• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Key to symbols
 Abstract
 Introduction
 Thermodynamics and the direct correlation...
 Two-parameter model for direct...
 Application to pure fluids-parameter...
 Thermodynamic properties of binary...
 Thermodynamic properties of binary...
 General formulation for multicomponent...
 Conclusions
 Appendix A: Microscopic model for...
 Appendix B: Carnahan-starling equation...
 Appendix C
 Biographical sketch






Title: Thermodynamic properties of high-pressure liquid mixtures containing supercritical components /
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Permanent Link: http://ufdc.ufl.edu/UF00097473/00001
 Material Information
Title: Thermodynamic properties of high-pressure liquid mixtures containing supercritical components /
Physical Description: xvi, 172 leaves : ill. ; 28 cm.
Language: English
Creator: Mathias, Paul Martin, 1952-
Publication Date: 1978
Copyright Date: 1978
 Subjects
Subject: Vapor-liquid equilibrium   ( lcsh )
Thermodynamics   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Includes bibliographies.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Paul M. Mathias.
 Record Information
Bibliographic ID: UF00097473
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000085309
oclc - 05320944
notis - AAK0658

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Table of Contents
    Title Page
        Page i
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
    List of Tables
        Page vii
        Page viii
    List of Figures
        Page ix
        Page x
    Key to symbols
        Page xi
        Page xii
        Page xiii
    Abstract
        Page xiv
        Page xv
        Page xvi
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
    Thermodynamics and the direct correlation function solution theory for liquid properties
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
    Two-parameter model for direct correlation function integrals in liquids
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
    Application to pure fluids-parameter determination
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
    Thermodynamic properties of binary gas-solvent systems; correlation
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
    Thermodynamic properties of binary gas-solvent systems; prediction
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
    General formulation for multicomponent systems
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
    Conclusions
        Page 136
        Page 137
        Page 138
    Appendix A: Microscopic model for direct correlation function integrals
        Page 139
        Page 140
        Page 141
        Page 142
        Page 143
        Page 144
        Page 145
        Page 146
        Page 147
        Page 148
        Page 149
        Page 150
        Page 151
        Page 152
        Page 153
        Page 154
        Page 155
        Page 156
    Appendix B: Carnahan-starling equation for mixtures
        Page 157
        Page 158
    Appendix C
        Page 159
        Page 160
        Page 161
        Page 162
        Page 163
        Page 164
        Page 165
        Page 166
        Page 167
        Page 168
        Page 169
        Page 170
        Page 171
    Biographical sketch
        Page 172
        Page 173
        Page 174
Full Text












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I .T E C Al iil' II. iL-F i l l :. linh. i.





PAUL l1. fl ti -L,--


A DIEE:F. .LI 1[1. FPE E .iLD iu TIEL E iAIiJATL C:'.L'iiIL ,:F
iHLE l FI;'EF.: iL ''r- FiL.PLDr
inl P,',PT I.,L ri.iLFILLri lh i,,t iris-. 1 l. ,I"il]f'l L l.,lI-, F,':F. THF-
EEf.r LEF I,,. T 11. F ,iF 'H!iLu ,iU .


iti.'n"E .!IT ,'iF FI.'Mr-DA


































*I'-' i F PREN- i3












ACKNOWLEDGMENTS


I wish to express my sincere appreciation to Dr. J.P. O'Connell

for his interest and guidance throughout my graduate studies. Working

with him has been a very valuable and enjoyable experience.

I also wish to thank Drs. D.O. Shah and A.A. Broyles for serving

on my supervisory committee.

It is a pleasure to thank the Department of Chemical Engineering

for providing financial support and a pressure-free environment in

which I could pursue my research.

I am extremely grateful to Jeanne for her excellent typing and

patience with my inexperience. Feeds' help with some of the figures

is also gratefully acknowledged.

Finally, I am grateful to the National Science Foundation for

financial support and to the University of Florida for the use of its

computing facilities.












TABLE OF CONTEI]TS




ACKNtWj l E.llCHE' 1 ....I ...............................................

LIST 'OF TABLES ............. .................... ...............

LISI OF rIUPES............................. .. ....................

.E'i TO SY IB LS ..................... ...........................

Ati ST ArCT ................... ...................................

C'ry.F.i:

1 It [F. DUL C i[ ir ............................... ........

Re t i en ~ .......... ...... .......................

S THEhIO[J i'UAiliCS ,"\I' THE ljIFi:EC Cl)PI:ELAT [I.'I FL),;-
T 1 'IO S,.iLtl[l iEJ FIiEuOP.'i ri LI.,UiD Fri,,l.L'ER lES .........

2.1 Int r.-.duc cti n ................... ...............

2.- a Stari ical Th ietiodJynarni, F.:.tii lisr. et i .3-
tlcn and DiscusSion of Fund.aental Equ3rions..

2.3 Der ,.,a Eion of Thermod,'rna ic Fquat ons.........

2.- iscus ion ........ .......................... ..

F eterencep ..........................................

3 [l.'O-i-[',AlAiE F HODEL F'.jR [ilFi-Ci CO' F FELAT I',:
FuNCTION'lr INIEiRRi'LS III Lii[UI ................... .

3.1 Pure Fluid Model ................... ...........

3.2 LE>.eniion to Ilixture . ........................

Re rf rences .................. .....................


iti




ix

ni

x i





1
8









11




-3

27



2,9



-,







CHAPTERS: Page

4 APPLICATION TO PURE FLUIDS--PARAMETER
DETERMINATION............................. ......... 43

References........................................ 50

5 THERMODYNAMIC PROPERTIES OF BINARY GAS-SOLVENT
SYSTEMS; CORRELATION ...................... ......... 52

5.1 Review of Existing Work....................... 53

5.2 Some Features of Gas-Solvent Nonideality--
The Carbon monoxide-Benzene System............ 57

5.3 Correlation of Experimental Data............... 60

References......................................... 93

6 THERMODYNAMIC PROPERTIES OF BINARY GAS-SOLVENT
SYSTEMS; PREDICTION ...... .......................... 95

6.1 Sensitivity to the Binary Param.:eer ...........

6.2 Comparison with Experimental Data ....... ..... 1ul

6.3 Solubility of Hydrogen in Coal i11s........... !i,,

References............................. . . ...... 1

7 GENERAL FORMULATION FOR MULTICOMPONENIT S i:.TE ..... 11.

7.1 Henry's Constant in Mixed Solvenrc.. ......... 11i

7.2 Discussion and Formulation of ai. rtbiltai.
System.......................... .............. 1. s

7.3 Application to the System: n-1.jLr.i -r.-r.: r.-
Methane ......................... .............. 12'

7.4 Application to Gas Solubility in io-rpie:..
Uncharacterized Solvent Mixture .............. 1 3i

References............................................ 1.

8 CONCLUSIONS. ............................ ......... 13,.

APPENDIX A MICROSCOPIC MODEL FOR DIRECT CORRELA 1lul.li Fui i ti
INTEGRALS ............................................ 139









AiFtL:' .[..: [ -E; .'iRt A., j-li-iA LiLM EiUA I FOR fi L :. IFLS ............. 157

. 'P L J[ i :-: .. .. . .. . ... .. . . . .. .. .. ... .. . .. .. . . .. 1 5.

B Il,-.iL: \ HICh L SKETC I .. .......................................... 112












LIST OF TABLES

Table Page

3-1 Universal Correlations for the Reduced Second
Virial Coefficient and the Reduced Hard Sphere
Diameter............................................... 33

3-2 Reduced Second Virial Coefficients and Its
Temperature Derivative........... .. ............... 38

3-3 t'redlcE[Lon of FedJ.j: .-j i.i chl'riT. C:.aprc,: 1tsb.il r.,
I-' I a0 a iun-ritio of F *du. d Ti^ r atur aiJ
Leni .. .. ....... ..... ...... ............. 3.

.-1 ic ting .:.ft Fure CoIEp-.nent Cn-.'.,prec i.:.n I ata...............

-2 '-.n:... [ihed V'.I 1':. *:.f Chjr cr,:-r L c tLC ariit.:- r f.or
the ic.rl-t Pa ra ffir. .......... ........... .

5-1 i:arbt.cn mon:-:*ide i i.-Ee : ne (2.1, t = 0.i10............ '
1.

c. rCarb.-n 'mIno>t ii-r,-1 l: 1ne' (-I. = 0. ............ r.
I'

-3 i.,jr.:.,gen (1 -Een.ene i = i1. ... .. .. ...... . .. ,

:-- H:,,J ri:.n 1 -n-:'ctane (' H = 0 .. ............. ...... *

S- H, .r.:,~in I11 -n-H >.a e I'2 . 12 = 0 .0 .................. .1

:,-: H, .rogen i-t1J, r '21 ,. .2 = 0. 0 ...................... .

I :, r,, _ro Ii tI ., n f a 2 . . . . . . . . . . .
1.



; n:, r.-.c II -Al rI rnia 1 2 I. 1 = 0 .0 .......................


-'9 CartLn rrno:.1i.le (ii -ul1cl.-n I:. i i, = -0. 1.1 ':..........




12
-1 u ; r, -'r, 1 -,jiiur,, nLt i_ l,1 = 0 0 .. .......... .... -q

5- 1 thane 1 l tJ er I 2 1 11 1 = 0 .0...... . ... ........ . ~

:-12 letl3 iI-tc r i,-', t,. = .. .................

-13 a rb n Jl iie I1 -l ter I = ..... ......... ',
12







'[I ble 1,

5-1- H. etih ne (I -Prp3ne (2), t:1 = 0 O .'i ..................... 8.

5-15 ehicrne (l -n-BuLn.r- (2) I. = 0.00 .................... .

5-l :lethane (.1 -n- et'e ne ('2,, : 'i = 0.05 . . . . ....... 8

5-17 i, Lhdr, e I'i -n-D :c re (2.), t; = 0.0 . . . . . . . . . .

5-18 urunmimar, ot Cor rei. tior, c,1 Binary Cas-Solvent LSstemi s .... a.

o-1 Tiripe rj ture Der i.;at iv oe thi? ir .jdijedx Second V'irial
Cii f ic i, nL ................................ ... ............ 99

6-2 Sensitivity of Henrir, 'i Constant tr. che Binary Incer-
.ac, or. Par meter ...................................... .. 100

6-3 Correl a ion of IHIydrogen solubill:t in Coal Oil .......... 0'

7-1 Coumpar i-n ct Predicted ,ran.d L\perimental Rcilcts tur
crie Sy,'temj n-But ran -rn-l ec c r-- i cl r . ................... 132

A-1 Co rcr l i on tor ReduJced Hard iSpri reJ Diameter ............... i2

A-2L Correl it i.mn ,j lure Fluid P-V-T Relat lors........... .. .. ..













LIST OF FIGURES


Figure Page

2-1 Molecular correlation functions for liquids............... 14

2-2 Schematic representation of the calculation of the
activity coefficient and comparison with conventional
calculation............................................... 24

3-1 Experimental variation of reduced hard sphere diameter.... 32

3-2 Variation of the asymptote of the reduced hard sphere
diameter with reduced temperature......................... 37

4-1 Smoothed variation of the characteristic parameters of
the n-alkanes with carbon number............. ............ 47

.- 1 E:.-i r t ..-ncli 1 i r i c t chi ii..ilJr i J i r i rb :. n
rp.: ,: .ide iir. i rt.:.rn m.:..-i.:-. de-b.enfere *-..lul 1... i ..........

.-r C '.' rr im-r ir. r.ri t *.il bl- ,. ..Ii. tirll- :[ b ren:enr .
i : j r L...n r.nn '. i. l -I:.ir; rne so I i:FI. ....... ... . ... . ..'1

; : -. l.ii E. ', *.* .i r.. n Ti :n.:.- i.j ir t r :r . . . . ... ,

-- .. l i.i I i ir. .:[ h' dr.:.r nri .ri I r ......... ... .........

..lutilcl t" .. ec- rei ir. it.r .... . .. ....... .......... 1




rr.. 1 i L y *.*. : =. a1st r1- zt.. J. in r n r.t.:..- r 1 c ..... n t:
r- t i [ i3 i ...' *..i i ,rL. :n..n i i.. .J I r r -rr.' c if ri t." cLdt u . t


r ..l i J r .. 1-1 i C 1n i jri Ir.. r..: .ei -j C l ed t r. tt.he
.: lutll r." ..t h .dr..g n in te. r . . ................... lit:





'-- F ,ucL u c l:,- ,t, H-r, :'; r .C' I- r ,L C' i. h re.L-t t Ejc r ,e :jl -
,:Ull j[,,j i th .l..- ,.llj ,1c: i. ,.1 1 i:f -C. e ve I tl 1 1:11:1.11 Cl ,jl
,. r ..:. . . . ', ,,,,. ... . . . . . . . . . . . . . 1. .,







Fiute PaD'e

c6-5 jlubilitcy c, hvydrogen in ,llnoline, ................... .. . li

:.- V"apor ..,:mpc.. it itc.n of h:y drogen-qu inc lin e o lu ions ....... 111

ti- o5,lubilLty o1 h.,d rogen in tecri lin c.a icii ed frofm t-e
soluii IlL o f Iydro'Sen in quinin E li ....................... 112

6-A. '.'a pr c,:.rt, pc, ition of H ,- c tr. 1 i i n iolut ,1nC : C l.ula t.3_,
Iron Lhe -solutillity o ~hy.dro..en in quiin.lin.I .. ... ...... 1 1

M-9 Solubilit:, of hydrognr inLl bi y-: ohe.-:y1] : alc. late fron
Lh-e .oliubilir, oLc h.,drog-n in quin-l ine .................. .11

'i-10 .o l.jbi il" .t hydrogen in m-Cresci :a3icuil3 ted from tihe
s,.lubilit',' c l hy dr.ogen in quini cline ................... .... 1'.

I-1 al. il -tion oi)" lct .,ry' *:onsLantL of -arbon di'-.ide (1
inl rthe minx.d :.i-ie.'r"c i 1mthi n l 2 -b --n , (,31i ....... .. . 20

7-I Calc:ulati,:.n of Henr',s -onslit nt of cijrbon dio id-e i)l
in rhe m l. i. d sol'.-cnt m,-l anol I '21-u ter t 3 (3 ........ .... . 121

7-3 C3l:ula icn of Henr: ,'' :onscanc ..f ci rbon ion-,. ide Sl
in Lhe: mixed sol ent nltr.c e -n nn (i.i-hcnzsenc ( ) ......... 22

7-- C:iii:l jL. t ion o Henr"' 5 on t;LntL it ar;go, (1) in the
mix jd .ol I.,ent IL th no3i l I.I n' .L r I j ................... ... 1

7-5 ;.:.:heinmat.i di ,gran of i.pp.roach C',o Iuilt :ImT'onil.EntL cI3e
at c.onrtit tL i l-n .-r L tur L. .... ........... .. . .......... .. .. 126

?-6 Exi.s ..oluJme cf n-uturn i'2.i-n-decarin and cacu lated
liiriry'_ constant of imecth.anr- in abco Tii/ zd sOl', nt. ........ 131

7-7 C.aiuljt cd ut.l- le press ur, of i metlhl -i; Ie 1I 1- L-but anl -ia--
dc.:anr, i: I :: + N,- i = 0. t,31) w'i t-hh i',rtar, e-deC:ar,

miL.ture tre ted aNs p eud,. :ronr.rn. it '2 i................ . 133

\- rI r-.ict-id nonrideau i lt ',' ,dr,.-, rz~-hei:ieni e ,c.ltl oric ....... 1-.












KEY TO SYMBOLS

A = 2-suffix Margules equation constant

a. = empirical constants
1

B2 = second virial coefficient

B = third virial coefficient

c = microscopic direct correlation function

C = direct correlation function integral

f. = fugacity of component i

g = radial distribution function

h = (= g-l), total correlation function

H.. = Henry's constant of solute i in pure solvent j

H.. = integrated total correlation function

I = identity matrix

K = Boltzmann constant

K.. = binary interaction parameter

n = number of components of mixture

N. = number of moles of component i in the system
1

P = pressure

Q = canonical partition function

R = separation between molecules

R = gas constant

R. = position vector of component i

R = location of minimum intermolecular potential
m







t = diuo. .' .'r rlible

i = jb ol ut t cLm opera ture

T = chat acttt3istic:: t[.iperatjurc

T = cr tijc l c. iripLr.cure

I' = i:h r .: c r ic ic 't.'.ni


S = critl: l I'. Iluim
L

= partial mr.lar '.*olumi, o l co ponInt 1

-m
' = partca iTr m lar .olumie oft ci:.mponcnt 1 at infinite dilution

'' '= .'c--:' '.,oluma e of iill.in.,

= liquid m:lt fr.c: ti: n of ciri i on,.ia t i

= ipr-or ii.:.le ftra ti.,r, ofi .o:mpnrT n ti 1

iir'



Gr .ek Letters
C. = 1 L tt r



'i total j- I.:cti rit, coC it i lc t ol comripori Lt i

I '-oiCip.'i iCro l .jit l'.'1t ccef fr t 11l nt of I'c~mi p:icii i

S = 1,-ard isph re diamei-r


S = :chim:,al potential of cco.mponant i

I = inrteiri.ale.:ul r I [,j p ir potential

i = numre-r den it-,

;rarnd :ari no i .- Il partitCior fui.:tior

S= i :,t h< ri- 1 c: -:mpr. i' i I ii :,

= ron ck r deta
11







Subscripts

i = molecule i

ij = ij pair of molecules


Superscripts

hs = hard sphere property

r = reference state

f = final state

L = liquid phase

t- = gas phase

MS = mixed solvent


Underline

- denotes vector, matrix


Overline

denotes reduced quantity


xiii







Ab:.-tt c of Dis :.-rt..tion re. -l.ncd to thihe Graduc, 3[ E Co-Incil
of the Universlit of Florida in Partial Filfilrlment f:, th. PLquirement-r
for ithi [-Dgr,- of Doctot of 'liiloo phy


1ilLP11i- iD'1tJAlC I FOPER.TI'I i.'F H [C-H-F' PEP .SULf, LIi'U .i)
MHi LIUF.Ec Ci:,-T.'. JIt [ Ill .Lit' ERi.F. TICAL Cip'' i', tlcr.

By

Faul M. M: tci--

June 1978

i'-, 1irian: iohn ,','Connell
;13j0r D.p.artrriien : ChemicAl Engineering


T-is .ork is concerned wil, a irmethoid tor dJiecribinr liquid-

phase fu3gaCiite-: in ord- r t.. FredicE higl-pr.;-s-ure v3por-liquid

equillbr uian f .yscteixs ccnEc lining supercr it ica cnmpoirient e.

A solucij icn tcleor-, fo'rmularted tc. pro,. idE. Ar.res:i on for

thle cli-nfl e in fiugacitCy v f C Cip.:,nenti o" a irmulticom.Iponent fluiI. duU

:to 3an' clangri. : inr the iC3a e of Ci, s:,-tei,, in tL rima ot inC egrals cf

the .61iSiCCiI '. ii ,-cla3nicdl direct c:.rrl'..ti ln f.ncri ion. (lhes

integrals are knownii c. t-e insen ltl .'.'IL t tfh. e,.act details of the

i[termaolecular forc.s f...r dien:e fluids and lhei.ct .- zimwpie emplric l

mrodl i5 prop.ed to de-crib, thirim which i based on fluid 'cruclure,

but .n.u.res tcli't. tl nrmerical calculations are crtr Ltable and rapid.

ithe rioidel Ia3s o:rnl) E p.r ailc[ Ers er for eii prur, comj-,:p nent c ,in a binar.

inc-Eraccion p.ar3marer for eajcri pair of coirmponent-. Ihe resiulcts i'

relarsci.el' ini: nsiti'. tc. the value of lih laJtI er, lthui e -eCntia ,

pre LicLin4 chhe result ofat isi.Lurbl ift ro pure component inf:,rmratiin.

Application of thn m(,odeil cc. pure liquids yields 3 good qquat on

of !a r-L for liquids. but nore imp.ortantit provides a means oft par3amer

determtinat ion.







The method provides an excellent description of the nonideality

of gas-solvent liquid mixtures including even those with highly polar

solvents like water, methanol, ammonia and m-cresol. Therefore, if

Henry's constant of the gas and the molar volume and pressure of the

solvent are available at any temperature, the method provides good

predictions of vapor-liquid equilibrium at elevated pressures.

The method also relates Henry's constants of a solute in two

different solvents. Investigations indicate that although these are

more sensitive to the binary parameter using the single temperature

independent value describes the temperature variation ot Henr.'n

constant very well. This method allows for instance a re.aonjbl-

correlation of methane solubility in n-decane using the Mnchin,,

solubility in water. It is also used to correlate the .ipor-liq.uij

equilibrium of hydrogen in various coal oils using the solut.iliL, of

hydrogen in quinoline. The method also gives good pred:iction for

Henry's constants in mixed solvents except for some aqueou solu.cin~; .

All the above results are used to formulate the apptoal.:h co

the general problem, i.e., the arbitrary multicomponen; -: -tem

containing many solutes and solvents. For this purpose at a Ci~en

temperature the input data required are the molar volumer ii d .apor

pressures of the pure solvents, the matrix of Henry'? c:o'ftrLnts i.

the solutes in the pure solvents (some of which may be et rminrd a~

above), and the excess volume of the solute-free sol.;it .i-.tute.

The formulation is tested on the n-butane-n-decane-meLthane ii ,Lte

and the results are quite good when a slight, but rea oi~ibl

C1i, uLmeni l [ .l h 11 n ht :,..1.en '. l J'd-etr .: JaCa. Thii Fre- lIJide







a definite statemenc, but overall Lndici:ions 3re that the formulatl.on

is ..alid for all multicnrmpynent -s.stemz .

Finally, a method is propc;ed to cjlculatt pllhae equilibrium,

by treatirng the aol',ent mixture as a ps;'udo-c mpon, nt. Tli 1: ;.'ery

useful fr ga siulubiliic in uncharacterized, complex stovent ,mixture- :.

Ine tigation. of Ltic n-buc lne-n-decan-it iLhane [crnar, indicates that

this 1i likely to be i viable appriacih.


:'-" i











CHAPTER 1
INTRODUCTION

Design of processing equipment requires reliable phase equilibrium

data. Experimental data are scarce and hence predictions must be made

in many cases. At present, there is no method which can predict high

pressure vapor-liquid equilibrium for systems containing complex

substances, such as water, methanol, and m-Cresol. This work is an

attempt to provide such a method. It requires a small amount of data

as a basis and is applicable to all types of systems containing at

least one supercritical component.

A thermodynamic analysis of multicomponent equilibrium shows

that at equilibrium the fugacity of each component is the same in all

phases. Thus all methods which predict phase equilibrium attempt to

calculate the fugacity in terms of the measurable variables that

characterize the phase. Methods that predict vapor-liquid equilibrium

can, in general, be divided into two categories: those that use the

same model or equation for both phases and those that use =;.-nr:i

equations for each phase.

The first category is best represented by methods that u e

empirical equations of state and by corresponding states .,tiho.'

Fugacities can be written in terms of Pressure-Volume-Temp=-rurc -

1
Composition (P-V-T-X) relationships of a fluid mixture. L-.ai..n-

of state provide these relationships and hence the vapor-!1quil.









equilibriumr can be :a lclaCed. lher'e ha..- been appiiiaicanon of this

appr:achl to mi.*:tures contatrining nonpolar and slich l.' polar -ib t ance '

i'hich ha .' Cet i- h ..ITme ijucceis- Ti'h drawbacks :4f chhe- m itihod are that

Che equations of siate contain man;, con.itarnt, -Arme of wi.hric.h jr

teTrmperjure dependence, and hat che miri g rule tfor obtaining g mri.:.cure

coCnslantEs -cntain Siiome paraimeters that require tinar,' or higher order

data for their e1.aluat3 1:-n. ibut it the externsl .e data required for

paramei ter *e.' aluati.-.n is a a..ilble, equation .of i tartc mrnc-tLcd LC-jn

prov.id, relia.bl- estirma e[ of prop., rti- for m ITlcicomrponent s .'stems.

Corresponding Esates methods ar.e u'tullU i special case .of

eqijations of Lt.at ,; thick. method emplo,'s cthi princi ple of corre.[-.jrndiLn

states to U-ite propertie. oat a mi..tIAre in t.rmiTi- of the properties of

a reference f lUid. The mrith':.d can nrrA.il l> charlccerize suhstancc

using f ,-uir firaireterS thniii equation-of-state methods as man.,

re eaarcherE in the area have shic.n. i" iii;..?'e er, it w..rI'= jell o:nl.

for mi,.rcures concaniirn subst anr.' which dre .imi la r to the reference

-ub ance.

Th- dicmtinant -.'eakness of t[he m-etrhodi decriLbed abo.e is that the.,

atct.mpt tr? des ribe the fluid in the t'..- -iaser region, ihe -:.pre slnri

for the fugJacity of a cornpon:rn in.ol.'e an integrati: n from .rr

denSity' to the den:it.' of ite phasE, at constant timperrature and

ci.:.apoit ion. Therefore. the irntegraicion path for the .:alculation oE

liquid-phasc fua,. it ies go.es through the two-phase region of the fluid.

The quaintiLtari.e dercriptior of a fluid in this region is difficult

when the fluid -:ontains similar substanc.:;s; ,h-n the substances or

the ni:tur-e are .ery different from each t:cher for example. h:dro'pn








and decane) this difficulty is greatly increased. Thus it has been

proposed that the equation of state description be retained for the

vapor phase, but that the liquid phase be described in terms of

departures from some conveniently chosen reference state.

In the following discussion we will be quite brief, describing

in detail only those aspects where our approach is different from

the one conventionally adopted by chemical engineers. (The reader

is referred to Prausnitz' book on fluid-phase equilibria for a more

complete discussion.) Also, we assume that the vapor-phase properties

can be calculated by methods already existing in the literature and

confine ourselves to a discussion of the liquid phas-.

Adopting the concept of using a liquid reference =ci-t r i.n

derive a well-defined thermodynamic expression for thet fu;a.ic'. o

any component in a liquid mixture as the product of ijedi .lue

and a factor; the factor, by definition, takes into 3. .ounE d..ij-

tions from the ideal value. The common practice among; -IhemicT l

engineering thermodynamicists has been to calculate Lhi: ri.:.ni.deiie

factor by invoking a two-step thermodynamic process. .*rmci' this

is a change in composition at constant pressure foll-:,c.. t. a *hai-

in pressure at constant composition, but other variar inc: m L be

taken. Thus we can write,



L f.r x r .(T,P,xi |
f(T,P,x) = ir x PTPx) exp{ i(T'P'df l-
Sxi Yi(T,prx) e R

P -


where the whole quantity in square brackets is the r:,nr.d-lic, Ia.cor,

', bt ing che .:.i. rpo= i i.:n .iorre t ion f : c.:.r and the -.ron:l ienr i ierm










Ic ht tamili a P:. ir ing c.: r r--CL rion act ,r i being cite pressure corir cLi. Ion

iact.:.r. Unlike Frausnicz iwe wrti; tlie rcierence quantity explicitly

r
as" 'I / I .1 The iimplicatiCa .ns :f this are discussed more fuil ',' In

tihapLf cr Here, ir is sUlfi ici rnt t.- ia'. rhi at Lith; ci rm arises

natirall, f romt a tunda. mental del t Iv i:'i n and thaL it s '.ver tL.-.i..ble.

Fir i nit, iic iL C,~te: l t, ih pure liquid fugiacit' in Llic cad,.: O: a

pure liquid reference stale. and L.: lenr.,''s con tant [or a coirijpcn.- nt

at nf LniLiL dillutionr. AIz.3, any mixed -ol-.'ent refi rne L.,ite i in

be handle l iciO.iL naiid li:at t i:n.

Equation i l-I hais t'ecn seilwn L i b a 3 r'opiti.:ous r=prcsenLacion

tor s.stenim at lo' pressure itt li il cL:.iTp.:ii L n termL ir s : min.nc.

in fiCtL, Li e ork of Prausnacz anJ co'wrk-:ers and F re.Jenrlun J nd

couctilers hails h.:-in thatl a large array :o liquid ri:nidealict can

be predicted to -.uffl- .i i aCi CJ'i i .:ur uIing r Jel L .lel: f pjrau etirL,:

bLa'cd on a large daaD SeE. Howcr'.el, chi SitU3it[-n i di[fccnt for

high pressure sYstems because thi P'o. iting i:Crrt.i. ion bCcOiiesc

re l. iv.l.-,' large and c u be ancounftd for accurately '. The wau,

..-rmc reT eai'Crct-r i .i'.i .LLtacked Lhie prot lem is b, using soej -._.'rr.cla-

Lion ..to ev .luaLd CE the PO.ntcing c..:rr.ct, .-.r and propo:i;ng a tlaujzible

but e--sentiall" empirical r.i:A el i-:., Lih, i-,:,,o -_'. i i n n,-.nidealit .

tihe par.ini.-Ler- in tLh c'-iTposiL ion r.'no deallr,' i.I bbs e:..,c'.a fri..

encrg.l i ':.i-del are then riLLed, anj subequcnil ,' 5.ne-raliz.-ed L.

using e.\perrimrental hign-pressure v.apor-liquid equilibrium data.

Prouunitz and Chueh achieved a large Jples *: success b chis

appro'acli r.* n.:np,.-iar SYistiemi.









Interestingly though, an examination of the actual values of

the composition and pressure correction terms shows that for most

systems the composition term is less than unity while the pressure

correction is greater than unity, and that the total correction is

relatively close to unity; sometimes less, but more often greater

than unity. For example, in the carbon monoxide-benzene binary

the total nonideality correction for carbon monoxide is within a

few percent of unity (< 4%) over a large range of liquid *:;.nd.ici 'n

in spite of a relatively large Poynting correction (up cr 3ji.'..

Therefore, from an engineering point of view it seems i .ii:niciL..

to calculate the total change from the same model. TI: chI- end

we write



f(T,P,x) = x.y(T,P,x;Pr,r)
SIJ I 1. 1



where yi (which we call the activity coefficient) is preciLei.

equivalent to the term in square brackets in Equation i-ii. in

Equation (1-2) we explicitly indicate that the activiEc c'.etficent

is a function of the reference state since, of course, L iEuL goC

to unity at the reference state.

In principle, the quantity yi can be calculated b:. 3n. derl.iaci.e

equation of state. One particular derivative equation ir. -:cItL i:

provided by the solution theory of Kirkwood and Buff f'.rrifJi -J in


By a derivative equation of state we mean an elujtr.in or -ie
of equations which provides the pirti-l deri"itives of in. [ii erni.-
dynamic quantity with respect t.: -ih i:f [cl. basis viri bles.









terms of JiriicL correlation fun-ction inte.ira1: linee ei hale the

macrix of iSjLlermi l partial deri.'ati :i oi the cheiiladi.a ptcelLijl

of all componCnLis of t he s:.ystedn vi'l recipe E co e3ch if ihi coim-

panent numr.er Je,~riti e. In this wor e i.,- in.:e ci4a e ct- alu.c of

chis cheor. as a incin. I ofr prediccin Jc.Aiati.on from idealic ,6f

iilgh-pressure liquid mi, .[ lu rL3 including those containing :omponien t

W1Lh I coFple' i in LTmolcrular forces.

'.~ ujily l le cheor., pro.'ide_: a more complete J-de.r ipLfion of

liquid nixturts [im n i.s implleJ b:, Equ. ion (1-2) ConiJed.r

L;inar.' mixture :f j s percrirical c':iinii[ ".. and a su.cri ical

component S ih |e most :orE'.'nienc lr rc'- tatce for thi :'.t5[er

is p'jure ,--the so-:cailled 'nri;,iTur. :ri. convEn'. tr in r r LhiS, yi: e

it would sm LthaE the cheor, dJ:i- noi a', 3nychinhlnt 3iaouE tf .: I)

= H H rii l K' con.sarn ; it cur'n ojut htia,' r that if ue kneu.

ilenry's coinS in t of in an:, ocler" sol'ent La the samie Cir.perrajcul

w- could. in principle'. calcutace HI In iic. -e ShO' Echac cii

can be done co uffic ini accuracy, in many : s i.t'or xa-.i ple,

,droarn in % ri.ous coal oil) lis; mrreans ihat nriy's c:, ;tani c

of a s -lute in an:, sol 'enE can b. c:onildered a "pure component"

fugsitcry -inci e alrlicouh IL is defined i-thi rcspecL ic oine pci:ricular

olW.cnc. its -.'3lcj in n', other soil' nc --or sol'.ent ixtll ure for lth

mrri3tr--can be calculated

Iinally,, this work repr' eresencE in atLtc .pi i use- _a ci.;it al

mi hanicz cjo dJescribe the complex syStems ercoiuntered. in chemiicalj

process' n l-reviously. PR.ger. ani Prausnir.2 ia-,3 e =.ho n th3t lie









11 12
Barker and Henderson perturbation theory,12 together with the

Kihara model of the intermolecular pair potential can be used to

give good predictions for systems containing essentially spherically

symmetric interactions. Gubbins and coworkersl3 have attempted to

incorporate angle dependent intermolecular forces into their

description of fluid systems and show, using a perturbation theory,

that many curious liquid properties can be predicted. But while

these investigations have produced results of theoretical interest,

they have not been applied practically. This is largely because

of the computer time involved: Rogers and Prausnitz indicate that

it took between 4 and 7 minutes to calculate a typical binir.

equilibrium point on a CDC 6600 computer. We have ( '.. r.::irce t

difficulty by formulating the problem in terms of quanEtici; chaE

are known to be insensitive to the exact nature of 1,n inte.. rM~il.Ji't

forces in a fluid (see Gubbins and O'Connelll4). Th.sj. %e hi

been able to obtain results of practical significance yhlil using

a relatively simple and tractable model.

Chapter 2 contains all the formalism of this work. It o-.. in

with a description of the derivative solution theor, for rhi;

particular application and then goes on to show how quantities oc

interest can be expressed in terms of direct correlaLioin fun:;ion

integrals. We also show the many advantages of usi:n Ecnp.-rjcure

and component number density as the set of thermodynr-ii b~iia

variables. In Chapter 3 we describe the model we e:4 i o. for the

direct correlation function integrals. Chapter 4 di;cu:z;i thc

.jppiC:ltion i0 .i Ltie mo:, del to the isothermrn compressitili i, of









rpur, fluid ; r.- is i r one means .,I pa r ameite r de ter iin a ic.n. The

app i icat3C in .f the method tc. high-pr. : u re, b ia ry, as- o l.'e nt

systems is discussed in Chapters 5 and r. Finjll w in Chjapt.r 7

we propose 3 general method Lo predic:r hil.-pre.sure Japolr-iLqIJUi

equi l ibr iuj fo r muIjlt ictompnrrin' r s'.:F tem s nd dJi oiis C iC o e r.f he

problem tih.,J risee when there is more chan stol.ienr in the ?,'ete..


Feferencej for Chapter 1


1. J.H. Prauj-ni -t "Molle. ulrl Thermi:.d'na3mic-. o l Fluid PhaF-e
Lquilibria" ii'r.rrii.e -l-131 Incr .. N.I. 19691.

S. Pi.'.'. ir',e, I. EC Froc. Des. Dev U. 57' '1 69? .

:i. B nedi t, C.E. lJebb and L.C. R'.u n, i heim. Lnf. Ir.. -7,
19? 1 19i1 .

I.W. Leyland and P.S. Clh3pp le.ar. [L',E Q', 15 i 19M.8,.

5.. J. h.:'lin :sor nd 1. D. 1'.as on. h'emr,. Eng 5 i-: 2 l n.",
119.j91; I.D. W.rison and J.S. FP.' lin-on. 'her. Eng. .-i..
1575 i 10r,' ; A.A Gunnwng and .5. 'o:jwl, in ..:,n, Cheri. Frn Sit.,
-. 521 11973I ; \.. 'leia .and J. TFUwlin ornI. Chhem. Fr,. "ci.,
:. 529 i I'j ; 1 HoLl ler up and J .S. r:c.wlinson, Chem. Eng.
Sci., .' 1373 i '1:- ).

u. Fredenslund, F..L. Jo..-nes and J.H. PrA u nit.-, AICh J., 21,
i10%l:. 111 5 1.

S A. Fr enslnd, G n rd Pf i=mu.-sen. "' ipFmor-L LqulJ
iEquilibr ium. U'ing Uir '.'.:--.-. .Gr-oup Conrtrib'ut ion Metlhod"
Llie''ier ,:entife ic Publi i hin.4 Co., 1'77..

i.Il. Praus rit: and P.L. Chueh, "C.:-mputc-r C.il.'l3tl c'ons tor
Hi;.h-Pres:.ure V'apor-LiqpAjid quilibria" (Prrntice-Hall, Inc.,
tNJ., 196 ).

9. J.C. O rEl..:wood anJ F.F. Buff, J. hei-m Phy.., I'_ .Th '1951).

1.'. E .L. ,iger' and M.11. Fr au niltz, Iran'. Far. Soc., Ub 3-7-
1 l97 1 .

11. J.A. Barker and D. Henders-:n, J. 'lC em. Phy ., -7. 2 5,6 i 3'l r, .




9



12. J.A. Barker and D. Henderson, J. Chem. Phys., 47, 4714 (1967).

13. C.H. Twu, K.E. Gubbins and C.G. Gray, Mol. Phys., 29, 713 (1975);
M. Flytzani-Stephanopoulos, K.E. Gubbins and C.G. Gray, Mol. Phys.,
30, 1649 (1975); C.H. Twu, K.E. Gubbins and C.G. Gray, J. Chem.
Phys., 64, 5186 (1976).

14. K.E. Gubbins and J.P. O'Connell, J. Chem. Phys., 60, 3449 (1976).












*:tL E'TLP
TIIL l i'Ii,',Ii .; ,:.l. THL [IF.t l i COF.F.EULA 'ION rUIril r;
'.'LLiTIuO' TtHEM'F.i FOIR L.LU iir PF.'OPERTIE


.1 Inc r.duct ion

in tlce pre-.ious chaprter we inrrc~u.luceJ1 cle idea thi it Uipliht

b od'.antag c us to ec.Oalate all dei.l-ac i.on IrCe TI deality idue tc:

changes in pressure and ccMnposition;) by a single mo..dl. Tiis

lchai pter S~hluS.i I how .. I soluclo'n theor, mr be emli lo'Tfed tfor thi

purp:.se. Lie de-" olop, clie ft'nrralii m r .liici iv; .'-- change in the

fulacity of iny component in tlie 5.'ternM due U. itc. ang in the

M.tate O: tlhe s tem. We note some Of the des iarble ft 1tures of

the represenctation before going into the dc11i;3 Of Lhe deriv.jiti:n.

1. 1he method erpli:/ temperaturee and compnc.rint number

density: .-a tlie bd.ais ariableb. Experience i ith ap plied stacattical

mechIanics indicates th.a [M ii. set Of 'j riablce is o fundar3ental

signift'icince. ,Also, and im erhgl s nore im.orc irt' l>, [ ? ., variible.-

are completely independent of each other, Iripli fyine tie mathematical

mITr.ipUlac13 n. iConcraat tihe Eicuation rLic [che usse Oa nolei frict iion

as part .**-f Cl baa is se ; in thiS cS tecl c i- str-Aint on cne :um of

the mole frctction s ma il it ip.o.:,- sible ct'. rtal 'i pf rctial deriv.acive

of arn, ,luantict:, with reSpecc to: one role frcticcin holding ilI tihe

oCtier nrA,,le' f rl'':tion-:i con-ttIL- L.









b. The formalism expresses quantities of interest in terms

of direct correlation function integrals. These integrals have been

shown by many authors to be very insensitive to orientation dependent

and nonconformal intermolecular forces in a fluid in the dense fluid

region. Gubbins and O'Connell1 have shown that if the temperature

and density are suitably reduced the direct correlation function

integrals of substances as different as Argon and water can be

2 3
superimposed. Brelvi and O'Connell'3 have demonstrated that in

the liquid region the direct correlation function integrals of a

wide variety of substances are insensitive to teT.pericure 3n.r1 m

be correlated on the basis of a size parameter nlr,. Tir.j i I.:.,

likely that a relatively simple model for the dir.:ct .:.[relIjti.r

function integrals will yield an accurate descrip-o.rion .i 2. :l ic',.

Section 2 describes the solution theory ari. ri qluinici.e

involved. Section 3 derives thermodynamic equate ionr fot -3.l l ir.g tt he

nonideality of general multicomponent systems. rinrll in '*.t.[i

we discuss some ramifications of taking the appil. jh.


2.2 Statistical Thermodynamic Formalism. Deri-. -itrn ari.j
Discussion of Fundamental Equations

This section shows how density fluctuati:.*r. r. .per. *. ,[re

may be used to derive expressions for fluctuation qurnit. i s in c.ro~i.

of direct correlation function integrals. The :-aior coriEr-i i:f [his

section has been reported previously4'5 but we r.~r ic it ir. irdr

to emphasize some points of interest.

Consider an open, 1.imo0,ene.s..j;, n-c:ionrieiE[ i =letm jetinc

t., the .jarit [ r, l e r I.; 'etit.re Lch.j ,ii t r....iL nti II o.' tli. n l PO,.| i ,n[3 .

n,3 [to1 -it ..'.,lu-e IT.u .,, [hri:,1J hiur c 'hi.;.st .JI i,:, c r it., r.









t:, an underline We define a lio&let number densict c li I 1
i -1
hdi tiimr-a'.'craged density of imoleculjr cancers io r:,pe i .t th,'

po-it i.n .:.B e Clhec Eie dei sie-i i15 a'eraged o'.'el 311 rii.nca-

tions in the, cas3e o0 as ruci:Uired par tici is since ie only,' cc.nsider

the l.'caciion of thc icercer of clie nmol cul. c. In similar fashion

we define rche pair number .Jenait', I% j ''i1. ) s he aJcrae
-2
densicr, of L molecular pairs ot ct''pe i at po-ri' i[in F and type j ic

F.. Inc grti c iO:n lt i i and p l , O r tch vlui e o:f che

s'.-tcic gi.es, I-% define cin, the cime-auveraged numbr.e f .i par czles

and Lhe time-'a.eraed nunibei r if i-j pairs in che system. Thlieefo',


V

Ri 1F: d_ Ii> ,2-1i



k. ~' lh F..__ d_.td[. : = i. C, N-'



whiere che usual nocacion his hi-en used for a imi-a' eraged quainity

and 6. ic t lhe Kr.:rn kel r delta.

A useful quanct;r, thI.t i.es- an indicaciorn of the inceraction

bc. ..wen the particles of th sycj etem it- tih pair corrcljciorn function

:.' tIL, radial d-iscr ibut n function. Ic is defined as,


2 J .

lj 1i h i ii
1 -L -


lhiii function can .ery easily he h-own co be the nJrm.alized conditional

probability: of finding particl j .*, i given that particle 1 1i5 .t 1 .









In a homogeneous system pi (Ri) is independent of the position

Ri and is equal to the average number density of species i. Hence,
-l1
from here on we simply refer to the quantity as p.. Also, the pair

correlation function depends only on the scalar distance R (=iRI-R21)

between RI and R Thus we can write,


S (2
1 (2)(R)-l)dR = 1 (2-4)
V ij



The notation of the radial distribution function in Equation (2-4)
(2)
has been simplified; g.. (R) gives the normalized probability of finding

a particle of type j at a distance R from a particle of type i. We also

note that Equation (2-4) gives an indication of the interacti..-n *: f the

particles of the system. If there is no interaction, then



=
l J l J


and the right hand side of Equation (2-4) is zero except for a

negligible contribution from the second term when i = j. In ti-i

case gij(R) = 1. However, in all cases of interest the interic,:ri,:.

between particles is significant. It has been usual to appr.:oiimace

the total interaction energy between particles as the sum of inter-

actions of all molecular pairs. This pair interaction is depenrder

only on the relative separation, and possibly orientation, of the

molecules of the pair. Figure (2-1) shows the quantitative for,.r

of a spherically symmetric pair potential and of the resultinr.























b




0

OJ
L..






a,
rU)



-0
c)
I"


I I








radial distribution function. We do not explicitly consider a pair

potential in this work--and in fact the equations derived in this

chapter are independent of any assumptions like pairwise additivity--

but it is useful to visualize properties as resulting from interactions

of this form.

An equation similar to Equation (2-4) but containing thermo-

dynamic variables can be derived from statistical mechanics. Consider

the statistical mechanical construct of the grand canonical ensemble6

(precisely the same as the open system we have already defined). We

can write the probability that the n-component system will contain

N1,N ...Nn molecules of the n species as


k=1
e Q(T,V,N1,...N )
P(N1,N2,...N) n= --



where Q and 7 are the canonical and grand canonical partiir.in ILfuncti.ns

defined by



Q = e-H/KT dpNdqN



S= Ne KT i Q(T,V,N ...N n)
N1,N2,...N



From the probability expressions the average number of p.ircl.:ieo

and particle pairs can be written as










.I = I. I 1 l ,... ni











SI' r,.. 1 n


i i 1:

l.iT
i P T.
ij- j r









T i






ir Equaions q -) nd 2-8 we .t .
: (,.* i T + lT




iI-tir -quations t2-'-4) had i2-n, we Cct


I i
*4 K > .vJ


= I i. g RJ-ldk + 12-')


.In.n:kce2 V is held con-tant we canr write the ;ibo.c, equiftii nr,


" -I 0 LT -


= .. )-1;F. +
1 j 1 j 1 1]


Equation (2-1i)1 is the b.a ic; equar. in,: of the Kiiri.o.:wod-Buf

soIlut ion chle.r... A fei. authors hai.re itl'teip1ti.d LO .-ipp': [tiii

equationii b., empliu:in: elemernntar:, tlhermJiodJ.n0mic I.-l ionE t*O ec:pr--e


t -- 7


i -dl








composition derivatives of the chemical potential at constant

temperature and pressure in terms of functions involving integrals

over the radial distribution function. This approach has led to

enormous equations of intractable complexity and very few practical

results. Further, since the radial distribution function is generally

a long ranged correlation, integrals of the form of (2-4), (2-9) and

(2-10) diverge at the critical point. Therefore, it would be desirable

to express Equation (2-10) in terms of some function which is the

inverse of the radial distribution function (or matrix inverse in the

case of multicomponent systems) and yet obtain a simple tractable

result. Fortunately this can be accomplished by defining a function

called the pair direct correlation function, c(r).



g (O,R) 1 c (0,R) + pk I k (O,Rl) [g( R, ) ldR

(2-11)

The function we have defined is very similar to the one defined

by Ornstein and Zernike.9 The only difference between the two is that

we use the angle averaged radial distribution whereas the original

definition contains the actual, orientation dependent quantity; of

course, in the limiting case of a fluid with spherically symmetric

interactions the two definitions are identical. However, Fpe i.:"i

results12 indicate that integration of Equation (2-11) .: it vii.Jld

the same results as integrating the original equation c.er orEii, --

tion and separation. Apparently in dense fluids, the ar-,ljr

indirect correlations (contributions from the integral ir, E.qjij rn

(2-11)) are unimportant and the integrals over the secor.l tr-r .:.n









thie riFlEt h3anl i ci are separdble. W. reitLratc hlre that this ielh .

workI is b si.]lc on this rathEr suiFci rising ret ult. 'hus, rfroi this i.oinL

on uWE 3.SUFfiT ELhil all contribut ion'r firoma p.rLticle correlate iions c:in be

mc delled as arising from ': sph Iricl s .,r etriL iniLCrtictioni.

The equal itac i'.e forr of [th direct correlacior ftunC:L on 1i

Ihown in Firure 2-1. It is q.ic E *Jift icule o ise physical

inL.-rpretatcion r o this correli tioin furn:tiin Iscv Percus0 ftor an

in.leph .J icu-sion). For the purpo_-es of Lhii chapter we assume

tha3t ltis funcitLin is onre wh.,ae ince._rai Is iLil. r .;d liable.

Ihus, an:.' pprojchi which iL prei;e- quantities .':f inccrc.t ir, terrim

.o direct corr ilL ion funcLion integrals -oul.] be implemenrctel

pr a, ic i1 .

We now go L.q H co lLie .J rivation at liar] incegrating

EquarLion 2i -111 ic. .r all P. ea s



i 'J i 2 '` ..i-i d. = p f C'" '2 lPJP + i i: i: I -, k 'i, d "








H = C + ikk j-1



wl.ere H and '. ..re the inLegrais o'.'Ecr the E ta3 corre lt_. ion unc-
ii1 i
i:,n g ( Rip.-1) and the direct cor rel.aicion function rEs [r.ectn ee .

In L'atri'. n Ltaraion Equation 1 -12 can be .'ricten as


H = F + i-


12-13)








where H and C are the matrices whose elements are H.. and C..

respectively, and X is a diagonal matrix with X.. equal to the

mole fraction of the ith component, xi. Equation (2-13) can be

written as


I + H X = (I C X)-1


(2-14)


Equation (2-10) can also be written in matrix notation as


1if i
X(I + H C) = A where Aj T .,Vk
S2-1 a 2-1 /KTJo giv


Equations (2-14) and (2-15) can be combined to give


(I C X)X-1 = A-1


where


1P ./KTI
(A )ij = 'P"-p J
S j Tp 'Pk


(2-15)


(2-16)



(2-17)


and therefore the individual elements of Equation (2-16) are


( i /KT

S 'kT,PAj


6.. C..
13 P


(2-18)


For any isothermal change, the Gibbs-Duhem equation is


dP = I pi dpi


Therefore, we can write










PT 1 .- (2-19
i I,p i t


Equations (2-1S) and (2-1'9 are the only ct.o basic equaLton-

ch3c i.te u:e. In the next section we show chac a simpl chermodynamic

approach can be used to calculate the change in properties of a sister.

due to ,an isochermal change in the measurable .ariables.


2.3 Deri :anin of Thermodj.namic Equarions

If any theojr. is co be applied in practical situations, it

must express prop.rtles of int ir in cerr m of conv'.enient m-ea-irable

variables For thE- desc riptiton of chemical proce--es these convenient

variables are cerperacure, pressure, and mole fraction of the ccm-

ponents of [he s[t.-m Eut Equation6 (2-181 and (2-19j are muc.-c

easily uted if che s3,srem is described in terrs of r-tcperature,

and number d.-nsit, or the n components. Also, trom the point Of

.:ieu of a model for the direct correlation function inctgrals

(Chpr r 3), the mosE appropriate set of Lariables is cLmper -iturL-

aid componenr niraber den.it'. The '-'.a w- e reconcile cheese two

requirements is to use the theory tc. c.-lculate the thange in

compi.nent number denitr, due co a change in pressure and compo-i-

cion, then changes in other thermodynaraic properties are calculated.

11
Some .iuthorg have aCtemirpted to use the equation o.if the

cype derived in tr-e previous section b. writing infinite series

expansion (in.'oling higher order correlation function)) about a

reference point in the solution. Herie, our approach is difference.

We assuu-.ie that Cie have available a method of calculating the second









order correlation functions for all conditions of the fluid. Thus

we obtain expressions which do not involve correlation functions

other than second order.

Consider a n-component, homogeneous fluid in an arbitrary

reference state characterized by temperature, pressure, mole fraction

of the n components, and specific volume (note that the system is

overspecified since, according to the phase rule, it has only n+l

degrees of freedom). We now wish to use the equation developed in the

previous section to calculate the change in properties when the fluid

state is changed isothermally to some other pressure and composition.

Let us denote reference state and final state properties by the

superscripts 'r' and 'f' respectively. Define a dummy variable t

such that for all components




P (t) = p + (Pf p)t i=l,...n (2-20)



The variable t operates in a similar fashion to the extent of reaiclrn

used in the thermodynamics of reaction equilibria.

We can now evaluate the change in any thermodynamic qujanrti

Q, a function of T,p,due to a change in the system from the initial

to the final state as a change due to variable t changing fromi, 0 LoI 1.

Thus,








1

S- = d
lo
cI'

1


i- r ,
dr




1










I -
t r n
)1 1 ] ,I t I t 1 t 2-21

















In f rac~ c ij easir [o deal rith the fugacit chjn =vich





nti r I
1 1 I



Nc.t,- hi,,r-. th t i.-. lie. charne.i oa er t'ronrii riolecu,,lnt to i ,Il r

repi cScntaria l- chl i .th repi n,1n K r ,by [,. We '.;e : t ,:. r.aie _i.i r'tl foi

molecular and molar *, .n* 1',' L-Fn lii;i-_ u=,*.; ljij rl- or no CC, nlu_ ipon.

In pFra.rE ice It i. e.-3le.r to d,:al '.'iti tilt iug'cit than ,i l'.

tit clihe'iti :l 'c.trflhl l.1' '1l2 p --" *'*.- r -' -, t iiiqu t oii (.2-1M) a3





3 -L K' -'



Fr.-.,. Eiutri,:.nS i2-21.i and 1'---3 '.: get


f I -- ," T ,i t 1':

SA .r I J. t


lTherefor ,








f ri f f (f r Ci (T,p(t))
Zn f = zn(. + kn x + Zn ( j) (t) (2-25)
xi 0 j


Equation (2-25) is precisely in the form of Equation (1-2). Now

if a problem is formulated in terms of T-P-x, it is easy to see that

Equation (2-22) can be used to solve iteratively for the final total

density and then Equation (2-25) can be used to calculate the change

in fugacity of all components. Appendix C describes the numerical

method we have developed to implement the procedure.


2.4 Discussion

The above development shows that if direct correlation function

integrals can be predicted reliably as a function of temperature and

component number density, deviations from ideality can be accurately

calculated. However, this method is new and thus will be unfamiliar

to most readers. Therefore in this section we discuss some of the

features of the approach.

First, Figure 2-2 is a qualitative representation of our

approach to the calculation of the activity coefficient of a com-

ponent in a binary mixture. Here, the reference state has

arbitrarily been chosen as pure 2. The integration path we have

adopted is a straight line in the p P2 plane connecting the

initial and final states. For comparison we also show the integra-

tion path corresponding to a method conventionally used. We :.te

here that the calculation does not increase in complexity as the

number of components in the system increase. We also note [chac

although Equations (2-22) and (2-25) may look formidable a:ctil


















TI 1'llPIn 1 '1 l El I'



11111 /1

I r E I I "
FI''.IIL C --- ----------I ----m a Firl n..L



i ,iiir "
II E

"' I,. 'T ,







I I I
I - - - -- --- -



I /1" 2^ "I-, /'




"/ '-





igre I. Sch-'ma ic repr esrnLLc I.n of Lhe' iCj Icll jL ionl
oft Lh act cL .1 cc.'-l t'ici nr_ nd c:.rn,'. ri ._.n
I-. Lh :onvner i.-an: I C c- Icula Lion









calculations are easy to perform. The solution procedure we have

adopted is stable and quite fast; on the average it takes about

one-tenth of a second to calculate a typical binary equilibrium

point on the Amdahl computer at the University of Florida.

Second, we note that the equations are simple in form and

independent of the reference state chosen. For a pure component

reference state f/xr is fL pure(T ,pr); for infinite dilution the
i i i

quantity fr/xr becomes Henry's constant. Further, the model and
1 1

equations are precisely of the same form for any reference state.

Recall that for methods involving the Gibbs excess free energy

different models are required--or perhaps more accurately, the

same model must be normalized differently. Probably of mouL s'lu ,

the representation of infinite dilution of a component in i n,..-e

solvent does not pose any problems; in fact, as we shc. betloi. ic

provides some useful requirements on the parameters.

Third, since the model provides a complete descrifrt ii, i:f

any change of state of a system, there can be only one refrenr:e

state for all components. Any other reference state vill t.. .-ith.,r

redundant or inconsistent. If the system consists of ri..-. uIer-

critical species and one subcritical substance, then chE mn.i

convenient (unique) reference state is the pure satura.:d UJit-

critical, i.e., all the supercritical species at infir.ire dilurticn

in the subcritical specie. For this system the above d. '1.i..I,. =L

can be used in a straightforward manner to calculate j.r,:.pEcit

of interest. But now consider the case of two subcri:ic al i uile'.nc.









sut.scanc s in the system. Hr. .i- ,.woul d hia'.- irntorimir in about both

O1 thi states :onrnE3in gi- pur '. llint- th i moi'd l can be I.sed

to Calculate the pripertles in onl :,il.A nt from those r in the ,other.

Thus., as s t ad earlier, the properties l, the second 'solveni are

-ithEr redunid ntr. or inconsiiStcnt WJe thc' in icter lih,]p.r thar

the irconsiste nc'y C:an [ us-d for par.i Tirte T determini-t ion; in ain

~ac this irconsistiencc must be rermnved.

FiOrthA.r. consider a SYstCm Wi.itn a super crit 1C sei'jtLnc' S

and two sub.crit ical subst anci.-i S, S .jnd '.. ." A we h shoi,i. th' mo.Jdel

can be used cie. *:CalcUlate the ratic' of Hent constants A S in pur.

S .. l 1 n2 d of 3- in pure 3 H 13 b, 'oin- fror pure 3, .a the initial

tcate to pure 53 a- the final scate. In similar fashion we can *caicul3t-:

che -.n]-r, rat-i tl tak hierng the re'.',-r pach. lNio becaus of the n:onex:'iac

nature of the miriodel, if hilic fil J State i- specifiEd in [terms of pressure.

these rat ,ios will not be .qual and cth innr ae S'rlt tr, .O'f t he s, tei m Will1

be lo.st. H .owe.,er if th.e f l aill s ta e is sp cif, id iA. r.rms f r..ta l

number densit:. the .r.iiecr:'. iill be rtA. inedJ in..e the two rat ios will

alwo,.'s be euatl. Tne symmetr.' requirement is rrore th.an ran est-htic

consideration. This Is shown tc adding another subceritical iuibsjia:ne

5 to the St.Lm. .Now, we can ShOw LhjLt iif ha4' v' good: predi :t iConi [r

the r.ati i:s inv.'iol.'iii s,.'.l a[ut nd W ., n m for 5, .i i.1 a then the

s iir et r injareos Lthat h3.'e a good prediction for the rutin' iin.o]'.:inc

53.iid Ilaori, impnrtaJn l 1,'. we can now pr ddict Henry'- conicant of

51 in the mx:ed saolvent; here. of course, we wcuid require the mi'-.,d

sol'.1 nt densit.y. or, in more familiar terms. the e:.. ss '5 volume e of mi.:ing

f he. mi- r- ,:d si .'nL stJ ate.









A final point we wish to make is that although solvent-solvent

nonidealities can, in principle, be predicted from the model, a severe

small-difference-of-large-numbers problem is encountered and thus the

prediction is very unreliable. Therefore, in the implementation of

the approach on a general n-component system while we use the theory

to calculate the reference state (solute-free mixed solvent) fujacicEi

of the supercritical components, the reference state fugacitie- *.:. r.h

solvents are assumed to be input data. We expect however that solvent

reference fugacities will be calculable by already existing methods.12

Most of the considerations in this chapter will be pursued

in greater depth in later chapters. The discussion given here should

only be considered a preview.


References for Chapter 2

1. K.E. Cubbins and J.P. O'Connell, J. Chem. Ph,-:., r., 3--' i '.

2. S.W. Brelvi and J.P. O'Connell, AIChE J., 18, iB23' 19'2.

3. S.W. Brelvi and J.P. O'Connell, AIChE J., 21. 15. i 4'15,.

4. J.G. Kirkwood and F.P. Buff, J. Chem. Phys., 1i. "2- i19"1'.

5. J.P. O'Connell, Mol. Phys., 20, 27 (1971).

6. T.L. Hill, "Statistical Mechanics" (McGraw-Hill. rz. 'ri 19'- I.

7. F.P. Buff and R. Brout, J. Chem. Phys., 23, ':. ,i.5i.

8. F.P. Buff and F.M. Schindler, J. Chem. Phys.. .. I''.'- i'i50,

9. Ornstein and Zernike, Koninkl. Ned. AA.d. Wer.cn-ri:hap., rod.. .r.
B, 17, 793 (1914).

10. J. F-r,,u_ "TI-,- FqJUil it-r. jj-, Tt- ,:.r; .-f C I z _--r. 1 -l ol.i.: :,
H L. tr r -:l. .ini ._. L. i..b.:- it:. F[ i -. . f l .R .i. i .i. In .
tH .. r I ., JA. i i- :'.

ii. SI. rEl, i. Ph.['. ui -.rr.aci.'.r L'ni.mr it; o f l.rid' 19,3.

12". J. 1 FI r -,jz n i. "Ii ,-J ,i r I hcrn.r-.d ii, -: '.f Fiui.i:l-[ h.- .Equi 1 JIbr I"
r r Iric.- H-l 1 loc. Iij, 1 .i '













:CAPT ER 3
IWO-fi'\PA IE[EfR IIDEL FOP DIRECT COri FI. A[F 0l N
FUNCTiON It'TEGFCALS IN LIQUIDS


This chapter describes tch model .. la'.,e de*'.el'- pEd tE

describe all the pair direct correlation function integral: of

a niu tic_[omponent liquid. The parameters required are a charic-
A
Lterisic teCirperaLure I ard a c:l rac3 eri. stc columnmrn V for each

c.;poinernt of the "slnem and a binary InteLact ion parrieter for

all unlike pairs. .Secticon 3.1 describec.s the pure iluid iodel ind

Section 3.2 slowi.- how the pure fluid moiJl s1 extended to mi:-:Lure-..


J3.1 Pure Fluid Mc.del

Three consideratcl-e went into thie choice of the mrod] for

thc pure fluid direct correlation iur.:cion integral.:

i The worl oqf Cubbins and O'Connell, and Brel.'i arid

u'C. nn ll hI a., sh.- n dhatl hi. quart ity: cra be moIdeled easily.

Tlius, we expect tlih t .om i tormn of Lto-parIam-tc r corresponding

-'taes theory would provide an adequate descript ion.

li) FuindarmeciLtl tat isLi.-- mechanic. ree aerchl on perturb.ation

tl-heo ies h3as sholiwn hlat in thi diene liquid region correlaLtion

functions are predominantly determined by r cuil ;i,. forces. These

in Lturn can be approx.imaLed as hard sphere interactions with a

condi tion dependent hard sphere size.








iii) We would like to have a pure fluid representation which

allows easy extension to the mixture case.

For a pure fluid, Equation (2-19) becomes



1-C = P/RT = 1 reduced isothermal (3-1)
T p pKTRT compressibility



Ai!2-- for Lhe .-rial e'pan i.:ron for hr.e ocn:,preE:=ibilic4_ factor






I I = i + ..b., + j3,i', + ..3. -2




i-r.e'e .. i r. h . c ..r,,l r i 1 *.:i:c ff ici r-nI.. i, [Eh: rhir r i l

.:oetl ient an, =.l .

'e niL- Frt p-_i ? that ar e>.pre 3s r, l:f r the pure flijid jire:t

.:orrelario:n fin.-r.i ri .r l 'hicli iz c.-i riz:r-.t : .ilh th.l ab.bvvc

c.:o iJd e r t 1



1-i; = I-I. + i L .. i 3-3



t-here 1= i- c.e .ld re .c nie.1 l ic i.n t'lJrn i -n .. 1 ,ir.i] ph.- rE t'fl I

anll .L i i l -. f.ri 1 '.: :efficient ci the t'ad hiere fluid.

IThri; or ,[o exr, t _-:i. .n h11 brn uIed b for,:.

Liu l.ion 3- -l i- inr qlual raci.e ag.reemernL iirLh consideration

i'' :in:e r. hi l dr : i r., .i. *Juan l ar nr :.r it-uL.i-n t, (r m i l t.e

I r:i . lie .a.- re.-l celd tkh: hard sphere second iral coefficient










i5, lthe i c jl l -cand '.'ir ii.jl cit Le ent .i.nce ac normrl 3l Ccond j cic il

the rec:ond .iriii co'efficienl i nepit ive', i.e., the attraci.'ve

force ouauCv'egh the repuls.ie c'rcO:. ihi- can be co.nsidJe d a ae,,

of accounting f.-.r the atttr ctcivi forc- .

Equation (3-3) als, s.atisfiec c.:.n~1.Jr.ILon li 1.1 since it

car. v'rcr ea-,L .i be generalized ro m ri\Eure.. lli L rrnahan-Srarlin,

elu'atii S Fr.r idei an iccurate cEt iLiLE i direct c rrel- i.:on

funcLiOi intr ral; Ln hard .pherl e mli'.aur5 s. 'I~: sac,: ld v' rial

;coiefici crn is a Luc,-bod.J term arnd hence i~ rhe 3jr,ie in principle,

in the Ii-.
Ui;.ed tC' i-lcjlaIe the ,c ros secondd virial c:oe ffcient-.

i' r-o. iasert t13li coinildera [ion ( )11 Jus ti ie5 a ':pos ulael.

C 1 i. L :

a. Ti. relJued hird Apler.. .diTm.iter i: =.:mU unL.Cr:31

funccion of tedJu.:ed teiperacLure and reduced jnLi.lt., and

b. Ie reduced second ''iriil coQ efI, ci.nt. i sCum untliver=.l

functionf of reduced terpin ra ture.

ini *.lh.r word, it we can find empirical correlac i.n= for the reduced

haril t;lpre Jdiametr and Lhe reduced secondd viral coefficient for

some refercncE fluid the. will h.ve un'.v:r ,,al v alidit,,. iW choa.e

.\ranr land Kr'ptcn ind :ennn iuici tl.' reduced i as tie reference

oublLancc since a large amount of reliatle Jdi ir e av liable.

For hiii- -ub-UL'nc,; 'e :-:~d thlie char.ac erJi.s ic par ,amet er, rc r the

corre-pondin c itic.al cor' t.ani s.


11 1









The correlation for the reduced second virial coefficient

was adapted from that presented by Tsonopoulosl8 (Tsonopoulos uses

a characteristic temperature and a characteristic pressure whereas

we use a characteristic temperature and a characteristic volume;

the change however is simple, requiring only that all parameters

be multiplied by a constant). The correlation is shown in Table 3-1,

Equation (3-4). For the direct correlation function of the hard sphere

8,9
fluid we use the Carnahan-Starling equation.89 Now at each point for

which data was available we found the hard sphere diameter which matched

the calculated reduced isothermal compressibility with the experimental

value. The reduced hard sphere diameters obtained are shown in Figure

3-1. From the form of the curves we deduced an empirical correlation

and the constants of the correlation were fitted by a least-squares

technique. The rather complicated function is shown in Table 3-1.

The particular form was chosen because it has the following features:

a) At high reduced temperatures the hard sphere diameter is

a function of reduced temperature only as might be expected.

b) At high densities the hard sphere diameter is a function

of temperature only. This again is to be expected since a hard

sphere fluid is a good model of a fluid at high density.

c) The first two exponential terms were chosen cr., r._prc-nt

the minimum in the isothermal change of the hard sphere Jlidi~acer

with density. The particular form was chosen since ti-: .-i 1L :,'i

of the minimum seemed to shift somewhat linearly with rsip,-riture.

d) The last exponential term is used for chang.- in ihe

hard sphere diameter required to obtain an accurate r.pr- i-ntjtr..-n

















'CP-









A\


I 1I I

p' L =
IJ.L' -
L 1. I


1.

C.i





















Os
U



7
Li'
:f

























',









-i







,L'


.L'
|L
I--










I .- 1






3
ij il



Sfc




r ;?





L1t

*'C
1Ll-


i



~










Table 3-1

Universal Correlations for the Reduced Second Virial
*
Coefficient and the Reduced Hard Sphere Diameter


Reduced Temperature: T = T/T

Reduced Density: p = p V

a. Reduced Second Virial Coefficient

3.2: 2 = 0.4966 1.134 0.4759 0.0416 0.00209
T 3.2: 0.4966 3 (3-4)
V T T T T


2 0.1376 1.972
T 3.2: = 0.3301 1376 9
V T T

b. Reduced Hard Sphere Diameter


T 0.73: f =
a


(3-5)


a8
a7/T


(3-6)


T < 0.73: fs = a4 expl- a15]


2l 3
-N 3
3 av
V


fs + a2/exp[a 4( + alT)2] a3/exp[a5(P + ai -


+ a9/exp[al0{(T a13)2 + all( a12) 2


a1 = 0.54008832 a8 = 0.1756588.

a2 = 1.2669802 a9 = 0.1887482-

a3 = 0.05132355 a10 = 17.952388

a4 = 2.9107424 all = 0.48197123

a5 = 2.5167259 a12 = 0.7669609',

a6 = 2.1595955 al4 = 0.809657E,-.

a7 = 0.64269552 a1 = 0.240628E'


0-76631363


!i..,cc tr.1-, .. .r-_- l i, i i .i ,plI ie in [r rmr- of r. du,.: d Ir LtblI.
and tI- i [l"' .'.r, -L jnc r r -i ina p-_idcn-. l *-r tl," pjrricular urij C .- li.- 1-
f.-.r .' jnd I


-?I









in the clitical region. it is SlA t S-urpri ing Eu Ui Cha3t

const tantE 3 a.d 3 13 ,are not clo- r c ..w u itt .

ihe fuficti''.il form ."-i the hish density a3 ,mptote of the

reduced hard il-rhre diameter It i, Equatioin (3-6,1) must be clho,- Er

within c.are- since argon data is available only over 3 linrited cempa'ra-

ture range i = uI. I = 2.81 and yec rany.' flu ltd o praictiial

iintrest are rormall, at reduced termpera cures .o.ucside cte range.

A pl..t :oi this asy:iptoce Lestim cted from Figure 3-1 against t reduced

telr.petr aurc indicate Cs that the l..garLthm Of cthe hlh djiie it, a'.imItote

ctr'"es linieirl '.. th che logritirm of the r,_'Juced ctorri[ ra ture. Thus,



i'n f i'n ., ., i'r I 13- I
S % a



We chi>:.s, the functit al1 [fori of L iqun io, r '3- ) b.y ui ring Equation

i -8 1 and i.' making our funcclon qujAitci'.e-- coaiStis cit Awith tci

Barker and H.nders-ioi ercurt i. aci 'i i Ch r.. lr I r t r s t tio use

results of thise cherry e' r h.e to dMfinae 3it iicernnlecuila pair oarcen-

Ltal, buj. for our purpo:ies, the de iniCtion does niot hla. to be quln-

ticitivc and w- need o'nly' assJuml chit the pair potential has the

quali[ti iv forTi sli iI 11i Figure 2-1. Tliie Barker-iir erder so hard

diameter is ,g en b"




UP .. 'z.1 ,3-.
J


*lher;r B i -. the Barker-Henderson hard sphere diameter, r i. an

inEcrm olcular zepar action which rrepre cents t .icn rcpulsti-l. iLicerdi tC olin









between the molecules (Barker and Henderson somewhat arbitrarily

assume r is the intermolecular separation corresponding to the zero
o
of the pair potential), and z (= T/r ) is the reduced separation.

An extrapolation of Equation (3-8) into the high temperature

region is in qualitative agreement with Equation (3-9) since Equation

(3-9) indicates that the hard sphere diameter decreases with increasing

temperature, but the effect of an increase in temperature decreases

with increasing temperature since the pair potential beccr.,.r "*r

steep at small separations.

Equation (3-8) predicts that the asymptote increase WichIoi'j

limit as the temperature decreases. But this is clearly in di. r.c i f-,c

with Equation (3-9) which indicates that the hard sphere Jiaui-c; r a.-

proaches a constant value of r at low temperatures. Thur jsIumT

that at low temperatures the asymptote has the essentially, -,mpiri~:

but qualitatively correct form,




fs = a14 exp [- a15T] I5i-


and we insist that the two functional forms have the same jiu- nd

first derivative at some change-over temperature T Thi cl t.

following relations between the constants in Equations (3-", ji. Il-11i:


a7
a1a = exp (a) -i'
14 a8
o


i 3- 1' !


a15 = a/To









The complete fu'ncltona3l forr. .as fiLted to compression dac3

b,' a lEi]ast-squares technLque. Tih values of the consr,; ants obtained

are shl.wn in T.abl 3-1. and the high dEns1rit aj,,-imptote of the reduced

hard sphere di',,ietcr i- plotted in Figure 3-2. We noe that rhe 1, I

cmpe.raiture ae'._ptcorte -*f the hard sphere fluid for irgon corr-espnds

to aj hjird sphere diaameter rof 3.6 S. n accur3ate -valuaion of che

'1
argon pair potential1 indicates that ttre zero and minimum of the

potential correspond r- i;eparati.ons of 3.'.\ 3a S1 respe:ti .'l..

Thus. our reasoning seems qaJl iLative.l; Corre't. HI e..'t..er, e would

nrc recommend the 1ue Mof this correlation at tempetratures below

i = 0.3B.

The repre sent-at ion should be Valid for supercr it cal substance.

which are at v'.ier high reduced temp.eratures; fo4r example. the re.due.d

temperature of hydrogen at roni te-mperaturue is about S. Although, a-

discusi.ed above, the reduced hard sphere diiaeter correjla-irn should

be .ajlid at thesc high reduced temperatures, the second 'virial coef-

ficient of i onopoulo;sn is not applicable. Hen,-e, .e adateed

correlationr of DeLing. and Sh.pe_ for h.,drogen and piece. the two

correlations together at the "Pcluein point", T = .2. The high

temperature fo.rm is shourn in EquaLion (3-'5) able 3-2 shows th,

*.'alue of the reduced second 'irial cLOfficiene and its temperature

derivatLive for the complex te crrrelar ion.

ablee 3-- shows the prediction of the reduced inothermal

c.-ompressibilit, 11-C) for a uide range of reduiied conditions. The

representation calculates the lhinr. in pressure corresponding Lo a

chance in density to an av.erag.e accurav of 0.i61: for argon.





-ut
w,
U




73





U
w

0 d

0 d





ci












Table 3-2

F.educd S-ccond Virial Cc efficient and ltI
Temp-pracure E:r i.'at r.'e


T = IT

-, = B,'V

Er,
T ,2 d 1 1


i5 16.8 30.,r,

0.- 4.15 l. IO

0.5 .5.. 5..8

S- 3.03 3.66

0.9 21.2 2.93




O. l.-l .38

1.1 .-lo 2.23

1.5 0. .3 1. -

2.0 195 1.6

3.2 + 0.95 1.


J3. + 0.'5 1.



1l.0 0.2 :'?; u.'5

:, 0. -l. i 0.33












o m o

0 -4 -1
C14M


00 -T r
1C o in r-
CM C14 (n -


0 0







o o4


C4 0
C:O


O 04
o --T
l co r-
r 0 0


0 o 0 '
mu i 0
I 0) 0 D
SE -1 0
ac E a


l Ox

0) c
0) 0
ol I-4


o I n m 3 1-4 oo0 m oo
o o O O o -l D 0 O
-0-
0 .0 I .
o H -1


N N
I I I


0 0 0 0 0 0 0
00 0C CO C C CD (O
0 -I -4 AH ,H C'J CM4


-v >






.fl d,
H a


O r3\






N -
O--
H









Appli icCi ion to orrher subsr.a:ncs i. dec'.r ibed in tihaptrr i. The reicons

of ni 'aet iv, comsprc sibiii ty--corrc ;pond.J in rto mei:chanic jl ins t Aic 1 --

are .-ll in tre rl- -phase re io-n x.:-:.epr for thl negati,.,: e lIjues in tl.,

c:r ti-al isoCheTr.. Tlhu che repre enr3tacio n :l :- nor pr-.ll..:c th

*:riti.al point ex:a,-c tl,. Bi[t rliL is nro a serious dra'b3,acl ,irni:.: .

,ji i'-::,t E*2: -ecL a tw,-.-p3r3imeter c..'rrl.latioL n tC:. L ic' rk in the critical

re~i ,n.


J.2 E-:tnsion cv tli:.ture=

W'i 3-juntl e hIIC the Va'snt'r u pair li.re:t rcirrelatrion furn ri.in

intr r-al in tlei r iit.ri e ar. gi'.'n b., logical e>:ei n i:, f Equ i[t:on

3-31. IThus 'e .ritc,



ht hc
C = 2i.lE I i 3-1 '1
Lj ij 1j 1I



The i-) 'c:.rin.i irial cc' fli. enr is c.al.:u acd b' as ur, ir n ihe

starndard iiix inar rule. U le aissumi that trie harl sphere mtiy.ure i' one

with add tli e diamiters and 1 J-: : ribh d, b' the .in(.'ii.j i.arn-'Ec r a linc

equj C icon f,- r Tii:.,[rE=t I AppF rnd i:: B) Further. .i, asunll- chat Lhe

hIrd sphere :ize cif e 3ch c ponent 1 gi'.. n b:., a icig:al e:. ir, io)r. o*

LiqEuar i n.: 3- I. An jn 3- i crac is. the reduced hard sphere sEie of

component 1 de-.rinds on thi ene-rg.' .: in: ter3c iEcn i T relati.' t*. the

clI:arccteristic energy cf the i -i inctra3ct ion ..nd il-co .In OMiei

.i..ag ra e relu.:- d,* nsit ,'' vhich is thc same t:.t .a11 components .

Thus We use tl-i rJducd ec.:-nd 'irial :;oef 1i.:ient correlation

Equ.-ri:c.rn I~3--) and t3-5) and the reduced 'ard sphere di3rriet[r

*Correlations Equiatcions i3-6) and 03- i) as









B = f (T/Tij) (3-15)
13


2r 3
2N 0
3 av i *
S f 2 (T/Tii PV ) (3-16)
ii


where fl represents Equations (3-4) and (3-5) and f2 represents

Equations (3-6) and (3-7). The mixing rules employed are:



1 *1/3 *1/3 3 *
V.. = (V +V.. ) ; V.. = V. (3-17)


1/2 *
T..j = (Ti. T. ) (l-K..) T. = Ti 3-i'


*
V = x.x. V. 3-19-
m J 13
ipj


.. = (a..ii + .4)/2 13--1



where K.. is a binary interaction parameter which must bE- .il,-.--d

from binary data.

This completes the description of the model for ti,: p-iir ,ir.lr

correlation function integrals. We can only make qualitati'. arumentii

for its validity. Quantitative justification (or rejection' re-:c *:'n

comparison with experimental data.

Incidentally, we had previously used a different 7..1J. I l.. r thl.

pair direct correlation function integrals which was ..i--.1 or, m.:,l..: ir

rh,.,r.'r ,- r-.:,._.jt. '- r,,.1 ne rI'rti- r... ,A l. *,- u. -_ given brier

--,'Cr if, i.: ,:. t i. ,n ApF..,Er i: A.










F.-er,- ces for Ch.pt r 3


:.E. Gubbins .and .. O. :'Co:nnell. .. .'.hem. Phv-. 6ji, $--9 (197;-<;.

S.W. Erel'.i and J.F. O',Conn:li, AlChE J., 18, 1239 119.2;.

S.L Brelvi and J.P. O'C'nn-ll, A :liE ..., 21, 157 (1975 .

J.A. et'Lrk r ani D. H nrderso n, J. Chem. Ph'js., -, 285r, .1907).

I .n. B ,rK r nd U. llHender :son, J. Ch-im. Phyc., L4 ;l- 196:).

J.D. Uee.- ,jd M D. Chandler. J Ch .. ht,., i-, 5-37 1971).

D.F. Pen..:;;.ski, H.,. '-n.-hol. J d '.C. lhuo, .bChE J.. 19
1.7 .197 .' .

1.F .C rnri -rn and I:.E. Starling. J. Chem. Fh:. 51, i, 5 11 ,9 i.

C.A. hjnr. ri, N.F. Cirnlhan. K.E. Starling and 1.E. Leland.
ji. I:t-i .. Phly 5-, 1523 1I ,711 .

U.S. S.ree[ r 3nd L.C.AM. FE l.'. '[ Ph.aic.,, -I. 5i '197? ).

J.tl.it. L-.'eil Ph,'sicq., "6, "1 I i'w1 ) .

1i.B. Stre tc Ph 'sica, ."_1, 51 :197-' .

W\. B. Scr.- t and L.A.K. ': Avel, J. Chl-,. Ph .s.. 5. 5, '4 5 .19l1.

U.B. Strr r.ct, L.S. Eagan nd L.A K. Sc, el.e I Chem. Thermr o.,
3, 0 _3 i1 ;3.1.

A. l -lh:-lI Hut. Jijl..-r Fid IL. Uj .ij:-r Ph.; i,i:d O. 11'9 9.

N.J. Tr ippeni.r i. i'js :n.n ir and C..J. W:.l'.-r Phi.'-ic.a, 32,
15,. k,19 ,).

i lic.h. 1 1. l'3 rni 3 a ar ud F. L.: 'wcr-, E Ph ,-i. *. 9 1' I .

C. isonopouIulos, AiCht J.. ".. C, 26.- IO7 ) .

V.E. n D iring and L.E. S, uShipe PIt,; s Pe'.., C. .6 I. i J31).

J.A. t.ar -r and u. Hendersr on. J. Chrt. [tI ., 7. 71. 19671 .

,.C. :l itld and E P.. Sn.ir ,:. . h .. A2, 861 19'71 .












CHAPTER 4
APPLICATION TO PURE FLUIDS--PARAMETER DETERMINATION


The model for the direct correlation function integrals can

be used as an equation of state for pure fluids. If the characteristic

parameters for a particular substance are known, then the change in

pressure resulting from an isothermal change in density from p to

p can be calculated according to

f
P
Pf pr = RT (l-C)dp '-
r
P


Also, if the change in pressure is given, p ca:jn t-e calculated

iteratively from Equation (4-1). We expect the use ot .:'ur icd.:l in

Equation (4-1) to be valid over a large range of conJiti:; r t:. r rinil

substances (e.g., nitrogen or methane) but to be limit. [;o ii dense

fluid region (p/pc > 1.5 except at reduced temper.ituit greater chin

2) for more complex substances which are nonrigid ,t;i n.a.ve tri:-o

orientation dependent forces. '2Clearly, Equation i--l carn tce uelj

in a least-squares scheme to correlate compression dara t., f.trin0
*
the two characteristic parameters, T and V But. ,,,r.= ip.,rtanti .

such a scheme provides a means of estimating the chara.cEcri-rl,:

parameters for use in predicting liquid mixture prcprLiEt .

Table 4-1 shows the results of fitting the *:.?iopre~l ion datl

of a wide range of substances; we have employed a ncnlen-.=r tiCLii.



























7 i i -i i- i . i
7 r- ..7 -i" ,- .


.." rI .* J 2 -, -I '- 3:
3- 7 r J .. .7 ,


r *- r -. a


.7 .r 0. Cr-


- 73 -, 2 Z 7 Cr 7 C '2 -. 4"C 33.-I
7 ~ -, -, IC. C. -7 C 7 4 2 -.7 7

I -.7 -. ~: :':. '- -' -' -. -, -, c 2


-.7 -I 3' ~ -3:- 3: C .-- -4 -. 2 '-- 4'; 3: '3

2,- -. 7 '2 -' a c.. .-I -r'. a3L _


a~~T

-'3 4-
C' -h C


3: 2 .- ri .. 2 .-, r 2 r- 1: 2 4- a. J~
4 4 -.4. 2 Cj 3 r. -' 2~ u-. C-c- 7


*73

.4,















~16





hi






"7 '.


I 2 a
4' C.


r. -. 2 .- 2 C -Z

-3 J 7 J 2 *" O- '- a **I r- 73 -1
C 3 27 i .r- C. *. 3 -4 -,
- r--, I 7 C' . r.j r 3


C




'2
-n
-'4 :


.4
-.1 '
44 "


1.' C
-1.'
C '3 '-.4
xl. 4 4 C
3- 14
44 44 .3
4.
4- '4. 7 c


-II .3!

(' Ii




C- C


*L


C J
,).,


,.1. -



C C.


* .7,,-. .3 C C C = 2 C 0-' -.

. .. 7 0- ,I C .I -
S ,' 0:- .'- .7 O J -. 2 r- 3 C -












c CN I "


N a m 4i co ao 0c 0 CO O 0 lun ul 40i n0


r-l NHH- N C(HN


U1
-4








0.


'z
0-


H N- 00 N C
40 N N -01

o o0 H


m n
r- -

I I
m n
oo ui
N N


cO\ -0 '0 -T 0-

0DI 0 N- NM


\o a0 0 Co N nCM
OM o0 N- oN m -1 -\D N DO \o
N C N O -A -4 -41 N- 0







r Ln O m o cn oo lH NO
N- 0 0o -H 0 OM N N OI D
vt '0 '0 L4 I0 N" m 4 mm


00 0 0 0 0
N N- 0 o H 4








0o oo 0o No 0 N
N- C0 N C H H -01
'^0 '0 N- 0 10 '0
U U \0~


'0 N

,--4







4 4

'0 -
8 t


0 0o0
01 00
0 00
0 u
*M *K
C
OH 4

4
0

o
u



r-1


01H
H
.0

H
H

U 0
u6
Bo

'0u






M

0i'















E
n!
z


COmm



m 01


0 0
n- nm m N 4 Cm
mp^ en 0 n n
N m -0 C) -c -t a
II I I I I uC0
0 mm D C 0 0 m 0

(n m cn c oo 0



0 0
00
aO

) M
X0
rm
i*

m 'O O NN 0 N O Nr
Io c r- ID o Dn

H NN N 0 0 H
CN







0 0
04 01

'0 '0 4^ 0' N-~ 41 '0 *

oT H N N H 0 CO 0C
r- O 4 m -.0 H N 0 0
r- u o 0 0 '0 n0


u u
ma


0 )




0 -
o H

0 W WH

U) 4 co. -H -mu
000 03 01
%I 1 1 1 I I I 0 _
1 1 1 1 I I I E *M X
O 2 f ri
.4-1 0v_.
0M I




I I I D I I I 0 0 -
I I 1 I I I 44 01
*- C -l II

0)



01 0 4 4 a 44
4 0 1 0 0 0 01

r0 0I c0 0 0 0 '-



.0 Hr 0 01 0 0 O C0 O 0 C


0 .0 0 4 4.0 044 0 .0 3 1
0-1 4 ,4 01 00 0 04 0.0
0H 01 H4 *H '00 .0 0 1 00 01


I 0 3 4 00 00 00 *4 H *
-0 44 4 m0 4 44 4 4 44 H


0Q 0w H
C 0 0
0 0) 0
4N N

0I 0. (01 0 -I 0
o H 0 'C 0 0 01
O *r-4 0 .0 01 0

0 T u.0u a
m < a 3 >









-ubr.iutince cLo find the characteLc icti pijara.iLet.ers which minimi-rzd Lihe

sum A.,f tcie Equ irt? of tLhe diff erence betA'e.ei the- calculated and

e..p rimrentajl presLure changes caused by isothermal densi., changes.

'he i.cerage error in moult cases is quite small and We 5eC that the

,model pr.:,.'ides a fjirlv ,good d,.--criptio.n Co liquid coimpiessi55 ib litieB.

The i.iracg errors for n-butine and n-pentane are quite htih, Ibu

Lhis is so minstl-, bec.jas5 tlie dact are close to the critical regicin.

Als.-, at high densiti'e the isothtrmal ::co.pressibiity prediit:td t-,

lthe mrdel is too high anrd therefore in the t[ictiLg we ha e ignored

dajt points where the reduced den-it' is tbrater than 3.,5.. H.we. r,

this normill:, Crerponds to preressures well bet-,.nd ti:-sc fir chemicial

processing. For e.aia.1le. at I = ,18 ,K and s. = 3.r.8 tih prt--3 Jurc of

n-jecane is 37=3 a1 m. let, the corr.-lac i n yields fairl. I :,od rnsult

e en in thi; i e.

As :We lha'e stated t.o..-. thie ni:.0 t imporcar.t purpose .- t itc ii,4

pure .componenri data i- .:,be iin aic-urate e stima5 e of the ch3a c-

Lcirimic parameters. But i.hile the method provides reliabic elstimates

of V T c.ji t-e .'ar td within a fairly li ice r3nge-. .T hii is .t. b-.

cxpc-ted since F.i e 'i and iO'Connel hs.i shown thac. toi a high degree

,,of *.ccur acy., liqJid compress ibi b1 t, data can be correlated b'.: a 5ize

parameeter onl.,.; Figure '-1 house a plot o.f the character sL iC

pjrairjaers of Lihe n-.al.j kneb.s .er uj carb...n number. Ihe dashed lines

are the' anal.,gous cure for critical conri tnts. ihc charalceri-.tiC

".'.lunies all fall --.n a an.o:thi curve: withI no no., cable fluctuation.

1htI are ,lightly, but s ignif ic antl', great r than Lhi 'em tIm t[ I J.I



























N I











*1

'1


'K


,fl
-5- e
H O
0
a)



-3m





'au
044

*tl 44a,
H H4-
44
00
00
0~o




H 0





u-O
4-3d)~

443-

0 L


3,-
mai~

44
H -










04-I
z Lc











Table- --2

S..::..rh. id L'aiuei oft Chiar .r : ri -tl Param.et.ers for
hr.he li.orrnl farar ff in


Fi .c Ed [ rram, t c= n,-:*Eh., Iaramr.Et r-
1T ,. 1 '. '

[1. th n,. -l 2 ., 9., I '2., i,9 8




Pr.-..n.- 37 1. i 1 32 1.1 1. 7

n--Eutane 0.2 256.9 -25 .' 25 2.

n-Pern ane 50 .. 321'..3 i-.t0 I .8

n-He:.:3ne 5:17. 372 52'7.8 372 0

n-Oc:tanc 58.. 5 -I.. 60,..'- ? .7

n- ,L nnne -. ?. 553. 6 0'j. .55-. _

r.- ,.-" n- c,7? .8 r.61 1 670. 61.*. 1

n-l.de:. ca3ne 73 . 7* '38. '

n-i-..
n-H pc[ -d,-_',:..:. e "?-5. 5 i05t,. . '.3. 1i57 .









critical volumes for the higher members of the homologous series.

The variation of the fitted characteristic temperature shows large

fluctuations (indicated by the points in Figure 4-1), so we have

drawn a rough average curve through the points. Again, as for the

size parameter, T is greater than the (estimated) T for the larger

n-alkanes. Table 4-2 shows the smoothed parameters for this series.

The predicted behavior of binary systems is generally not

sensitive to T and hence an accurate estimation of this parameter

is not crucial for this purpose. But for multicomponent systems

we also have to insure that Henry's constants of a solute in many

solvents are mutually consistent. In this case, if the values of
*
T are not appropriate, the values of the binary interaction para-

meters from Henry's constants will be different from those for the

correct deviation from Henry's Law. Thus, for multicomponert 1,-t.-i:c

a reasonably accurate estimate of T is necessary.

One way of obtaining consistent values of the parameters is

to adopt the method we have used for the n-alkanes. Another .lethod--

one which we have employed to a limited extent (see Chapter r '--i r.,

evaluate the parameters to insure consistency between pure c.-:.p.:nrn

and mixture data. In any event, if only pure component data ar- u-Sd,

they should cover a wide range of temperatures; pure componcnr Julc

over a single isotherm will not be sufficient to obtain the iuhra.:-

teristic temperature.
*
For the n-alkanes T and V are greater than or ecua.l c.:. EhV

corresponding critical constants. This appears to be a g-n=-ri.l cr-n










f lr all noap armolecules. F...rI LronA l .' polar molecules, 'V.' i.

significantly: sI mallcr than V, and I is, in general. ic.5, han I
C C
Tlihec are Iaowevy.r onl qualiiative crends and there do not appear

to be an, quantl tE3.ti.e relations fior our paramiretor ii tarmsr of tan.

other paran. cters. Because of the Ter.1it icy of thi result-l to the
*
.alue of V we are con.'in ed tlhia predaictior ot r. iil qui:n, ic., ill be

u.ele s and ..nl. compression data will Lt. adiequdae to obtain :appropoprte

:.ls.
'. 1 I ,i^F .


Re erer-ces for C(hapter -

1. I..L. ',ubbt n .nd J.P. u'Connel l, J Chenm. Phys., 61I, 3.-, 1 97.,).

S.. 4. Brel'.'l anid J t. u'Connell, AIChE J. I 1239 II 72'.'

3. W.B. Street, Phli- : 3 76. 59 119'- .:

4. U.W '. GtreecI anJ L.A.K. Sta.eley,, J Chem. N'nv ., 35, .2493 (1971).

. .b. Str.r LL. L.S. .'.- ran and 1.A.I .'-O.,elev J. Chem. lahermrru.
5, 633 (1973).

I:.. A. Micihel fHub. Wijker and H.i. Uiji-er, Fl-'.,ica, '*." .,,27 il .

7 N.J TraJ peniers, 1. wr .ernraar 3.J LUolkers, Pl- yica. 32.
1 503 j 9ti'i:, .

-. A. ltichela, i. Wa'.s rnaar and F. Louuerse, Physici :._; 99 1 195 1.

Q'. I-':chard i'.. Tiag1'e. Ph.D. Di ssertatio.n, C i lifornni Insticute of
1 b-in .-.lo YL: t. 1I'r. I .

lu. W.E. Deming and L .E. Shupe, Phi:.~., .e'.. '. '., 11932).

11. E. bender, Fifr.tl Snipoc.rsum on TI herophyri'.sic l Propcrties,
C.F. Bonilla, Ed.. ASMC. New a orn 1970, p. 227.

12. B./. Si,- a. rd L.IH. La.:e y, "Theri.,d:, namic 'Pro.pertlie of tih
Lighter Hy:dr,:carb.jon jnd liricro er," Amer. ri'cr. In t.,
New. tYor r 190.

13. D.E. S.tewart, B.H. Sage arnd U.tN. Lace!, i LC, ', 2529 (19'.y).









14. J.F. Connolly and G.A. Kandalic, J. Chem. Eng. Data, 7, 137
(1962).

15. Thomas Grindley, Ph.D. Dissertation, Stanford University (1971).

16. P.S. Snyder and J. Winnick, Fifth Symposium on Thermophysical
Properties, C.F. Bonilla, Ed., ASME, New York, 1970, p. 115.

17. P.W. Bridgman, Proc. Am. Acad. Arts & Sci., 69, 389 (1934).

18. C. Chen-Tung, R.A. Fine and F.J. Millers, J. Chem. Phys., 66,
2142 (1977).

19. F.G. Keyes, J. Amer. Chem. Soc., 53, 965 (1931).

20. A. Kumagai and T. Toriumi, J. Chem. Eng. Data, 16, 293 (1971).

21. R.E. Gibson and O.H. Loeffler, J. Amer. Chem. Soc., 61, 2515
(1939).

22. R.E. Gibson and O.H. Loeffler, J. Phys. Chem., 43, 207 (1939).

23. S.Y. Wu, Ph.D. Dissertation, California Institute of Technology
(1972).

24. R.C. Reid, J.M. Prausnitz and T.K. Sherwood, "The Properties of
Gases and Liquids," 3rd Ed. (McGraw-Hill Book Co., Ne6 York.
1977).

25. J.M. Prausnitz and P.L. Chueh, "Computer Calculations Lor
High-Pressure Vapor-Liquid Equilibria" (Prentice-Hall, in i.,
Englewood Cliffs, NJ, 1968).

26. J.P. O'Connell, University of Florida, Gainesville, Fi.riJ;,
personal communication.

27. W.B. Street and L.A.K. Staveley, Physica, 71, 51 (197-1.

28. J.E. Lund, California Institute of Technology, Pasadei,
California, personal communication.

29. R.E. Gibson and O.H. Loeffler, J. Amer. Chem. Soc., 63, ,',
(1941).














CHAPTER 3
THEPJ .Fli'jj.IIC PRi'PERTIE.> OF FblJI GAS.-;_LLVE IJ[
5'i'iTL15; 0i -RELAT lo


CGa.-sol.enrt binaries pr,'. ide a sLrilghLforward applica1i-*n u

hre mechod i e h e ev el'i .'1.o.pe1. b,, ja ,a5-s-~i ,rL s.,stem .e m-adn an.,

'ty temi hic ,h cntains a supercri lica3i :-'Tmpo en[ It(li-e I .i s >.,r 3..JSol e)

and a subcricicc l C*:ompoinenn (die SI'.,elinc. For i II.t- Lp ..of -,.Lster

che ~ ':>. cioneniencr unique I rerer'en2nce sc.t is cte purle. S curIcel

o01:.'enL at3 the Lemrperature of the s.,stem. Let subscript I refer c.,

the s,..luLt Indl *.ibscripr 2 t. tii. e solv.iint. rl,:I EqJua ion i'-2; Lcar be

'rit [ein is



Vr f (P. .: = n H Tr. l' l= l + in i + n 1 : ) I.3-1



I =lo.) = in f i r + I', +PT x




-here H is Henry,'s ccnsr'nr cf [he gds in -he pure, gatLur.ir.-J [.l'.'eit

and f 13 the cugaclt:,, :f the 'ol'vent at the reference 3Stlte. ,''ur

methi..J i:.In e i eJ t.' predJicc cie I.:ci'.'ic, coefficients so if thi refer-

ence fu_.icicLiS are .'iiiilablie .e can c3alcu13te the iquilid-phlsie ifugac-

;Lti-s. These can then be used toi et her :illh ,.ame '.',ir..r- ph sei equate ciro

v.f staJL Lt predict apor-i iquid eq, iirliLria.

In tlil3 hli.pter w e describe e 'our c,:rrelat i.n o" gas-s.'lvs nt

vapor-liquid equilibrium b, using our rmech.-J LiC cil-.:ula e iquid.J-









phase nonidealities. Before describing our work, we give a review

of other work on the same problem and also show some experimental

results to demonstrate that it is advantageous to adopt our approach.


5.1 Review of Existing Work

The first attempt to correlate the nonideality of gas-solvent

systems was that of Krichevskii.1 He reasoned that since in these

systems the solubility of the solute is normally small, the com-

position nonideality can be neglected and deviations of the solute

fugacity from Henry's law are entirely due to the pressure. Therefore,

we can write,2


P
L 1 -
an fl = n (1112x1) + -R dP (5-3

P 2s


Further, if we assume that the infinite dilution partial muir.r

volume of the solute is independent of pressure (a reasonable i.'uip-

tion if the temperature is not close to the critical tempteraure oc

the solvent), Equation (5-3) simplifies to



L V s
n f = n (H2) + j (P P2)



Equation (5-4) indicates that in (fL/xl) varies line-arly icli

pressure which is so for many gas-solvent binaries. Howe.er, EIquation

(5-4) also indicates that the slope of the linear variation ri e*ul
-00
to V /RT which is not generally observed. The partial moijr *olumoe

obtained from this slope is invariably different from the :.r.p~mental










parLtil nol ir .olu'i'E AL tLhe SOluLe. The appar.i nc .3ainoal, lihas been

e:-.plaine.d b: trienclicher and rausnitz, ard b. Gibb ii nd Van .e:s- "

wi.)ho snoil 'edj that the icompo i[ -. n nnonilea d iL nm t als 3 be aCcounLe.1

for. lliese authors emplo' the simple but cie n.J,niril L., ons i scent

expression. i.>:



4
RI Yn r A : 11 5-51




here A (the tWo-suftix liatgules e*qulaLi.n cn inLanLr I IJuperind ori

Ltemperatjure, but no on pressure i ,t Ci'.iipoSi Lion. follo. ".rl n

P-'rLun-ntz :"e use e the supersccript t indiate that is" nOL-

malii:ed in tie uns:mrietric conv'.nttini.

The reason 'wh,' is signiii-ant e'ven at small value; iO :.

is that its first derivacti.e at the re reniMe State is non:erC



O,
Si = -0i
i;1 *U 1

i
An e prrssibin for can re obtained frcirom that for I b:.

us il tlic- I.'bb t -Liuhem equalti.on




hi 4 ** i = -.*7 t -l.
S 1




lihe quantiLt nec: --arily I,. a ze'r., c:mposit:ion Jeri.'atci.c jt

the re erence sctae.



3 '. n'i ,
lim -- = u I5-a.
x-0 1









Therefore, it is reasonable to expect that an equation of the form

(5-1) would be valid for the solvent component. However, for the

solute, the composition nonideality must be accounted for. Using

Equation (5-5) we obtain the Krichevskii-Ilinskaya equation,



en f = n (H12x + ) (x2 1) + C (P P) (5-9)
12X n RT 2 RRT 2



At small values of x1 Equation (5-9) reduces to



-I 2A 2,
kn = in (Hx) + V RT (5-10)
812 1 2


where pl is the fugacity coefficient of the solute a: infinlce .Jilucion

in the saturated solvent. Therefore, even if the conrsiu i.:n non-

ideality is taken into account, in (f /x,) varies li~iir i.iclt

pressure, but the slope is different from that given Fq.,tic.n if.-..i

Orentlicher and Prausnitz3 have used Equation 1)-1I.'l LO cor-

relate the solubility of hydrogen in cryogenic solveicr. a iti;i

pressures. Their correlation of the quantities V1 arnd Lt.

essentially deduced from the experimental data. Ther.-.fr,. it 1-

specific to the solute hydrogen. Further, their appt'o .: is [.:.

correlate the two quantities separately. But, as their re.ulL: ah.-.,

the composition nonideality provides a significant cfncelt.ir..n ,.

the pressure nonideality. For example, for the system i,..dr gen- --rr..'n

monoxide at 88K:













= 'A5. c .g mole
tIl2.j ,


jnl Ior the s., tleem hydrogeo-prop.ne sr E tli.' :




V. = c : :c/. mole



--- = .:.'. cc,*'g moIl

l ,l I



IThe above reslltis were :'brained f rom, Table 2 ,r F. eferenc:e 3.

Tliirefore, 33 .e hav'. m ntil oned in Chapter 1, flom the engi-

neerirng point of cic' A i-. -kiS ..'art",eous. to b.,E single r'.dj lI

.I.ich ac.:collnts fr toth thefe correct on:'.

A further dis d'.v'3a e o f a repLeenlcitl'r.TIn o the form of

Lquatonl (15-7) is that at high cempar cr re (T.'T' 'he t : 0. ) the

anJptli'o [Chat the partial rmo.ar '.*olume i izndepeidenic of prce sure

is not *.al'ld. t-h under t i.,s ronIiion s, chl- '.v3ri tl n f the

partL31 mil3t ..'olume Wit-h preSi ute--jnll iLh Cti .,=-i c it Ion--must be

taker into, a.jccoui If such a de. cript on i=. thermod.Inami,311.

eon: il :en it mu-t b, C'orlai vCtil with tie dJesi cri.in ofl t trie -Ic..-

rpoiic ion nonid.clelitv. ,Burprisin6l ., uJr e.".periice is that the

total Idet. a;. on.f: rom,; idLalicC* tl -in relait '. 'iy snim ll ev'.een .en

a11 of thes "arilation occur.









5.2 Some Features of Gas-Solvent Nonideality--The Carbon monoxide-
Benzene System

The features of fugacity variation of gas-solvent systems we

have been trying to describe are elucidated by the experimental data

on the Carbon monoxide-Benzene system.6 Figure 5-1 shows the varia-

tion of in (f /x co) with pressure. The full line represents the

experimental quantity; the dashed line shows the prediction of the

fugacity if the pressure correction alone is assumed, and the

horizontal line gives the fugacity if Henry's law were obeyed. The

main feature to be observed is that the total nonideality correction

is very much less in magnitude than either the pressure correction or

the composition correction alone. The 5&3.15K isotherm is a good

illustration of the opposing effects of the two nonideality correc-

tions. At lower pressures, the Poynting correction is slightly

greater than the composition effect so the product is greater than

unity. As the pressure increases, the composition correction in-

creases relatively faster and the product becomes less than unity.

Yet, over the whole pressure range the total correction to Henry's

law remains quite small.

Figure 5-2 is the analogous plot for the solvent component,

benzene. In this case, the correction is simply due to pressure

since the composition does not change materially. Our theory

(Equation (5-8)) describes the effect very well. Recall that the

solvent saturated density is input data and our parameters ar.

obtained by fitting compressibility data. Whatever small char nes

there are in density due to pressure will be correlated very -Eil

















-33.1 '-.


0 L- -.- -- FI-











P P :ri nH n l
r- '














































F, L E,,' S- E '.p:r irri. "ir i [ ,-, n i of 1- ,-oni ,loo ,- .
a --b-o -- m, iHenr.'s .- i- n












.. C n n ll
iI/\























t.2'J -I







FC' j. 11-1. \.p.erurer 1 'lr iin rf 'oni.le1l1Ev A
\lulS Cnnl I










;uiuL1ilr'i i C'Tnncjl l, '1


















3.6






3.4



523.15K

3.2






3.0


2.2 -






2.0

433.15K



1.8
20 40 60

PRESSURE, atm.
Figure 5-2. Experimental variation of the rL:.ril.-l 't ,:.
benzene in ca l-rt:.r, i. :r: :.i --1.i-r, -.'l. -. .l.A- l r :i r
(Connolly6). Th, Pr. C rn. : : .rr C r:.i .:- : r
and activity _:**. .l 'ic. c :r Lri.; l i.i chr,
experimental i'' rr










and w.'. can alwa..s expect 3 ve'r, 3CcujrtSE c tiErmate or tie ol .Ient

activic. t cocf flcl,.it if the solute concelctratl onr is small.


5.3 Correlation .1 Lipqer mental Data

One va' to tet the ,valid ity ot our method i- to a lculdatee

the nonzidealitc' from exp.'rim ilenii t data and t nhe comnpjre it 'itth the

predictlor. ofi the miriecl. BEi thiIs could lead 0 a li .'aJlid Concluj ilorS

nicee it is difficulty to eglgC tlhe et fect o:f an error in the accil 'it',

coetfi ic ent on the predicteLd 'equilibricum. Beclides, the aiim o oujt

model is co obtain predict._ ii.. o phase equilibrium. Thus ie tcse[

ouir model b., uzing it as part of 3a s:lhe i, to pr.--Jic:t .'apcr-liquid

equjiltblium. W' Jo uo at the risk tLhart hen errors appear. ,ie clnnot

be sure 'where tc attribute tl r,.

Ie., use ot our method Li obtain liquid-phae ftugacitlie

iequijire iriLfO iTiatcun abou-i the r[eere. L _i tacre: the s cauration

denJsit' and pressure of the solv'.'nt and Henrt.," consEanCt i.r Lhe ga-

in the -olvent .

[lic dea-r cipion of the vapor ph3 .e muir t bLe Olij3necd itom a

.'ald equation of -tate. hIe modii.Jiied Pdlich-i..ong equation' provides

ain adequate de-criptiun of nonpoljr jnd sli thtl: Folar m-i:;cuCres. Lev'.eral

e-ccllent 2nd triala l coefficient ccr tel tlc ons''' le a3 ila ble. tr he

'.'i ial :-.paniiorn crijncated jat rhe secondd et'm is '.3lid on, n jr 3 1.r

der Cnitite A*n cqu tion called lie p r turbed--hard phere '.jjcuatio

has been proposed foi gas mi>.tCte cintainri ng polar componentc- ithi

equation appears to provide a reaso.inbl. *ac crate dJecr ipt Ion of gas

ri.tures containing co-mpunents Lih,-_ pfar.meter_ are. listed in the









original paper, but there is no way to obtain accurate estimates of

the parameters for new substances. We have used the most appropriate

of the above three vapor-phase equations depending on the binary

system involved.

The method we have used to correlate gas-solvent data is very

similar to the one proposed by Barker.11 We employ a simple empirical

form for Henry's constant:



in H12 = a0 + aT + a2T2 (5-11)



where a0,al,a2 are constants to be determined from P-T-X data. Values

of xa and yl were calculated from each P and T data point using the

various equations for vapor and liquid fugacities, then a nonlinear

fitting subroutine was used to find the value of the constants

a0 a2 which minimized the sum of squares between the calculated

and experimental liquid mole fractions, X1. When available, the

experimental vapor mole fractions were compared with the calculated

value. This provides a further consistency check on the data and

the correlations.
*
All the pure component parameters (T and V ) used ha = tenr

obtained from pure component compressibility data (see Tablc .-1

and 4-2) except for carbon monoxide where we assumed the par maccer' ire

equal to the corresponding critical constants. The binary irL.!rnLr'ri

parameters must be estimated from the binary data. Fortunately in

many cases the predicted activity coefficients are insensitl. o ici

value. For these systems we have set Kl2 = 0, thus actually: [prcdli,.ing
12









the li.iuid nun ideal :, from pure ccimponern data i. 1-.,-. In car -s

Wi-'er a nionczero .aluLe of K.1 i; required. we 13a.',e stiMijated it1

v. lu- bv firing [hi- dacj uWit ..'arioui- v,'alue of [le paramn-e er arnd

picking the on.e ie tl.u Ch.lit 'avc the beSt overall tit.

Table 5-i1 5-17 l.ow n conmpar i-orn of Ltih calculated and

experience l liquid and vafp- r C:i.mp..- itionr fr a wide varirLe '. of

a-i' -:flen hinary v. iteiri.4; ie als o h;I the prdteic d e act viy~.

cofifticientc[ Ro,, in tci E- tbl1.- ibi ich are d notedd by a cor-

retpond tc- data 3 foincz hwi-ii were not consid. rr d in t-1 fiicing: dw.

co 'r.iider Lhe;e pcin t ti be. beyond che range of r ppiicabilicy of h'e

t eor., 1.e., to close the critic l This i; .uail'., detected

by the experimental value of yl dJcreaing e thi inc:rea.-ing pr--i--ure,

Sgreacer ctan U 4 or T/TT : U. Ia.ble 5-19 iiis.arie= thle te=iilc

of the correlation.

The cieorv pro.'ide, ec' el e lenrt e action t of it e It1 tIld-pilh se

nonidealities orf ill the ;v:i en,; c-n-.idered ec,:epc for che carbon

m no.id.j'- -et- arcanol .term Figure 5-3 sho- E[he calculat icln oif Che

liquid mi ole fractcon- for thie systeme carbon irono ..iJ. -bcn/ene.- w .ee

th -u che re-zuilc ate '.'r-rv cln~ce to tcli- Ee xe.[e ria~i.nll .'1 i i. Fr:-.Tm

[able 5-1 we zee that the predicted -.'vpor composi tonsr are q'uice

cloi-e to the experr imental valueIs, but the m le fracictin of the ;ga3

i'" -ilititl:. high. iti3 i- a feature cou~inon to mCst y:,tems we ha",e

inrivec l'c-t d, b, t b ce c rjnnc.- account ftc.r cEle preciEce reason Why i Clii

is I.o. W'e tia'.'e used a n.on ero .'a Ie j of tie binary irnera c ion

p.aranriie r for chi- binary. It' 'sliE ic equal to C he value obtained


















































a1JI- c C a . = II c l

A AA & -'1. I1
* U L 'I I L .f- r.. II
S .. WIN.,5>


II I. I m .i Irr.
*II -.I I







j al:. l 3. 1

arlt.on mc.rr:. de I i -Fen-ere 12 1[2 = 110i

1 2




[. ,j r m :.,P ...c lc '' i V calc 'i 'I


433.2 0.9 .00309 .00 320 .2585 .25 1 1.0) 1.01
12.6 .00615 .00637 7 4037 0oA9 1.00 1.02
16.1 .00993 .0 I C26 .5202 .52.-6 1.01 1 .'3
21 .C015 3 .016 3J .6186 .6242 1 .01 .05
37.b .03228. .013313 .75o2 7r2 1 .0 2 10
71 .1 .0o714 r.797 .8377 441 1 .04 .22

443.2 12.1 .00J 02 '0* 13 .2585 .Eco2 1.00 1.01
15.7 .00796 .008 1 .0"J3 .4308 1.00 1 .02
20.3 01301 ..1334 .C202 .5239 1.01 1 .4
(.0 .0 2042 .02088 .61 o .6245 1.01 I .Co
48.7 .n04 04 .'0446 .7562 .7.i-01 1.03 1.1 3
103.2 .1 0340 .10220 .8377 .84 72 1.07 1 .34

453.2 12.3 .00253 .00 5'15 .1 175 .1499 1 .00 1.01
14.' .00518 .00529 .2535 .262 1.00 1.01
19.1 .01 032 )53 ..037 .1-068 .01 1 .03
25.0 .01 70C 1 732 .5202 .52 9 1 .01 1.05
33.f .026l8 .02730 .(6186 .,'2" 1.03' .07
63.5 .0 99 .69 1 11 .7 62 .7615 1 .03 .18

463.2 14.7 .00324 .20331 .1475 14 9 1.00 1.01
I 7.f. .0C635 .00t.79 .25o5 .2627 1.00 1.02
23.2 .0 33j .0 136,3 .4 037 12 2 1.01 1.03
30..b .0.2226 .072261 .52) 2 .52,3 1 .0 1 .06
41 .3 .03 .63 )3'01 .'180 .o2-' 2 1 .10
P5.8 .08837 ..03732 .7562 .764 1 .05 I .2o

73.2 1 7.5 .00418 .0?42.6 .1 .75 .15' 1 .00 1 1
21.1 .00 8 360 .00 77 .258C1 2638 1 .00 1 C2
28.1 .01741 1 7 7 .4037 1 13 1.01 1 .04
37.3 02900 .-29?37 .5202 .5251 .. I .07
52.1 .04792 .O4C l .t1'86 .62. 1 .02 1.13

483.2 2C. .00537 0549 .1 4 5 .1-.13 1 .0 1 .0
25.3 .3 01102 ..1 1 .25A5 636 1.00 1.03
33.9 .02270 .C230C .4037 .41 I 1.01 .05
45. .3P3 .0339 883 .5202 2 .5,4 1.01 I .09
b6 .5 .O. 58H . 593 .6186 ..'-,9 1.02 1. 7

493.2 2 .5 .00298 .00304 .06 la) .0722 1.00 1.0 1
24.3 .00693 -0707 .1.75 .1510 1.00 1.02
2 *. T *1440 .0)14 6 .2585 .263'7 1 .00 1 .03









Table 5-1 (Continued)
T,K P,atm xl,exp x ,calc y ,exp y ,calc yl Y2

40.9 .02993 .03040 .4037 .4117 1.01 1.07
56.9 .05239 .05283 .5202 .5272 1.01 1.13
87.3 .09591 .09514 .6186 .6302 1.02 1.24

503.2 25.0 .00381 .00388 .0698 .0711 1.00 1.01
28.5 .00900 .00916 .1475 .1504 1.00 1.02
35.2 .01890 .01924 .2585 .2638 1.00 1.04
49.6 .04014 .04071 .4037 .4110 1.01 1.09
72.3 .07434 .07471 .5202 .5275 1.01 1.17

513.2 28.9 .00501 .00507 .0698 .0710 1.00 1.01
33.3 .01185 .01203 .1475 .1498 1.00 1.02
41.8 .02524 .02562 .2585 .2630 1.00 1.05
61.0 .05592 .05651 .4037 .4097 1 .00 1.12
98.1 .11897 .11889 .5202 .5330 0.98 1.28

523.2 33.5 .00672 .00673 .0698 .0708 1.00 1.01
38.9 .01594 .01601 .1475 .1483 1.00 1.03
50.0 .03487 .03510 .2585 .2596 0.99 1.07
78.1 .08504 .08529 .4037 .4079 0.97 1.18

533.2 38.6 .00913 .00886 .0980 .0683 1.00 1.02
45.8 .02252 .02207 .1475 .1459 0.99 1.04
61.0 .05145 .05083 .2585 .2536 0.97 1.10


RMS deviation in xl = 0.0004










fIbl. 5-2

Ca r .:n monox. iJe I i-n-0liC L i i 1,2'1, ., = ). -'1

T,' ,_ t _. ,_ x1 ,c.l: ':' ,e .p 1, a '1 '2


463.2 6. .0049 006 7051 .2655 1.00 1.01
9.3 .00~83 .r0O 90 .-05 .4130 1.01 1.72
10.s .'01 3Jt .r1364 .5092 .5161 1.01 1.03
1 .4 .02 7 .2 7 02280 .6277 .6361 1.C2 1.06
23.3 .042iS .24259 .7.~9) .7577 1.03 1.11

473.2 9.1 .00635 .063t .2750 .28.05 1.00 1. I
1D.3 . 1O O0 .11161 ..35J .4131 1.01 1.03
12 .? .017n 7 .01788 .5092 .5167 1.01 1.0,
13.1 .03011 .,:30 5 27 7 .0369 1.02 1.07
29.7 .05707 .05678 .7.-99 .7582 1 4 I.1

4 s3.2 '. 0823 .C0327 .275C .28, 3 1 .01 1.02
12.6 .C 1509 .C 1515 .-G59 .4130 1.0 1 .03
16.0 .0?336 .0'23-4 .5092 .5170 1.01 1.C5
22. 03C76 .03981 .?277 .6378 1.03 1.00
33.6 .C07o06 .07768 .74199 .7620 1C.5 1.20

493.2 9.7 .OC500 .0050' .1 5I 15.5 1 .00 1.01
.I 3 .01073 .?10o0 .275: .2804 1.01 I .
15.4 .01976 .019 q_ 05? 1 3 1 .0 1 1.0"
19.7 .0307j .03094 .5092 .5187 1.02 1.C7
28.5 .03297 .05314 .b 77 .6100 1.03 12
51.1 .1 0 12 .10385C .74 o 7666 .06 .29

503.2 10.2 .r00274 .30276 .0732 .072 1 .00 1.01
11.6 .0C65'. .Cr:o6o .1 52' .1556 .'j0 1.01
14 .5 .01405 .,01 16 .2750 .2 -9 5 1.01 1. 3
Id. .025'98 .:2 22 .4059i .41 i 3 1.01 1 .06
26.4 .04080 .04 11 .502 .5 .'1 1.09
36.1 .07187 .*7223 .0277 .6411 .0 1 1 7

513.2 12.1 .0036) .00. 362 .0702 .)719 ..0 1 .01
1 .00862 .00663 .1528 .1551 1.00 1.02
17.3 .01 60 .C 1873 .275) .279 1 .01 1.04
23.0 .03473 500 . 405 I l 1.0 2 1.07
30.3 .05518 .0'560 .5092 .5198 1.02 1.12

533.2 16.7 .00651 .00637 .070.. .06q7 I O 1.r,
'19.5 .01578 .0 1554 .1528 .15 1 1.00 1.03
25.3 .03453 .0342. .2750 .274A 1.01 1.0
35.5 .06717 .06729 .405 ) .4077 1.01 1. 13


FI1 de;.i ion in .s = i .I=,'.2










Table 5-3

Hydrogen (1)-Benzene (2), K12 = 0.0

T,K P,atm x ,exp x1,calc y ,exp y ,calc yI y2


433.2 20.7 .00890 .00925 .6147 .6186 1.02 1.04
32.0 .01615 .01673 .7376 .7418 1.03 1.08
51.4 .02846 .02931 .8285 .8295 1.06 1.15
89.0 .05145 .05242 .8897 .8904 1.10 1.28
113.0 .06601 .06644 .9081 .9085 1.14 1.38

443.2 18.8 .00715 .00739 .4921 .4990 1.01 1.03
25.5 .01182 .01219 .6147 .6189 1.02 1.05
39.8 .02149 .02207 .7376 .7415 1.04 1.10
64.9 .03822 .03893 .8285 .8292 1.07 1.19
115.6 .07053 .07071 .8897 .8904 1.14 1.38
150.2 .09159 .09087 .9081 .9089 1.19 1.53

453.2 22.8 .00938 .00967 .4921 .4982 1.02 1.04
31.4 .01562 .01605 .6147 .6191 1r.3 1.07
49.3 .02853 .02913 .7376 .7413 1.0r. .13
82.1 .05139 .05187 .8285 .8285 1. 1 i .24
151.7 .09734 .09600 .8897 .8909 1.2C J.'3

463.2 22.2 .00801 .00826 .3902 .3946 1. 01 1.03
27.7 .01233 .01269 .4921 .4980 1.02 .0O
38.4 .02061 .02113 .6147 .6189 1.04 1.C .
61.1 .03787 .03850 .7376 .7410 1. '7 1.1I,
104.2 .06937 .06948 .8285 .8287 1.13 1.32

473.2 33.5 .01621 .01672 .4921 .4982 1.0 1.r,
46.8 .02718 .02787 .6147 .6187 1.0" 1.10
75.8 .05042 .05112 .7376 .7408 1.:- 1.21
134.0 .09478 .09401 .8285 .8299 1.17 1.43

483.2 32.0 .01387 .01439 .3902 .3956 1..2 1.05
40.4 .02136 .C2208 .4921 .4979 1. 3 I.06
57.2 .03605 .03701 .6147 .6188 1.0.6 1.13
94.5 .06764 .06836 .7376 .7408 1.11 1.27
175.7 .13175 .12894 .8285 .8326 1.2- 1r.,

493.2 38.2 .01825 .01902 .3902 .3947 1.C3 1.C2
48.7 .02824 .02928 .4921 .4970 1.0- .0C
70.0 .04811 .04937 .6147 .6177 1.r7 1.17
119.2 .09212 .09243 .7376 .7405 1.'15 1.30

503.2 45.6 .02425 .02535 .3902 .3934 1.03 1.08
58.9 .03779 .03924 .4921 .4974 1.05 1.12






T -.3 rb c
"it.-l. .-3 "..*ricirnued i


m6.4 .06510
15, 3 .' 1 2 63

513.2 36.7 .01253
54.4 .03282
71.7 .05147
108.1 .09030

523.2 35.1 .00741
-3.2 .01725
85.5 .063c39
140.1 .13163

533.2 41.) .01041
il .6 .06602
113.2 .10802


.06668 .61.7
.1 27(.6 .7376

.01323 .2027
.03431 .3902
.05333 .4 21
.09165 .61"7

.33779 .00 3
.)193C .2027
.07038 .-n21
.13071' .6147

.01078 .0 114
.06661 .3902
. 1 064 2 .4 21


.6169 1 .09 .22
.7 38 1 .20 1 .50

.2 n' 1 .02 1.04
.3 3j0 I .Oa 1 .10
.4Q63 1.07 1.16
.6176 1.12 1.30

.1003 1.01 1.02
.2C37 1.02 1.05
.48.76 I.OR 1.20
. 2 7 1 .1 7 1. 2

.OoI0 6 1.01 1.03
.3"30 1.07 1.17
.4'042t 1 13 1 .30


k lt .J i j'.L i.. n 1i .I O.jOl


:* I -1 LC 'I eL'P ,IC-c I *










Table 5-4

Hydrogen (1)-n-Octane (2), K 2 = 0.0

T,K P,atm xl,exp x1,calc yl,exp y ,calc y y2


463.2 9.9 .00869 .00913 .4954 .5015 1.01 1.03
16.2 .01850 .01937 .6713 .6807 1.03 1.07
27.2 .03554 .03697 .7959 .8026 1.05 1.13
43.1 .05933 .06116 .8629 .8706 1.09 1.24
62.5 .08741 .08916 .9004 .9074 1.13 1.37

473.2 12.3 .01152 .01204 .4954 .5019 1.02 1.04
20.1 .02454 .02554 .9713 .6813 1.03 1.08
34.4 .04755 .04905 .7959 .8045 1.07 1.17
55.2 .07952 .08103 .8629 .8718 1.12 1.31
82.0 .11837 .11882 .9004 .9100 1.19 1.51

483.2 15.1 .01525 .01595 .4954 .5027 1.02 1.05
25.1 .03263 .03389 .6713 .6829 1.05 1.11
43.6 .06361 .06527 .7959 .8069 1.09 1.23
71.4 .10762 .10851 .8629 .8758 1.16 1.42
108.6 .16150 .15938 .9004 .9140 1.26 1.73

493.2 10.6 .00538 .00568 .2107 .2129 1.01 1.02
12.6 .00914 .00963 .3117 .3148 1.01 1.03
18.5 .02024 .02125 .4954 .5037 1.03 1.06
31.0 .04317 .04487 .6713 .6832 1.06 1.14
55.0 .08493 .08679 .7959 .8094 1.12 1.30
92.9 .14584 .14537 .8629 .8804 1.23 1.59
148.5 .22479 .21691 .9004 .9209 1.38 2.11

503.2 12.8 .00716 .00762 .2107 .2132 1.01 1.02
15.3 .01228 .01303 .3117 .3171 1.02 1.C3
22.6 .02693 .02842 .4954 .5039 1.04 1.C5
39.0 .05848 .06090 .6713 .6876 1.08 1.1I
70.1 .11466 .11662 .7959 .8130 1.17 1..0
123.7 .20080 .19702 .8629 .8885 1.32 1.e6

513.2 14.1 .00701 .00755 .1635 .1654 1.01 I.0C
15.2 .00956 .01027 .2107 .2127 1.01 1.01
18.4 .01643 .01761 .3117 .3168 1.02 1.C4
27.7 .03617 .03843 .4954 .5059 1.05 1.10
48.3 .07814 .08153 .6713 .6889 1.11 1.24
91.2 .15784 .15923 .7959 .8231 1.23 1.5t

523.2 18.2 .01304 .01415 .2107 .2127 1.02 1.03
22.2 .02233 .02413 .3117 .3172 1.03 1. -;
34.1 .04926 .05262 .4954 .5064 1.06 1.13




71.



TaLle S5-4 ,Cion i nued.1

.aI 'p '. C, '' ,E l' co 'I c


01.7 10862 1315 .6713 .60i9 1 .14 1 .32
123.9 .22573 .22385 .7959 .8413 1.34 1.85

533.2 17. 9 821 .00897 .1080 .1078 1.01 1 .n0
1 .9 .01323 .014.4 .1635 .1620 1.02 1.03
21.8 .01815 .01'77 .2107 .211 7 1 .02 .04
26.9 .03112 .33372 .3117 .3152 1.0. .08.
42.5 .0692) .073r3 .4954 .5115 1.08 1.18
30. .15525 .16005 .6713 .7169 1.20 1.46

543.2 21 .2 .01193 .01298 .1080 .1054 1 .0 1.,3
26.2 .02610 .02833 .2107 .2082 1.03 I.t,
33.0 .04454 048 4 .3117 .3137 I 0 1 .10
54.3 .10204 .19748 .4954 .5259 1 .II 1.26
115.8 .25001 .240 41 .o713 .7658 1 .32 1 .6

F'l' de i.iitilor in x = (O.U iiL2
.1











Table 5-5

Hydrogen (1)-n-Hexane (2), K12 = 0.0

T,K P,atm xlexp xl,calc yl,exp yl,calc y1


277.6 34.1 .02800 .02980 .9960
68.1 .05400 .05779 .9980
136.2 .09900 .10914 .9980
204.2 .14000 .15504 .9980
272.3 .17900 .19642 .9980
408.4 .25600 .26805 .9980
544.6 .33400 .32821 .9980
680.7 .42200 .37976 .9980

310.9 68.1 .05900 .06068 .9920
136.2 .10800 .11411 .9950
204.2 .15300 .16124 .9950
272.3 .19600 .20314 .9950
408.4 .27900 .27417 .9950
544.6 .36600 .33216 .9950
680.7 .46100 .38083 .9950

344.3 34.1 .03400 .03371 .9610
68.1 .06400 .06591 .9780
136.2 .11900 .12377 .9860
204.2 .16900 .17406 .9880
272.3 .21600 .21814 .9880
408.4 .30900 .29138 .9880
544.6 .40500 .34976 .9880

377.6 34.1 .03600 .03681 .9040
68.1 .07000 .07353 .9460
136.2 .13200 .13861 .9660
204.2 .18700 .19419 .9710
272.3 .24100 .24208 .9740
408.4 .34500 .31987 .9740
544.6 .45000 .38037 .9740

410.9 34.1 .03800 .03981 .7970
68.1 .07800 .08353 .8860
136.2 .14900 .15975 .9270
204.2 .21300 .22347 .9390
272.3 .27400 .27727 .9440
408.4 .39300 .36228 .9460
544.6 .51100 .42653 .9440

444.3 34.1 .03700 .04097 .6060
68.1 .08600 .09578 .7720


.9973 1.05 1.21
.9983 1.10 1.47
.9988 1.21 2.15
.9990 1.33 3.13
.9990 1.45 4.56
.9990 1.73 9.57
.9991 2.04 19.95
.9991 2.40 41.25 *

.9932 1.11 1.42
.9956 1.22 2.03
.9963 1.34 2.87
.9967 1.47 4.05
.9970 1.76 8.01
.9972 2.08 15.66
.9974 2.45 30.42

.9628 1.05 1.18
.9791 1.11 1.39
.9871 1.23 1.94
.9897 1.36 2.69
.9910 1.49 3.72
.9923 1.79 7.03
.9931 2.13 13.13 *

.9052 1.05 1.16
.9476 1.12 1 .37
.9688 1.24 1.88
.9759 1.38 2.57
.9795 1.53 3.50
.9833 1.85 r $.78
.9855 2.21 11.50

.7913 1.05 1.14
.8849 1.12 1.34
.9334 1.26 1. E3
.9496 1.41 2..'
.9580 1.57 3.34
.9675 1.92 5. i5
.9725 2.31 10.~7

.5938 1.05 1.1
.7714 1.12 1.31








itlie 5-5 Conrin i-ued)


1,1 ,au ri :'.i, e:

13b.2 .17200
201 ..? .'4900
272.3 .32200
4R08.4 .4610)

477.6 34.0 .03200
b8.1 .10300
136.2 .22400
204.2 .34100
273.5 ..< 900
335.6 .55500


:-1 il ;1 ,exr

. 18970 .8510
.26653 .8730
.3J004 .8820
."2713 .8840

.03530 .3100
.10769 .5680
.22835 .7000
.32335 .7360
.3 79 7 i
.6157c- .7470


.~l12,
"1 1' '1 '2

.86sR 1 .27 1.79
.9032 .43 2.a3
.9221 1 .61 3.26
.9.36 2.00 5.76

.2967 1 .0 1.08
.56t 5 1-12 1.2e.
.7'95 1 .2 9 1 .79
..9231 1.4 7 2.47
. 611 1.69 3.37
.9'015 1.41 o.21


f'S deJ.'LtiLOn. in t.f = 0.0099

IiraGtes d3a poir.e ignored in tltring iad cial.ulajcon
ot aj.,erace error.










Table 5-6

Hydrogen (1)-Water (2), K12 = 0.0


T,K P,atm x ,exp x1,calc yl,exp


273.2 25.0 .00040
50.0 .00090
100.0 .00170
200.0 .00330
400.0 .00640
600.0 .00930
800.0 .01190
1000.0 .01430

293.2 25.0 .00040
50.0 .00070
100.0 .00140
200.0 .00280
400.0 .00540
600.0 .00790
800.0 .01020
1000.0 .01240

313.2 25.0 .00030
50.0 .00070
100.0 .00130
200.0 .00260
400.0 .00500
600.0 .00730
800.0 .00950
1000.0 .01160

333.2 25.0 .00030
50.0 .0C070
100.0 .00130
200.0 .00250
400.0 .00490
600.0 .00720
800.0 .00940
1000.0 .01140

353.2 25.0 .00030
50.0 .00070
100.0 .00130
200.0 .00260
400.0 .00510
600.0 .00740
800.0 .00970


.00045 .0
.00090 .0
.00177 .0
.00342 .0
.00646 .0
.00920 .0
.01171 .0
.01401 .0

.00039 .0
.00076 .0
.00151 .0
.00294 .0
.00560 .0
.00806 .0
.01036 .0
.01250 .0

.00035 .0
.00069 .0
.00136 .0
.00267 .0
.00512 .0
.30740 .0
.00954 .0
.01157 .0

.00033 .0
.00066 .0
.00131 .0
.00257 .0
.00495 .0
.00717 .0
.00928 .0
.01128 .0

.00033 .0
.00067 .0
.00133 .0
.00262 .0
.00507 .0
.00737 .0
.00955 .0


Ylcalc Y1 Y2


.9997 1.02 1.02
.9999 1.05 1.04
.9999 1.10 1.08
.9999 1.22 1.17
.9999 1.47 1.38
.9999 1.78 1.61
.9999 2.14 1.88
.9999 2.57 2.20

.9990 1.02 1.02
.9995 1.05 1.04
.9997 1.09 1.08
.9998 1.19 1.16
.9999 1.42 1.35
.9999 1.69 1.56
.9999 2.00 1.80
.9999 2.36 2.09

.9970 1.02 1.02
.9984 1.04 1.04
.9992 1.09 1.07
.9995 1.18 1.15
.9997 1.39 1.32
.9998 1.63 1.52
.9998 1.91 1.74
.9999 2.23 2.00

.9919 1.02 1.02
.9958 1.04 1.03
.9978 1.08 1.07
.9988 1.17 1.14
.9993 1.37 1.30
.9995 1.59 1.49
.9996 1.84 1.69
.9996 2.13 1.93

.9808 1.02 1.02
.9901 1.04 .03
.9948 1.08 1.07
.9972 1.16 1.14
.9984 1.34 1.29
.9988 1.55 1.46
.9990 1.79 1.65








Table 3- ,.:nLi nuC d .

T, P,., 1 .e:'-p .0 1 :.-. '90 ,-- p ,5 "i

1000.'0 .011[0 .01163 .0 .,r902 2.05 1 .* 7


373.2 25.0 .00040 .00035 .0
50.0 .00070 .C'0072 .0
100.0 .00140 .00144 .0
200.0 .0028) .00284 .0
-400.C .00550 .00552 .0
b00.0 .00800 .00804 .0
600.0 .01030 .01044 .0
100O.0 .01250 .. 127, .0


.95'0r, 1.C 2 1.0 1
.Q7Q 1 .04 1.03
.9891 1.07 1 .0
.g' I 1.15 1.13
.9q 7 1.32 1.27
.o ,76 .52 1.44
.993 1 1.7 1 .t2
. Qg 1.9q 1.82


i\l', *. ,- iati o irn :i = I. il1
1











Table 5-7

Hydrogen (1)-Ammonia (2), K12 = 0.0


T,K P,atm xl,exp xl,calc yl,exp


273.2 50.0 .00250 .00243 .0
100.0 .00510 .00495 .0
200.0 .00990 .00967 .0
400.0 .01820 .01800 .0
600.0 .02520 .02523 .0
800.0 .03120 .03162 .0
1000.0 .03640 .03734 .0

298.2 50.0 .00340 .00328 .0
100.0 .00740 .00719 .0
200.0 .01500 .01448 .0
400.0 .02820 .02740 .0
600.0 .03920 .03859 .0
800.0 .04890 .04849 .0
1OCO.0 .05680 .05736 .0

323.2 50.0 .00390 .00376 .0
100.0 .01010 .00982 .0
200.0 .02180 .02113 .0
400.0 .04240 .04110 .0
600.0 .05970 .05837 .0
800.0 .07410 .07364 .0
1000.0 .08670 .08736 .0

348.2 100.0 .01230 .01208 .0
200.0 .03050 .02981 .0
400.0 .06290 .06136 .0
600.0 .09060 .08893 .0
800.0 .11390 .11365 .0
1000.0 .13380 .13633 .0

373.2 100.0 .01180 .01120 .0
200.0 .04160 .03944 .0
400.0 .09650 .09107 .0
600.0 .14550 .13848 .0
800.0 .18830 .18448 .0
1000.0 .22780 .23291 .0


Yl'calc y1 Y2


.9073 1.05 1.06
.9482 1.11 1.12
.9686 1.24 1.26
.9785 1.54 1.59
.9818 1.90 2.CO
.9834 2.33 2.50
.9844 2.83 3.13

.7783 1.04 1.05
.8773 1.10 1.11
.9270 1.23 1.24
.9516 1.51 1.55
.9597 1.84 1.94
.9639 2.23 2.40
.9664 2.69 2.97

.5513 1.03 1.03
.7504 1.09 1.C9
.8525 1.21 1.22
.9039 1.47 1.53
.9211 1.7- 1.89
.9299 2.15 2.33
.9352 2.57 2.87

.5538 1.07 1 .
.7336 1.1 I .:0
.8271 1.42 1 .50
.8595 1 .70 1 .
.8756 2.02 2.30
.8855 2.39 2. 3

.2830 1.03 I.
.5545 1.13 1I.I
.7080 1.33 1 .-9
.7602 1.55 .97
.7889 1.78 2.34
.8062 2.00 2.-5


IMS deviation in xl = 0.0021

exp
Y1 = 0 indicates no vapor phase composition availio1..










Tat-.l 5-3

H',idJro an Ill- [tarrnnoIl 12 K, 2 = O.i

T, P. t i 'p Xl.cal / :.p .calc '1 '2


294.2 76.0 .01150 .)1153 .0 ." 3 1 .06 1.13
178.0 .02710 .2;2681 .0 .,9-v3 1.14 1 .23
170.0 .02740 .02696 .0 .99q 3 1.14 1.35
272.0 .04070 .3'4076 .0 b96 1 .22 1.57
284 .0 .04 40 42545'. .0 9 9 1 .22 1 .

362.2 71.0 .01500 .01 460 .0 .9'033 1.05 1.11
i.. .0 .01650 .11532 *. i Es. 1 .05 1 11
134 .0 .02860 .n2302 .0 .r811 1.10 1.22
18 6 .C4 10 .03-39:4 .0 .0 6 1.14 1 .31
284.0 .05920 .054 7 .0 91 1 .21 1 .52
300.0 .00200 .06284 .0 .9923 1.22 1.55

413.2 75.0 .02210 01 03 .B900 .' 446 1.04 1.10
1 6.0 .05400 .05227 .9530 .i4 12 1.12 1.,0
281.0 7610 .0765 9 .9650' 10 1.17 1 .46
300.0 08094 .06:213 .9700 .9638 1.16 1 .50

rFJV duL'. Ati: n in 'x = 0.l 012

y = 0 injicate no *.3pOr phise c.omiir r available










Table 5-9

Carbon monoxide (1)-Methanol (2), K2 = -0.10
12


T,K P,atm xl,exp x1,calc yl,exp


298.2 60.0 .01910 .02098 .0
67.0 .02170 .02338 .0
110.0 .03580 .03771 .0
180.0 .05560 .05882 .0
186.0 .05590 .06044 .0
241.0 .07570 .07313 .0
243.0 .07710 .07350 .0

363.2 50.0 .02220 .01740 .0
100.0 .03910 .03471 .0
150.0 .05270 .05061 .0
200.0 .06450 .06471 .0
250.0 .07390 .07662 .0
300.0 .08150 .08609 .0

413.2 86.0 .03820 .03623 .8870
90.0 .04030 .03814 .8930
145.0 .06150 .06408 .9250
291.0 .09100 .12482 .9630


Ylcalc Y1 Y2


.9970 0.98 1.11
.9972 0.98 1.12
.9982 0.98 1.20
.9987 0.99 1.36
.9988 0.99 1.37
.9989 1.03 1.51
.9989 1.03 1.52

.9438 1.02 1.07
.9703 1.04 1.16
.9790 1.08 1.25
.9832 1.13 1.35
.9857 1.19 1.45
.9873 1.27 1.57

.8480 1.01 1.11
.8542 1.01 1 1.L2
.9039 1.03 1.22
.9453 1.12 1. 52


RMS deviation in xl = 0.0029

exp
yl = 0 indicates no vapor phase composition avaliit.ie

Indicates data point ignored in fitting and calcul[irn
of average error.










T.blc 3-11.1

liL .og. ii Ill-A, monI I' .I ,
12

1 ,. I ,uL[T :<1.. 1' : llc '. ,c:-.:p


273.2 50.0 C.00 31 .3028?7 .0
100.0 .00600 .005J9 .0
200.0 .01030 .0001 .0
3CO .0 .01330 .01 1 .0
"00.0 .01550 .01422 .0
600. 0 01 960 17 91 .
6C0 .0 .02',r'') .C2090 .0
1 000.0 .02210 .)2347 )

293.2 50 .0 .00410 .00392 .0
10').0 .00 840 078' .0
200.0 .01530 .01407 .0
300.0 .02050 .01 79 .0
400.0 .0245) .02265 .0
600 .0 .02980 .02?88 .0
'00). .03320 .03393 .0
1000.0 .03560 .013825 .0

313.2 50.0 .00480 0.0 90 ..
100.0 .01133 .01112 .0
00C .0 .02'240 .021 b .0
300.0 .03130 .02'01 .0
00.0 .0,3810 .0359 .0
600.0 .0l 730 '.5'93 .0
800.) .05350 .05436 .'j
1000.0 .05720 .Ofj61i' .0

333.2 50.0 .0)483 .05 11 .
100.0 .01430 .01481 .0
20 .0 .03170 .03115 .0
300.0 .04630 .04.36 .0
403.0 .05880 .05540 .0
00.0 .07770 .0732e .0
800.0 .08850 .09772 .0
100 .0 .9O540O .1001 .0

353.2 50.0 .00200 .00274 .
100 .0 .01580 .01777 .0
200.0 .4 00 5*.. .n
300.0 .07050 .067 0 .0
400.0 .09460 .08648 .0

373.2 ICO.0 .01530 .i 1 31 .0
FJ1 *I l:i evi--i.n in :< = x ..00-3


' 1 1 r1 '

.c1 15 1.07 1.06
.9533 1.1 7 1.12
.974G 1.38 1.26
.'i l 1 .6" 1 ...4 2
.0853 1.95 1.5r
.9198 .74 2.00
.90q2 3.A2 2.50
.9933 5.27 3.12

.8152 1 .06 1.05
.9025 14 1. 11
.:4 73 1.3" 1.25
.9628 1.C8 1.40
.q626 1 . 1 1 .,.0
.070 1 .87 1 .'r
.' 7,i 4 2.568 I.
.9238 3.5. 2.12
.9866 4.81 2. 9

.6 6 b 1 .05 1.0.
. 203 1.12 1.10
.9C32 1.30 1.23
.'322 1 .: 1 .38
.9 2 1.77 1.5-
. -9 2. 2 1.91
.4711 3.27 2.-6
.9761 I .3r3 2. 0

.4373 1.03 1.03
. 6,r 57 1 1 0
.8349 1.24 1.22
..P851 1 .42 1.36
.'1112 1.65 1.52
.9353 2.21 l 3
.523 2.0 2.32
.960`1 3 .' 1 .85

.1410 1.0 1 I 1
.51 1.05 1.07
.7322 1.10 .20
.861 0 1.31 1.3 *
.0578 1.50 1.51

.2q47 1.01 1.05


I' = 0 indr icJl no \%ap:r co-,mpiosiition avl3ilabl

I i.j'_.Les data point ILniaord in fittin; ind c .l,:ul t ion of
dJtr. e erroi .










Table 5-11


Ethane (1)-Water (2), K12

T,K P,atm x1,exp xl,calc yl,exp


310.9 13.6 .00031 .00029 .0
38.7 .00065 .00065 .0
131.0 .00082 .00083 .0
212.1 .00089 .00089 .0
342.8 .00102 .00096 .0
430.9 .00111 .00100 .0
643.6 .00113 .00106 .0


344.3 13.3
34.9
53.4
135.1
222.9
332.5
441 .5
656.9

377.6 14.4
37.3
76.2
133.8
172 .6
235.2
477.2
676.3

410.9 14.3
37.9
66.6
143.3
244.0
343.4
440.1
665.4

444.3 15.4
36.0
67.5
135.1
247.8
342.8
451 .3
635.5


.00015 .00020 .0
.00042 .00046 .0
.00057 .00062 .0
.00079 .00085 .0
.00089 .00095 .0
.00101 .00104 .0
.00108 .00112 .0
.00117 .00123 .0

.00013 .00018 .0
.00039 .00043 .0
.00066 .00072 .0
.00094 .00093 .0
.00104 .00101 .0
.00113 .00111 .0
.00133 .00138 .0
.00151 .00153 .0

.00016 .00016 .0
.00046 .00045 .0
.00080 .00073 .0
.00121 .00117 .0
.00153 .00146 .0
.00170 .00166 .0
.00187 .00182 .0
.00200 .00211 .0

.00014 .00014 .0
.00050 .00050 .0
.00104 .00094 .0
.00167 .00160 .0
.00232 .00220 .0
.00252 .00254 .0
.00279 .00286 .0
.00320 .00329 .0


= 0.0


Yl'calc Y1 Y2

.9951 1.03 1.01
.9982 1.08 1.03
.9992 1.32 1.10
.9993 1.59 1.16
.9994 2.12 1.27
.9995 2.57 1.35
.9996 4.08 1.57

.9751 1.02 1.01
.9901 1.06 1.02
.9934 1.10 1.04
.9969 1.30 1.09
.9974 1.54 1.16
.9978 1.91 1.24
.9981 2.36 1.33
.9984 3.58 1.53

.9169 1.02 1.01
.9667 1.06 1.02
.9830 1.14 1.05
.9893 1.26 1.08
.9908 1.35 1.11
.9921 1.51 1.15
.9944 2.33 1.33
.9954 3.30 1.50

.7612 1.02 1.01
.9070 1.06 1.02
.9451 1.11 1 .C
.9717 1.26 1.08
.9799 1.49 1.15
.9833 1.75 1.21
.9854 2.05 1.28
.9886 2.96 1.45

.4707 1.01 1.00
.7675 1.04 1.0c
.8713 1.09 1.03
.9303 1.21 I .C
.9553 1.45 1.1-
.9634 1.68 1.20
.9690 1.98 1.27
.9749 2.63 1.40


IMS deviation in xl = 0.00005
.exp =
1 = 0 indicates no vapor composition data avaii4tt%










i NIl1_ 5-12


S1 1.


TA P, F. rr. p 1 a l ',' F:.p


298.2 23.2 .0005 .'39056 .0
44.9 .00100 .. 0101 .0
8i7.8 .00C 16 .001 72 .0
169.8 .0 8 58 .i02~1 .0
234.3 .00311 .00309 .0
327.4 .00366 .00356 .0
*30U. .03417 .00403 .0
633.1 .00445 .00467 .0

310.9 22.5 .00044 .000, C
'5.2 .00034 .00089 .0
b6.b .00144 .00151 .0
175.3 .00229 .00241 .0
140.6 .00276 .0)r, 25 .0
33..2 .00333 .CC335 .0
4- .2 .00391 .0038' .0
b73.6 .00465 3045 .0

344.3 22.5 .00034 .00036 .0
.9 .0006 3 )JOo c. .0
89.4 .0 118 .00125 .0
173.9 00 1 2 .C020C 0
240.. .00238 .00250 .0
335.3 .,0277 .00302 .C
441 .2 .00342 .00352 .0
671.6 .0042. .j0) 35 .0

377.6 22.7 .00032 .00033 .0
'.. .4 .0006nt .0e063 .0
d89. .00119 .0011R .0
172.6 .00198 .001 9 .0
24J3. .002S .00;52 .0
336.0 .00317 .C0311 .0
44 .2 .0036 1 .0, 367 .0
672.2 .0)451 .00..4 .0

410.9 :.i .00033 .00033 .0
43.5 .0.067 00C67 .0
80.2 .OC133 '01 3' .0
1362 .6 .00235 .0 216 .0
242.. .0)301 .00292 .0
338.7 .00.380 .OC3o8 .0
44 .2 .0049 .03439 .0


" 1 Ft],: '"2

.9985 1.03 1.02
.9 9 1 I.06 .03
.9994 13 1.07
.9995 1.28 1.13
.99 6 1 .43 1.19
.9996 1.65 1.27
.99?9 1.97 1.38
.9996 2.69 1.50

. 68 1.03 1.02
.9 2 1.06 1.03
.99 1 1. 13 1.06
C.991 1 .28 1. 13
.902 1 .2 1.19
.9992 I.63 1.27
.9903 1.93 1.37
.9993 2.73 1.60

. 84. 1 .03 1.01
. 9 I 5 .06 1. 3
.994 9 1.12 1.06
.9 .-. 1.2t I .12
.99c6 1.37 1.17
.9971 1.56 1.24
.99 3 1.81 1.33
.9976 2.-6 1.54

.9,4 1.0 3 1.01
.9700 1.05 1 .03
.9 2'4 1.1I I .C5
.98"7 1.23 1.11
.990 3 1 .35 1 16
.9914 1 .52 1.23
.9022 73 1.31
.9933 2.2 1.50

.3460 .02 1 .01
.9165 1.05 1.02
Sb33 1.10 1. 05
.9691 1.20 1.10
.9755 1.32 .15
. 79: 1 .47 1.21
.9813 1.66 1.29


.'II




81


Table 5-12 (Continued)

T,K P,atm x1,exp x ,calc yl,exp y ,calc y1 Y2


669.5 .00574 .00566 .0 .9842 2.15 1.46

444.3 22.0 .00032 .00031 .0 .6231 1.01 1.01
45.1 .00079 .00079 .0 .8067 1.04 1.02
92.6 .00173 .00169 .0 .8959 1.09 1.05
175.6 .00302 .00287 .0 .8892 1.20 1.10
243.7 .00383 .00394 .0 .9468 1.29 1.14
343.4 .00487 .00509 .0 .9557 1.44 1.20
444.2 .00595 .00610 .0 .9608 1.60 1.27
630.4 .00775 .00768 .0 .9667 1.96 1.40

RMS deviation in xl = 0.00009

exp
1 = 0 indicates no vapor composition data available










iibie 5-13


Cjrb.t'n .t -;iJ.: i i l-Qli.;r (. = -l. ,i

.1 . Lm 1 Ne:p . : lr i .:' p .c ic 1 '2


273.2 1.0 .00145 .00147 .0 .930 1 .02 1.00
5.0 .00,77 .nOfb73 .0 .9 87 1 .0 1.C.C
10.0 .01269 1216 .0 .9993 1.1" 1.01
20.0 .02097 02066 .0 .9 1 .31 1 .01
30.0 .026 4 .'644 .0 .9997 1.3 1.02
34.0 .02 ?23 .02835 .0 .9-9?7 1 .47 1.02

276.2 1 .00183 ..011 7 .0 .9913 .0 1 .00
10.0 .01056 .01001 .0 .99 1 1.14 1 .01
20.0 .01 810 .01717 .?90'5 1.27 1.01
38.0 .0253u .02573 .0. .999(6 1 ..2 1.03

283.2 1.0 .00095 .00095 .0 .9678 1.01 1.00
5.0 .00460 .00450 .0 .9975 1 .06 .Or:
1 3.0 .008 54 .00,33 .0 a9'7 1 .12 1.01
20 .0 .01535 .01'.52 .0 '4104 1.21 1 .C.
38.0 .C2191 .02219 .0 .'995 1.35 1.03

285.6 I .0 .00081 .00087 .0 .9857 1..01 1.00
5.0 .0'413 .00412 .0 .99q7 I .06 1.00
10.0 .00771 .0n766 .0 0i4 1.11 1.01
20 .0 0 ..0 0 1O 3 .0'991 I .20 1.01
30.0 .01 63 1 700 .0 .900q 1.27 1.02
45.0 .02203 .02272 .0 .99 5 1 .37 1.03

288.2 1 .0 .00078 .00079 .0 .9q 31 1 .0 I I .C00
5.0 .0: 363 .00375 .0 r-. 65 1 .05 .CO0
10.0 .00693 .00702 .0 ,9982 1.10 1.01
20.0 .01211 .012-.0 .0 .9 g9 1.1- 1.01
30,.0 .01610 .O106o .9qQi 1.25 1.02
45 .0 .'? 020 .C S1 1' .0 .8850 1.31 1.03

298.2 1 .0 .0031ol C0056 .0 .9664 1 .01 1.00
20.0 .00976 .1,: 0C37 .09 .9f 1 1 13 1.01
45.0 .016.5 .01666 .0 .9989 1 .26 1 .03

iIlS' .1 ,*.'ij icn In .in = O.l.lori..

..' = I'i irn i LtL i.' -'.jpor *.:.mr-.ic-, n J-u, -',ai' luabie










Table 5-14

Methane (1)-Propane (2), K12 = 0.015


T,K P,atm


310.9 13.6
20.4
34.0
51 .0
68.1
81 .7
91 .9

344.3 27.2
34.0
40.8
47.6
54.4
61.2


x1,exp xl,calc yl,exp yl,calc yl


.00490 .00451 .0521 .0456 1.00
.04600 .04496 .3255 .3117 0.99
.12350 .12403 .5209 .5268 0.98
.22160 .22110 .6210 .6260 0.96
.32710 .32083 .6635 .6636 0.92
.42260 .41296 .6779 .6742 0.87
.56100 .55588 .6087 .6829 0.73

.00630 .00610 .0276 .0275 1.00
.04330 .04444 .1550 .1606 0.98
.08130 .08372 .2392 .2492 0.96
.11990 .12496 .2983 .3101 0.93
.16180 .17165 .3414 .3546 0.89
.20810 .23501 .3656 .3741 0.78


RMS deviation in xl = 0.0088


Y2

1.00
1 .03
1.09
1.18
1 .30
1.45
1 .86

1.00
1 .03
1.06
1.10
1.15
1.23


T .










-.t. l, 1 -- .

l LLh,;arC ])-n- but,ar.e / i'i- = 1 0 f l:

1 : ['. n :. i,,-. : :.'i,,.:,-al,: e:-..p : .c l. .


310.1 4 .1 .00210 .00316 .1 335 .1331 1 .0') .CO
6 .8 .01 750 .01 7993 .4596 .455: 1 .00 1.01
13.u 0531:0 .0543 9 .7027 .7037 1 100 .0
27.2 .1 160 .12442 .8239 .8268 1.00 1.11
5 .4 .25400 .255-37 .8774 .8807 0.qQ9 .2 J
81.7 .38100 .3'3116 .A738 .7387 0.?7 1.40
1O8.9 .51680 .51375 .9570 .886~ 0.13 1.80 *

327.6 0. .00640 .00677 .1 791 73 1.00 1.01
13.6 .0 040 .04 127 .5 31 .548% 1.0 1.03
27.2 .10600 .10777 .7390 .7379 1 .00 1.01
5 .4 .23180 .232'b .8193 .8247 I .00C .24
81.7 .?5560 .3'52 4 .8263 .84 2 0.9 .-3
1Ou.') .48880 .4 7693 8 4 .8622 0.95 1 .72

34-.3 10 .2 .00900 .00981 .1 722 .Ij 1 1 .00 1.01
20.4 .05730 .05874 .5318 .52 13 1.00 1.05
27.2 .08850 .0 '- 05 .c233 .62 0t 1 .00 1 .08
5. .4 .21190 .21 24C .7451 .7507 1 .00 1.22
S1.7 .33?80 .33019 .7504 .71 1 1 0.91 1.
108.9 .46960 .45611 .7285 .7877 0.9'5 1 .69

360.9 1I3.6 .00790 .00824 .1110 .1035 1 .00 1.01
20.4 .03870 40-? .3626 .3 470 3 .O0 1 .04
27.' .0 6950 7130 .4848' .4 723 .00 1.0o
54.4 1160 .19257 .6529 .o537 0.99 1.20
81.7 .31540 31 393 .6745 .6,89 0.97 1.39
10,' .46850 .46101 .6265 .7175 0.90 1.73

377.6 20.4 .01752 .D OlR84 .1599 .1479 1.00 1 .C
27.2 .C 759 0- 96 .324l .3057 .On .05
40.5 .10940 1 1 2 .4 799 .4642 0 .9 1 .1
54.4 17220 .1728'. .542' .54.15 0.98 1 .19
C.. I1 .23560 .23616 .5630 .5901 0.96 1.27
81.7 .3) 11 0 30529 .5613 .601 0 .93 1.39
95.3 .33190 .39675 .3370 .6218 0.35 1.58

394.3 27.2 .02170 .02240 41i 12 6 1 .00 1 .2
34.0 .053"0 .0534.4 .2"50 .2328 0.99 1.05
.0.8 .09500 .a-~o4 .3411 .3077 .98 1 .CO
54.4 .15190 14834 .4147 .3960 0.6 1. 16
68.1 .21920 .21741 .4327 .4361 0.9? 1.26
81.7 .30200 .31300 .4101 .4631 0.92 1.43

ill de,'.i tic.n in = i..',]

IriJi.cT.e: li[. inT. ign'..ir d in ficn ,,g and .ilC:J laLior,
of : 'r,' error.




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