssumptionIIIwasusedheretocofiinethe-pu-ameters

andthesameprocedureofpart(A)wasfollowedwiththis

variationfcrtheacentric factor.
a. (o.1-o.1‘*1)1/2 - (0.1- 0.1 "LAP/2
I 2:l

+ X: (001 " Osld‘.’ 1) 3/2

d. 1 1/2
be d Oel ' 0e].
( ax ) (0.1.41 d+l)ll./2

 

. (0.1 - 0.1“ 1)1/2

d 112
c. g _ 2(O.l-O.l 1+ )/ -(O.l 0.1 J

2.303(0.l - 0.1r + 1) -1/2 0.1' I + 1

+ 191/2

. Assumption V was used to combine the parameters with

the following modifications .

s. (111413) _ 1 1

dx 1313 " 033
.. -—- ewe-Big)
c. $5) ~;l-§c(§'__c)W)('1-'c:__:)J

 

a. d(T:Pc) .éc(§-c)- 2:3. fi-i) w?“ Z) (1;)
°- -- --— «E3 (9] (2). (Z)
*- e- .{éz [GEL (Ti-fl 6%) T3}

33

D. Liquid Volume Predictions for )Iethanol-Noruel Butane Mixtures

This calculation was initiated with a comparison of
experimental data with volumetric predictions obtained with the
Pitzer correlation. The large deviations from the experimental
results prompted a study of the volumetric properties of several
pure compounds which are discussed under Calculation (B).

The volumetric calculations for the methanol-normal butane
mixtures were continued to include predictions by (a) Amagat's

 

Law, (b) a pseudocritical method with the "Chemical Process
Principles Charts“ of Bougen and Watson and (c) lhxwell's method

 

using a correction factor with compressibility charts. A
comparison of the values obtained with each method was made in
relation to the experimental volumes .'

Calculations were made on mixtures containing 0.25, 0.50
and 0.75 mole fractions methanol at ho psia and temperatures of
200?, 2110’1", and 280?. The same calculations were repeated
with these mixtures at 125 psia ad 280?.

1. Pitser Correlation

a. Determine the reduced temperature and the reduced
pressures from the experimental temperatures and
pressures.

b. Determine the values of 2° and z' for each reduced
temperature and reduced pressure by interpolation
with Tables II and IV of Pitser's work.

c. Calculate the value of 2 with Equation (1-15).

z-z°+u)z'

2.

3.

h.

d. Calculate the volume by the following equation:
v - ZHT/P
at“ Law
The volumes were calculated by applying Equation
(1-éo). ' .
V. =- xiv, + IJVJ
Pseudocritical lethod
a. Determine the pseudocritical temperatures and pressures
with Kay's rule.
Pg. I X1Pc1 + XJPcJ
Tc. . X1T¢1 + XJT¢J
b. Determine the reduced temperature and reduced pressure .
c. Calculate the compressibility factor, Z, from page
103 of "Chemical Process Principles Charts" by
Hougen and Watson.
d. Calculate the volume in the same nnner described
in step (d) of the Pitser correlatim.
thwell liethod
a. Determine the reduced temperature.
b. Calculate the reduced pressure for each comonent
by the following correction.
Fri ' P/Pe(!1)1/2
c. Determine the compressibility factors from pages
152-3 of lbxwell's "Imta Book on mdrocarbons" for
each component.
d. Calculate the volume by the following equation.

v - RT/P(Y121 + rJzJ) ‘

35

E. Pitzer Correlatigs for Liquid Volume Predictions

These calculations were performed to determine the accuracy
of Pitzer's correlation when it is applied to the calculation of
liquid volumes for pure compounds. Benzene, normal butane and
normal pentane were the selected pure compounds. Several vapor
pressures were chosen and the calculations followed the procedure
outlined under "Pitzer's Correlation” in Calculation (D).

P. Pseudocritical Properties of Btlnne-norlml am

1. The Benedict-Webb-Rubin equation of state was used to
determine the relationship between the critical constants for pure
compounds and the pseudocritical constants for mixtures. The
eight arbitrary constants of the equation were combined in. the
manner suggested by Benedict and his coworkers and reported as
Equation (at) in the theory section.

K. - X113, 4- XJKJ + ----
when x _ A0 1/2, 30: col/2’ g1/3, bl/B’ c1/3’cc 1/3
‘ or x 1/2.

Five significant figures were used with each constant as
recs-ended by Bendict et al. They pointed out that six figures
should be used in calculating the properties of liquids and four
or five figures should be used for gases. They detected that a
changeofone inthesixthplaceofoneconstantbyitselfwould
alter the calculated pressure of a liquid by approximately 0.01
atmospheres. However, they stated that it is possible to clnnge
all the constants simultaneously by about five percent without seriously
affecting the calculated results.

36
Critical densities were assumed for the mixture and the
critical temperature was obtained by a trial and error calculation

elploying the nthentical require-eat presented in Equations (9)
and (10) of the section on theory.

(ear-o

 

jet . . 1
ad

m _
lnthiscasethevolumeisreplacedbythedensitysincethe _,
Benedict-Hebb-Rubin equation expresses the pressure as a function I

of mm and density.
’ The Benedict-Webb .mbm equation of state was differentiated
and equated to zero to obtain the following relatims:
0 - c0 ’"2(342 + 38d“ - 232d6) - 2an +
(- 2Aod - 3ad2 + 6a‘d5)T2 + (R + 3de2 4» 23030133

0 . ce' '42(6d + 6 Xd3 - lBY‘Qd5 + 1933i?) - 20° +
(- 2A0 - 6nd 4- 30a¢¥Cdl‘)1l.‘2 + (6de + 2308)T3
These cubic equations in the absolute temperature were solved
with a trigononetric solution using canon logarithm to obtain
the required accuracy. These equations have the fore:
A13 + 8:2 + D - O
and were solved as follows
Let sin 0 - (3n/23)(3/n)"/2
'or logsinO-logl.5+O.Slog3+logD-l.SlogB
then the required root . 203/3)”2 sin 0/3
- log 2+ log s.in 9/3 - 0.5 log 3 - 0.5 log 3

37

The antilog of the required root is equal to the pseudocritical
temperature. The correct pseudocritical density is obtained when
the pseudocritical temperatures calculated by Equations (1-9)
and (1-10) are equal. The pressure was then calculated at this
point by employing the original form of the equation of state.

These calculations were carried out for mixtures containing
0.5 and 0.8 mole fraction ethane. In addition, a comparison was
nude of these results with values obtained using Kay's rule and
van der Waalsf combinations. Also the values were used to
calculate the compressibility factors for the pure compounds and
the two mixtures.

2. The recent work of Prausnitz and Gunn was employed to
calculate pseudocritical temperatures and pressures for the ethane-
normal heptane system. Difficulties were encountered in evaluating
the corrections for the characteristic critical temperature be-
cause this system was not included in the graph presented for
determining these corrections.

a. Calculate the characteristic critical temperature
and volume by the following equations.
TciJ " (TciTcJ )1/2 ' ATci‘j
VciJ = 0.5 (vc1 + m)
The correction, are“, my be obtained from a graph
of this correction vs. the critical volume ratio
vci/vcd included in the authors ' work .
b. Calculate the pseudocritical volume.

ch = 212Vc1 + affidvcu + YJZVCJ

 

d.

f.

38

Calculate the acentric factor .
Q. - 2101 + I“):
Calculate fl .
,6 - 212(vcirci) + 2311.1 (VciJTciJ) + 1:2(Vcd'rcn)
Calculate I .
l = 112(Ve1‘1'gi) + 211!J(Ve13T§13) + 332(chTgJ)
calculate the pseudocritical temperature.

Tc“ I figfia + erI’ll/e
8V3;

Theconstants, rands, nybedeterminedfrom
Table 2 of Prausnits's work.
Calculate the pseudocritical pressure. ‘
I Po. - mcm/ch (212.1 + IchJ)
where the critical compressibility factor, Zc, uy
be calculated by one of the following relations
zc - Pare/arc ‘
Zc - z: + a) Zé

G. Miscellaneous Calculations
1. At the conclusion of the calculations with the Benedict-

Hebb-Rubin equation, an attemt was ads to establish an empirical

relationship for determining the pseudocritical temperatures and
pressures for a mixture. The values obtained in Calculation (I)

were used and the resulting relationship was employed as
Assumption IV in determining the relationship of azeotropes to
the reduced vapor pressure line.

2. m effort was nude to use the hrtin-Hou equation of state,

39

Pa m- + A2+32T+Cze '5'1’79rr+ A3+B£+C£'5‘hm
v-s (V-S)§ (v-s)

Al;

to calculate the critical compressibility of an ethane-haul

 

 

heptane mixture containing 0.5 mole fracticsi ethane. The arbitrary
constants for each component were calculated by the method outlined
bylhrtinandBmandtheywerecombinedforthemixturebythe
rules employed with the Benedict -Webb-Rubin equation.

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M88

 

Pitzer Cprrelations for Liggid Volume Predictions
Critics; temperatures, critical pressures and acentric
factors. (26)

Specific gravities and vapor pressures. (20)

 

 

 

 

 

Benzene Volumes, n3j1b no1e
Teupereture , ’1? Actual Pitzer
110 1.1.63 2.327
125 1.076 2.289
150 1.503 1.9.6
175 1.530 2.230
Home]. Butane
o 1.507 1.901
10 1.519 1.920
20 1.531 1.862
30 1.5% 1.891

60 1.826 2.269
70 1.8h0 2.061
80 1.863 2.23h
90 1.87h 2.215

57

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8&6 mmmmmd oméh £38: 02 33
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008.0 P830 HN.mN 8.3.3 .ndnsmnnnoauvufiouflom

00mmm.0 00.8 0018+. 344034.405

 

03M 4404.2 3H! “.0 .N

 

 

 

 

80.0.0 4.42.4.0 8.5 44.30 339m 4.8.8:
0000.0 083.0 8.3 $.08 8830
Na. M13040... :4 .00 54 8.4 a. as . 0008.408 0.43 .4

443 00.00.48 41.402
3 . 00¢an
38

 

33000 80am. 48.82.0083 «0 0049104480 400404800300

mg

58

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mm»mm.0
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038.0
oom0:.o

 

moifioa fl 30

No.0m
8.0.
wa.wm
sd
09.0

 

34 am

>m.m:m
mm.mmm
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ho.m:m
mm.mo:

 

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annulupaququhm
nanbmwnnmsnvOAGQaum
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3.50 83an ”do: 040

DISCIBSION OF RESULTS

59

DISCUSSION OF RESULTS

The animate ob.) ective of predicting vapor-liquid equilibria
from an equation of state was not achieved. The principal
difficulties seem to arise from a failure to determine the exact
relationship between the properties of a mixture and the properties
of the pure compounds in the mixture. This relationship requires
an accurate equation of state and a precise method for cosbining
the arbitrary constants of the equation of state when it is applied
to mixtures.

The fact that these calculations proved unsuccessful does not
minimize the importance of predicting vapor-liquid equilibria by
Method 11 with an equation of state, nor does it repudiate the
generally accepted belief that an equation of state that applies
accm'atels'topure compounds canbeappliedtomixmres ofthese
compounds provided they do not react chemically.

Method II is potentially the most advantageous way of
accurately predicting vapor-liquid equilibria because it does
not require experimental activity coefficients for the mixture.
ThesearedifficulttodeterMneandareseldomavailableinthe
literature.

Table 9 depicts the many variations in pseudocritical
properties for the ethane-norm; heptane system obtained with the
listed methods of predicting the properties of mixtures. A
significant factor that is revealed by these calculations is that
the critical coupressibilities of the two mixtures are generally

 

outside the range of those for the pure compounds. This contradicts
the basic assumptions of these methods, that a single phase mixture
behaves like a single compound with properties intermediate between
thepropertiesofthemrecompoundsinthemixture.

Values for the pseudocritical temperatures, pressures and
densitiesarealsoshowngraphicallyinrigures I, IIandIIIto
further illustrate these differences. A plot of experimental
critical properties from the work of Kay (33) is included in each

figure for comparison.

 

Figure I shows that present methods of calculating pseudo-
critical temperatures yield values which are of the same
mgnitude. The experimental actual critical temperature is higher
than the pseudocritical temperature at all compositions.

Figure II illustrates that each method for calculating pseudo-
critical pressures give values that show mrked differences from
one another and are all considerably lower than the experimental
actual critical pressures.

Figure III shows that the pseudocritical densities follow
the same general pattern as the experimental actual critical
densities.

The vapor pressures of pure compounds calculated with the
Pitzer correlation were generally in good agreement with accepted
experimental values. The maximum deviation was 5.h7$ while the
average deviation was less tlmn 1.01». i

The prediction of liquid volumes of pure compounds by the
Pitzer correlation is presented in Table 8. In each case the

predicted volumes were higher than the actual volumes . The volumes

Temperature, 'K

550

500

150

1:00

350

300

ETHANI - HORIKL HIPIAEI

Figure I

 

 

 

0 0.1 0.2 0.3 0.h 0.5 0.6 0.7 0.6 0.9 1.0
Home Fraction N - Heptane

lxperimemtaI.-———o————- Benedict—lebb-Rubim -————--—-—-————

 

 

van der“waal-———--————- Frau-mits-Oumn x

 

Kay's Rule

100

90

60

70

60

So

ho

Volume, ftj/lb mole

30

20

10

63

3mm - 1(an BPTAIII

Figure II

 

 

‘—
V——#

 

 

0 0.1 0.2 0.3 0.1; 0.5 0.6 0.7 0.8 0.9 1.0
Ilele Fraction N - Heptane

Experimental : 9 Benedict-Webb-aubin

 

 

van der Waal ———-~ —-— Prausnits-Ounn —————- x
Kay's Rule

 

nun-my, 1b mole/ft3

0.5

0.1.

0.3

0.2

0.1

ITHANE — NORMAL HEPTANE

Figure III

 

4r

\
\

.,\
i.

.,‘\\‘\‘
\
‘=::====

 

 

O 0.1 0.2 0.3 0.h 0.5 0.6 0.7 0.0 0.9 1.0

Isle Fraction N - Heptame

kperimemtal 4 Benedict-Webb-Ruhim

 

—-*

 

van der Weal --—- -- —-—— Prsuenitz-Omm

 

 

Key's Rule

65

of benzene were from 29.0 to 59.0$ larger, for normal butane from
6.0 to 225" larger and for norul pentane from 12.0 to 2h.3$
larger.

The application of the Pitzer correlation to azeotropes
produced some significant deviations from the experimental pressure
at the azeotropic point. The best result was obtained when using
Assumption v, consisting of van der Weals' combinations for the
pseudocritical properties, and combining the acentric factor as
the reciprocal of its cube. The deviations ranged from 0.35% to
12. 5%. with the average deviation from experimental values
approximately 6.7%.

There were also large deviations encountered when calculating
(313/8 1):. along the saturation line. This value should be zero
at the azeotrope composition but the results obtained varied from
-o.699 to -o.333.

A comparison of measured boiling points with boiling points
predicted from the calculations discussed above, as shown in
Table 5, revealed um: the predicted values were in rm agree-
ment for the water-ethyl alcohol system only. The predicted
values for the norml hexane -ethyl acetate system were higher
than the experimental boiling point for all assumptions. For
the benzene -etlwl alcohol system the predicted boiling points
werehigherthantheexperimentalvalue exceptinthecaseof
Assumption V.

A similar comparison, in Table 6, of measured pressures with
_ predicted pressures showed that the calculated pressures were
always lower for the normal herane-ettnrl acetate system.

66

The benzene - ethyl alcohol and water - ethyl alcohol system
showed both positive and negative deviations for the five
assumptions.

The results or the calculation of liquid volumes for the
metm'lalcohol -nomlbutanemixturesarepreeentedinTab1e7
and. also in Figures IVI- VII. The Pitzer correlation see!- to
follow the experimental volumes closer than any of the other
three methods employed.

Amgat's law exhibited definite deviations at the higher
pressures chosen for this calculation and it always gave lower
values than the experimental volumes .

Both the pseudocritical and the Well methods predicted
volumeswhichweregenerallymuchhigherthantheexperimental
values. Considerable difficulty was encountered in accurately
reading the compressibility charts associated with these two

methods, hence some error was probably introduced at this point.

67

11mm. ALCOHOL - NORMAL BUT”!

Figure xv

172

171

170

169

168

167

166

165

161;

Volume, 29/1» eds

163

162

161

 

 

160

159

 

0 0.1 0.2 0.3 0.1; 0.5 0.6 0.7 0.8 0.9 1.0

Mole Fraction II - Butane

 

Experimental—F— Pitser -- Maxwell— -' ‘—

 

Pseudoeritiea‘l. —-—+u-—— Angst's Law

IITHIL ALCGOL — IOWA]. BUM

 

Figure V
ho Psis
2&0 °r
18h I
183

 

182

18]:

180

179

Volume, ft3/1b m.

178

177

 

 

176

 

0 0.1 0.2 0.3 0.1; 0.5 0.6 0.7 0.8 0.9 1.0
IslelreetieaN-Butene

mmm“ : fimt‘OP ..___m11_______
Peeudeeritieal —-—m-——- Amsgat's Ln

 

 

 

195

19h

191

190

69

METHYL ALCOHOL - NORMAL BUTANI

 

Figure VI

110 Psie

250 °r

 

0 0a]. Oe2 0e3 Ooh 0e;

Isle Praetien N -

hperimental + Pitser

 

 

Pseudocritisal—M— Amagat' s Law

0.6 0.7 0.8 0.9 1.0
Butane

~—--Ilaxee11——----—-

 

 

7O

METHYL ALCOHOL - NORMAL mm

 

 

Figure VII
125 Psia
60 280 °r It]
59 y.
.3 55
.I
Q
"l.
H
g" 57
E
56
SS

 

 

0 0.1 0.2 0.3 0.1; 0.5 0.6 0.7 0.8 0.9 1.0

llele Freetiee N - Butane

 

 

prerimental e pit";- . . Maxwell— _ .. __

 

Pseudoeritieal—w— Angst's Law

 

 

COMLUSIONS

72

COM-0810”

The fellowing conclusions have been made as a result of

this investigation:

1.

2.

3.

h.

6.

Present methods of relating the properties of pure

L 1&3}

compounds to the properties of their mixtures are 3

inadequate.
The Benedict-Hebbanuhin.combinations, van der Haals'. l~

 

combinations, Kay's rule and the Preusnits-Gunn correlation
predict pseudocritical teaperatures of the same magnitude.
These four methods predict pseudocritical.pressures that
show'marhed.differences from one another.

The pseudocritical densities Obtained with these four
methods follow the same general pattern as the experimental.
actual critical densities.

Pitzer's correlation.predicts vapor pressures of'pure
coupounds within engineering‘accuracy‘but there are
significant deviations when it is applied to the
determination.of‘the vapor pressures of azeotropic mixtures.
Pitser's correlation is sore accurate than.Amagat's law)

the pseudocritical method.and the Maxwell method.when
employed.to calculate the liquid.volumes of mixtures.

W083

73

 

W016

It is suggested that the use of a high speed digital computer
be considered for future work on the application of an equation of
state to the prediction of vaporeliquid equilibria. This machine
will enable engineers and scientists to extend.their studies to
innumerable mixtures and it‘will yield.a degree of’accuracy seldom
achieved.with.manual calculations.

It might also prove to be beneficial to investigate the
correlation of Lydersen, Greenkorn and Bougen (8) to determine

its accuracy in describing the volumetric properties of mixtures.

7h

 

50:30

a, b, c
A.» 8., c.
a, b, c,¢c,3
52: *3: Mb
32’ 33: 35
02, C3, 8

w‘maH-ez'ef'rf'ae

76

Heldmltz work function

Mela]. Kalaholts work function

Constants in Beattie-Bridgenn equatim or
state

Constants in Benedict-Webb ~Rubin equation of
state

Constants in hurtin-Hou equation of state

Constants in van der Haals' equation of state
Halal density, moles/unit volume

Critical density

Reduced density

Free energy

Partial modal free energ

N63611:!

Equilibrium ratio, y/x

number of moles

Pressure

Critical Pressure

Reduced Pressure

Vapor pressure of pure component at temperature
of system

 

Universal gas constant

Parameters in pseudocritical temerature equation
of Pramnits-Gunn correlation

Absolute temperature

Critical temperature

Reduced temperature

Prausnitz-Gunn correction to cmracteristic
critical temperature

Volume, unit volume/mole

Critical volume

Reduced volume

Halal volume of liquid phase at temperature
and pressure of system

Mole fraction in the liquid phase

Mole fraction in the vapor phase
Compressibility factor

Critical. compressibility factor

Graphical functions of reduced pressure and
reduced temperature in Pitzer correlation

Greekgymbols

2 121112ch (Prausnitz-Gunn Correlation)
Riedel parameter

5 1211I2(VcTc )1.2 (Prausnits-Gunn Correlation)
Liquid phase activity coefficient to correct
for deviations from ideal solution behavior

 

78

Gas phase activity coefficient to correct for
deviations from Amagat's law

Pugscity coefficient of pure component in liquid
phase at vapor pressure which con-espouds to the
temperature of the system

may coefficient of pure component in liquid
phaseatthetemperatureandpressureofthe
system

Total pressure

Acentric factor

3......
Critical state
Components i and .1
Liquid Phase
Propertyforamixtureanditsignifies that
the quantity is formix‘laxresasawhols
At vapor pressure of pure comment
At total pressure of system
Reduced, dimensionless property

Vapor phase

F's-‘1

 

79

mom

1.
2.

5e

7e
8.

9.
10.

13.

mm

inept, n. 11., Ann. Chin. mm, 2, 19, 38!. (1880).

Besttie, J. 4., and BridgeIIn, o. c., Proc. AI. Acsd. Arts
Sci., fig, 229 (1928).

Hattie, J. A., an, o. J., and aim, c. L., An. Chen. soc.,
g, 92'. (1939).

Benedict, 11., mm, c. 3., and Rubin, L. c., J. Chen. mm,
9. 3311—316 (191mm 13. “(In-758 (1912).

Benedict, u., He», 0. 3., and Rubin, L. c., friend, 1.,
cm. ms. Pro... 1.1, 1.19-1.22, nae-us!» 571-578. 609-620

(1951)-

Berthelot, P. 3., coup. rend., 126, 1703, 1857 (1898).

Cullen, n. J., and Kobe, x. A., 1.1.0113. J., _1_, use (1955).

Dodge, 3. 1., "Chemical Wing Wes,” W-
Hill Book Cm, Inc., new York, 19%.

Galitzinc, flied. Ann. msik, E, 770 (1890).

Hirschfelder, J. 0., Curtiss, C. 1., Bird, 3. 3., ”W
Mdmsumauquide,” Johnmeyandm, Inc.,

New York, 1955.

Hansen, 0. A., and Watson, K..M., "Chemical Process Principles
chem," john wiiey and Sons, Inc., New York, 19.6.

Jordan, '1‘. E.,'-."Vapor Pressuru of Organic Coupounds," Inter-
scienoe Publishers, Inc., New York, l95h.

m, H. 3., Ind. mg. Chem, gg, lOlh (1936).

 

1h.
15.

l6.

17.
18.

23.

211-.

25s
26.

27.
28.
29.

Kay, w. 3., Ind. mg. Chem, 3, 1.59 (1938).

Keenan, J. 11., and Reyes, F. 6., “Thermnenic Properties of
Steal," John wiley and Sons, Inc., Kev York, 1951;.

Kendall, B. J., and Sage, B. 3., ”The Volumetric Behavior of
Carbon Dioxide ," An. Petroleum Inst. Research Project 37.

Lorentz, n. 1., Ann. Ptvsik, 13, 127, 660 (1881).

Iydersen, A. L., Greenkorn, R. A., and Hougen, O. A.,
Generalized Thenodynanic Properties of Pure Fluids, ,7

 

Univ. Visconsin, hag. hpt. Sta. Rept. h, 1955. r
Mei-tin, J. J., and Hon, Yu-Chun, A.I.Ch.E. J., 1, 11.2 (1955).
Maxwell, J. 3., "Into. Book on marocarbons," D. Van lostrand
cm. 1951-
nelson, L. c., and Obert, 3. 2., Trans. Am. Soc. neeh. 3:13.,
15, 1057 (1951;).
Optell, J. 3., Schlinger, w. c., and Sage, 3. 11., Ind. Eng.
Chem, 56, 1286 (1951.).
Perry, J. 3., "Chemical Engineers Handbook," 3rd ed., McGraw-
Hill Book Compamr, New York, 1950.
Petty, L. 3., and smith, J. 14., Ind. Eng. Chem, 31, 1258
(19%).
Pitzer, x. 3., J. Am. Chen. Soc., 11, 31.27, (1955).
Pitzer, K. 3., Lippuenn, D. 2., Curl, R. r., Huggins, c. .14., and
Peterson, 0.3., J. Am. Chem. Soc., 31, 31.33 (1955).
Prausnitz, J. 11., and Gunn, R. 1)., A.I.Ch.E. J., g, 1.30 (1958).
Riedel, 1., Chen. Ing. chh., g, 679 (1951.) and 31, 209 (1955).
Schiller, P. c., and GanJar, 1.. ll., Chem. Eng. Prog. Synosiun
lo. 1, 67 (1953).

30.

Snith,-K. A., and Hatsm, K. M., Chen. Mg. Prog., 132, 89h
(19159).

saith, L. 3., 3eettie, J. A.,end m, w. J., J. An. chen.
Soc., 59, 1587 (1937).

Stotler, H. H., and Benedict, M., Chen. Eng. Prog. Symposiu-
1'0- _5_. 25 (1953)-

Van der ween, J. 3., Dissertation, Leiden 1873.

Van Leer, J., z. meik. Chem, 13, 723 (1910).

 

SAMPLE CALCULATIONS

33

 

SAMPLE W10”

1. Limiid Volume Predications for )iethanol - floral Butane mixtures
A. Pitzer Correlation

.00. _ X1“), . 130., . (0.7s)(o.201) + (0-25)(0.567) - 9.22.23

Pc- ' X1P¢1 + 1.1ch " (0-75)(37-h7) + (0°25X78-57)
:- h1.77 at.
are. - rice, + We, - (0.75)(h25.2) + (0.25)(513.1)
- Mme ’K
P, - P/Pc. - 2.722 g- 87.77 . 9:929};
13. . r/ra - 366.1. 9 M72 . £25
zc - 2° +0) 2' . 0.9605 - (0.293)(0.0239) .- 9:25;:
v . Zen/P . (0.9535)(10.73)(660)/h0 . 168.81 n34» sole

B. Pseudocritical llethod
'ihetaperahmesandpdessmsareidmticaltothose
used for thelPitzer correlation since Kay's rule is used
in both calculations for conbining the pseudocritical
constants .

Zc - 0.966 (Page 103 - Chenical Process Principles
Charts)

v - ZcRT/P - (0.966)(10.73)(660)/ho . 111.03 ft3[lb nole

C. mmll method
3,, - P(Y1)1/2/Pc1 . (2.722)(0.5)/78.67 - 0.017;
Pr: - P(r,)1/2/Pc, = (2.722)<o.866)/37.u7 - 9,1622

 

Tr . 0.8122 (Kay's rule)

21 ' .1292
(Page 152-3 - Will's Btu Book on
2.3 - 0.22 W)

V - lair/My;1 + 1323)
- (10.73)(660)/ho (0.75)(o.985) + (0.2s)(o.951)
- 169.88 ft3/lh nole ~52,

 

 

2.88 3.8.8 gamma 02.88 mo." u n
988a 8.48“ «928 $.88 m8 awn
81.8.. deemed .1938 $.36 v.8 u e
pamm.mm mmom.ma mpmmm.m mmmmm.m h
8.68 2298” «4.8% «.088 n
c.1628 «2.88 8.82. 8.18.8 83.360
85.2 880.8 08.24 3.884 on
288m 0.288 0888 p.053 04
a 885m 8.38 «Sean 3:380

 

 

 

 

dowvg cg oo.d gflfig 03 8.0 swag 331 8.0 Bdflg 0H8; 8.."

 

cannon-.380 Havana

 

snpnawoommmnm dunno: u «gonna no aoavmmmwnm Hacavunooonenm .m

87

gemmture and Density

(GP/ad).r . 0
ce" “2(302 - 35 d1’ - 282d6) - 2dco + (~2Aod - 30:02 - 6aard5)'1‘2
+ (R+3de2+ZBoRd)T3.-o

 

(a) 0 0.30100 (11) -21205 0.02.8.1
(b) d2 0.09282 (0) (l)+(m)+(n) 0.37072
(c) 03 0.02810 (1)) c(o)(k) 631.2.hx105
(a) 0“ 0.00851. (q) .2ch 8.58.6005
(e) 05 0.00260 (1‘) (p)+(q) 1883.8x106
(r) . «15 0.00079 (a) .2100 43965

(3) <17 0.00021» (t) can? -16609

(h) Yd? 0.51176 (11) 6a°¢d5 2163

(1) (lave-3026 0.22225 (v) (a)+(t)+(u) 281110

(J) antilog (1) 1.66821 (3) 3 10.731.

00 1/0) 0.59910 0:) - 3m? 17.733
(1) 3:12 0.27725 (y) 23080 9.101

(a) 3M 0.11.100 (z) <v)+(x)+(y) 37.878

(er/30).], e 0

 

(as) 103 (1') 9.275035

(Db) 103 (V) 11.853871

(cc) 103 (2) 1-578387

(dd) (as) - (bb) 8.821561.
(ee) (cc) - (ea) -8 + 0.303352
(ff) 4-5 log (dd) 7.232%

(83) 103 1.5 0.176091

(hh) log 3.0 0.1177121

(ii) 1.5 103 3.0 0.238561

(.13) (11) + (83) 0.1111652

(kk) (ee)+(fr)+(JJ) 9.950350 - 10
(11) 0 63' 7* 18"
(on) 0/3 21‘ 2' 26"
(an) Ice sin 0/3 9.555129 - 10
(00) log 2.0 0.301030

(pp) - 0.5 103 (dd) 21.10588

(qq) (pp)+(nn)+(00)-(11) £79318!)

(rr) antilog (qq) 621.132 '1!

(ss) T 71.9h3 'C

(32P/3d2)T II 0

ce-“2(6a . 6103 - 188405 + 1.5307) - 200 + (6de + 2303).,3
. (.210-6ed+30eccd")12=0

 

(a) e 0. 301.00 (11) .181 205 4.1.3311.

(b) 82 0.092112 (0) . 1.33.17 0.16298

(c) 43 0-02810 (3‘) (1)+(n)+(n)+(0) 1.88731
(a) 8” 080858 (0) c000) 25.15.5206
(e) .15 0.00260 (r) .20, 1h666.hxlo6
(r) 85 0.00079 (a) (q)+(r) 107790005
(8) 8" o.oooeu (t) 21, .5933

(h) “2 0.51176 (11) 6a 109272

('1) (h)/2.303 0.22225 (v) aorta“ -35573

(J) antilog (1) 1.66821 (10 (t)+(u)+(v) 119637

00 l/(J) 0.599115 (2) 615m ' 116.67

(1) 6d 1.82100 (y) 28.8 30.96

(a) 6183 0.93387 (2) (x)+(y) 187.63

(321>/ad2).r e o

(as) 108 (8) 10.032579

(Db) 103 (V) 5.077866

(cc) 108 (2) 2.169175

(dd) (ad-(W) 8.958713

(«3) (ed-(an) - 8 + 0.136596
(tr) - 1.5 log (dd) ' M32070

(33) 103 1-5 0.176091

(hh) log 3.0 0.1077121

(ii) 1.5 103 3.0 0.238561

(.13) (11)+(ss) 0.111.652

(kk) (ee)+(ff)+(d.1) 9.933313 - 10
(11) 0 7’9 13' 15"
(m) 0/3 215' w 25"
(an) log sin 0/3 9.621701 - 1o
(00) 103 2.0 0.301030

(PP) - 0.5 108 (ad) 4.1177357

(qq) (pp)+(nn)+(00)+(11) 2.793187

(11') antilog («10) 621.136 '3

(as) ‘1' 71.9116 'C

Pressure

P =- m + (3031 - no - co/Ie)d2 +0531: - e)d3 --ced5
+ cd3/‘1'2 Bl +Xd2)e' 36.2.]

 

 

(8) RN. 2026.8

0)) BoRT 96111-2

(c) -A° -22968.9

(d) ‘ -c,,/'r2 49007.11

(e) (b)+(c)+(d) -32362.1

(1) (8)82 -2990.8

(8) WT . 39728.7

(8) . -a _ -59907.7 _
<1) (e)+(h) ’ 20179.0

(.1) ' (1)83 -5669

(k) .206 109.6

(1) cd3 001.0 x 106
(In) (1)/'I'2 2078.2

(11) 1 +6 (1.2 1.51176
(0) x 02 0.51176
(3) (0)/2.3026 0.22225
(0) entiioe (p) ' 1.66821
(1‘) 1/(0) 0.599115
(a) (r)(n)(n) 1883.3

(1:) ’ (8)+(k)+(.1 )+(a)+(f) 1.62.0 you

(n) P ‘ 2.1.1. Ata

 

“xii ifi\(fl\)1\)‘(\\fi)flf)( hi ((1)111)?