Equations of State: Theories and Applications (Acs Symposium Series)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.fw001 Equations of State Theories and Applications In Equ...

Author: Kwang-chu Chao | American Chemical Society. Division of Industrial and Engineering Chemistry

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Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.fw001

Equations of State Theories and Applications

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.fw001

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ACS SYMPOSIUM SERIES 300

Equations of State Theories and Applications

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.fw001

K. C. Chao, EDITOR School of Chemical Engineering, Purdue University

Robert L. Robinson, Jr., EDITOR School of Chemical Engineering, Oklahoma State University

Developed from a symposium sponsored by the Division of Industrial and Engineering Chemistry at the 189th Meeting of the American Chemical Society, Miami Beach, Florida, April 28-May 3, 1985

American Chemical Society, Washington, DC 1986

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Library of Congress Cataloging-in-Publication Data Equations of state. (ACS symposium series; 300) "Developed from a symposium sponsored by the Division of Industrial and Engineering Chemistry at the 189th meeting of the American Chemical Society, Miami Beach, Florida, April 28-May 3, 1985." Includes bibliographies and indexes.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.fw001

1. Equations of state—Congresses. I. Chao, Kwang-Chu, 1925- . II. Robinson, Robert L., 1937- . III. American Chemical Society. Division of Industrial and Engineering Chemistry. IV. American Chemical Society. Meeting (189th: 1985: Miami Beach, Fla.) QC173.4.E65E65 1986 541 86-1109 ISBN 0-8412-0958-8

Copyright © 1986 American Chemical Society All Rights Reserved. The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owner's consent that reprographic copies of the chapter may be made for personal or internal use or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to copying or transmission by any means—graphic or electronic—for any other purpose, such as for general distribution, for advertising or promotional purposes, for creating a new collective work, for resale, or for information storage and retrieval systems. The copying fee for each chapter is indicated in the code at the bottom of the first page of the chapter. The citation of trade names and/or names of manufacturers in this publication is not to be construed as an endorsement or as approval by ACS of the commercial products or services referenced herein; nor should the mere reference herein to any drawing, specification, chemical process, or other data be regarded as a license or as a conveyance of any right or permission, to the holder, reader, or any other person or corporation, to manufacture, reproduce, use, or sell any patented invention or copyrighted work that may in any way be related thereto. Registered names, trademarks, etc., used in this publication, even without specific indication thereof, are not to be considered unprotected by law. PRINTED IN THE UNITED STATES OF AMERICA

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ACS Symposium Series M. Joan Comstock, Series Editor

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.fw001

Advisory Board Harvey W. Blanch

Donald E. Moreland

University of California—Berkeley

USDA, Agricultural Research Service

Alan Elzerman

W. H. Norton

Clemson University

J. T. Baker Chemical Company

John W. Finley

James C. Randall

Nabisco Brands, Inc.

Exxon Chemical Company

Marye Anne Fox

W. D. Shults

The University of Texas—Austin

Oak Ridge National Laboratory

Martin L. Gorbaty

Geoffrey K. Smith

Exxon Research and Engineering Co.

Rohm & Haas Co.

Roland F. Hirsch

Charles S.Tuesday

U.S. Department of Energy

General Motors Research Laboratory

Rudolph J. Marcus

Douglas B. Walters

Consultant, Computers & Chemistry Research

National Institute of Environmental Health

Vincent D. McGinniss

C. Grant Willson

Battelle Columbus Laboratories

IBM Research Department

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.fw001

FOREWORD SYMPOSIUM SERIES

The ACS was founded in 1974 to provide a medium for publishing symposia quickly in book form. The format of the Series parallels that of the continuing except that, in order to save time, the papers are not typeset but are reproduced as they are submitted by the authors in camera-ready form. Papers are reviewed under the supervision of the Editors with the assistance of the Series Advisory Board and are selected to maintain the integrity of the symposia; however, verbatim reproductions of previously published papers are not accepted. Both reviews and reports of research are acceptable, because symposia may embrace both types of presentation.

IN CHEMISTRY SERIES

ADVANCES

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

PREFACE

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.pr001

THE EQUATION-OF-STATE APPROACH to model and correlate fluid-phase equilibria has been emphasized more and more in the eight years since the first symposium on this topic. In 1979, we edited the volume entitled Equations of State in Engineering and Research (Advances in Chemistry No. 182), which was based on that symposium. Meanwhile, research activities have been continually robust, particularly in building new models and in extending established equations for new and improved applications. The present volume is based on the April 1985 equation-of-state symposium held to present a comprehensive state-of-the-art view of progress in this area. The term "equation of state" is used in a broad sense to include mathematical description of volumetric behavior, derived properties, mixture behavior, and phase equilibrium of fluids. The main thrust continues to be the description of fluid-phase equilibrium, a phenomenon of enduring interest because it is basic to mass transport and separation operations. At the present stage of development, nonpolar fluids are modeled almost exclusively with equations of state, and active research is extending to polar fluids. The twenty-eight chapters in this volume, reporting work and progress on a broad front, are arranged in seven sections. Contributions are made by authors from diverse disciplines, including chemical engineers, physical chemists, and chemical physicists. The division of the volume into sections is not rigorous; some papers can fit easily into more than one section. Nevertheless, the arrangement should serve as a helpful guide to readers in their initial encounter with this substantial collection. We greatly appreciate the encouragement and support of the Division of Industrial and Engineering Chemistry of the American Chemical Society in sponsoring the symposium. Special thanks go to the authors of the papers. In addition to the regular task of preparing their manuscripts, they have gone the extra mile by preparing them in a camera-ready form. Their care and devotion are clear from the quality of the finished product. Robin Giroux of the ACS Books Department worked with us throughout the development of the volume.

K. C. CHAO

ROBERT L. ROBINSON, JR.

School of Chemical Engineering Purdue University West Lafayette, IN 47907

School of Chemical Engineering Oklahoma State University Stillwater, OK 74078 December 2, 1985 ix

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1 Equations of State and Classical Solution Thermodynamics Survey of the Connections Michael M. Abbott and Kathryn K. Nass

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

Department of Chemical Engineering and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590

Many contemporary researches in equation-of-statery focus on the representation of properties of liquid mixtures. Here, the connection with experiment is made through excess functions, Henry's constants, and related quantities. We present in this communication a review and discussion of the apparatus linking the equation-of-state formulation to that of classical solution thermodynamics, and illustrate the key ideas with examples. The c o r r e l a t i o n and prediction of mixture behavior are central topics i n applied thermodynamics, important not only i n their own right, but also as necessary adjuncts to the c a l c u l a t i o n of chemical and phase e q u i l i b r i a . Two major formalisms are available for representation of mixture properties: the PVTx equation-of-state formul a t i o n , and the apparatus of c l a s s i c a l solution thermodynamics. I t i s well known that the two formalisms are related, that a PVTx equation of state i n fact implies f u l l sets of expressions for the quantities employed i n the conventional thermodynamics of mixtures. Only with advances i n computation, however, has i t become possible to take advantage of these relationships, which are now used i n building and testing equations of state. The formulations d i f f e r i n at least two major respects: i n the choices of independent variables, and i n the d e f i n i t i o n s of s p e c i a l functions used to represent deviations of r e a l behavior from standards of " i d e a l i t y " . In the equation-of-state approach, temperature, molar volume, and composition are the natural independent variables, and the residual functions are the natural deviation functions. In c l a s s i c a l solution thermodynamics, temperature, pressure, and comp o s i t i o n are favored independent variables, and excess functions are used to measure deviations from " i d e a l i t y " . Thus translations from one formulation to the other involve both changes i n independent variables and conversions between residual functions and excess functions. 0097-6156/86/0300-0002$ 10.75/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

ABBOTT AND NASS

3 Classical Solution Thermodynamics

Simple as a l l this sounds, we have found i t f r u s t r a t i n g not to have available a single source i n which the connections between the two formulations are neatly l a i d out i n purely c l a s s i c a l terms. P a r t i c u l a r l y vexing i s the lack of a f l e x i b l e but clean notation. To meet our own needs, we have synthesized a system of d e f i n i t i o n s and notation, p a r t i a l l y described i n the following pages. The system seems adequate for both research and classroom use.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

Deviation Functions Rationale and Definitions. It i s rarely p r a c t i c a l to work d i r e c t l y with a mixture molar property M. For example, M may not be defined unambiguously, and may therefore not admit, even i n p r i n c i p l e , d i r e c t experimental determination. Thus, neither U nor S nor H nor G i s defined at a l l , i n the s t r i c t sense of the word. Both U and S are primitive quantities, and H and G are "defined" i n terms of one or both of them. Moreover, property M by i t s e l f may not admit physi c a l interpretations, except i n a loose sense (e.g., entropy as a "measure of disorder"). For these and other reasons, one finds i t convenient to introduce such quantities as residual functions and excess functions. These quantities, themselves thermodynamic prope r t i e s , are examples of a general class of functions which we c a l l deviation functions. Deviation functions represent the difference between actual mixture property M and the corresponding value for M given by some model of behavior: M(deviation) = M(actual) - M(model) The choice of a model i s to some extent arbitrary, but to be useful the model must have certain a t t r i b u t e s . I t s molecular implications should be thoroughly understood, so that deviation functions defined with respect to i t can be given clean interpretations. Real behavi o r should approach model behavior i n well-defined l i m i t s of state variables or substance types, so that the deviation functions have unambiguous zeroes. F i n a l l y , to f a c i l i t a t e numerical work, i t i s desirable that the properties of the model be capable of concise a n a l y t i c a l expression. Once a model i s chosen, the conditions at which the comparison ( r e a l vs. model) i s made must be s p e c i f i e d . There are several p o s s i b i l i t i e s here, but two are p a r t i c u l a r l y f e l i c i t o u s . Thus, we may define deviation functions at uniform temperature, pressure, and composition: M° = M -

od

rf" (T,P,x)

(1)

Here, the notation signals that mixture molar property for the model i s evaluated at the same T,P, and composition as the actual mixture property M; superscript (capital) "D" i d e n t i f i e s the devia t i o n function as a constant - T,P,x deviation function. Alternat i v e l y , we may define deviation functions at uniform temperature, molar volume (or density), and composition:

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

4

EQUATO INS OF STATE: THEORIES AND APPLICATIONS mod, M (T,V,x)

M

M

(2)

mod Here, M i s evaluated at the same T,V, and composition as the actual mixture property M; superscript (lower-case) "d" d i s t i n guishes this class of functions from that defined by Equation 1. The two kinds of deviation function are related. Subtraction of Equation 1 from Equation 2 gives M

d

mod, M (T,V,x)

D mod, - M - M (T,P,x)

v

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

whence we find that

Here, pressure P i s the pressure for which the mixture molar volume of the model has the same value V as that of the r e a l solution at the given temperature and composition. According to Equation 3, deviation functions and M are i d e n t i c a l for those properties M for which M * i s independent of pressure. D

moc

Residual Functions. ideal-gas mixture: m

M °

d

The simplest model of mixture behavior i s the

= M

ig

Deviation functions defined with respect to the ideal-gas model are c a l l e d residual functions, and are i d e n t i f i e d by superscript R or r . Thus, as special cases of Equations 1 and 2, we define i

(4)

i

(5)

M

R

= M - M 8(T,P,x)

M

r

= M - M 8(T,V,x)

and

Residual functions M and M are related by Equation 3, with the assignments mod = i g and P* = RT/V. Thus R

r

(6)

Ideal-gas properties U*§, H*8, of pressure. Hence M

R(

r .M

M

C^S, and

are a l l independent

(7)

U,H,C ,C ) v

p

i

On the other hand, ideal-gas properties S^S, A*8, and G 8 are functions of pressure. In p a r t i c u l a r ,

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

ABBOTT AND NASS

5

Classical Solution Thermodynamics

R P 1£ 3P

RT P

T,x

and hence, by Equation 6, (8)

R£nZ A G

r

G

R

+ RTfcnZ

(9)

+ RT£nZ

(10)

where Z = PV/RT i s the compressibility factor. Expressions f o r either M or M are found from a PVTx equation of state by standard techniques: see e.g. Van Ness and Abbott (1_)« If the equation of state i s e x p l i c i t i n pressure, then T,V (or molar density p), and composition are the natural independent variables and the M are the natural residual functions. If the equation of state i s e x p l i c i t i n volume, then T,P, and composition are the natural independent variables and the M are the natural residual functions. Tables I and II summarize formulas f o r computing the M from a pressure-explicit equation of state, and the M from a volume-explicit equation of state. Conversion from M to M , or vice versa, i s done by Equations 7 through 10. Note that V and P are i d e n t i c a l l y zero.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

R

r

r

R

r

R

R

r

r

R

r

Residual Function A : a Generating Function. Most r e a l i s t i c equations of state are e x p l i c i t i n pressure; T,V, and composition are the natural independent variables. These are also the canonical variables f o r the Helmholtz energy A, so the constant - T,V,x r e s i d ual Helmholtz energy A plays a special role i n equation-of-state thermodynamics. I t can be considered a generating function, not only f o r the other constant - T,V,x residual functions, but also f o r the equation of state i t s e l f . The relevant working equations assume a pretty symmetry when the independent variables are chosen as reciprocal absolute temperature x ("coldness" = T ~ l ) , r e c i p r o c a l molar volume p (molar density = V~*), and composition. They are summarized i n this form i n Table I I I . Application of these formulas may be i l l u s t r a t e d by a simple example. We choose f o r this purpose the van der Waals equation of state, f o r which r

r

A (vdW) = - RT£n(l-bp) - a

p

(33)

where parameters a and b depend on composition only. The residual pressure P (= P-pig = P-pRT) i s found from Equation 27: r

2

r

P (vdW)

bp RT

2

so the equation of state i s

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

6

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

P(vdW)

pRT 1-bp

(34)

ap

The other residual functions follow from Equations 28 through 32, Two results are r

(35)

r

(36)

S (vdW) = R£n(l-bp)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

U (vdW) - - ap

Equations 35 and 36 support the common interpretations of the hard-sphere (repulsive) term as representing an "entropic" contribut i o n to the equation of state, and of the van der Waals "a" as an energy parameter. The material of this section can i n fact be taken as a point of departure for the development of a c l a s s i c a l l y inspired generalized van der Waals theory, motivated by Equations 33 through 36, but unrestricted by the assumptions attendant to the o r i g i n a l van der Waals equation of state (2_). Excess Functions, The conventional standard of mixture behavior for condensed phases i s the ideal solution. Deviation functions reckoned against this model are c a l l e d excess functions, and are i d e n t i f i e d by superscript E or e. We define id

(37)

id

(38)

M - M (T,P,x) and M - M (T,V,x)

where superscript i d i d e n t i f i e s the i d e a l solution. Equations 37 and 38 are special cases of Equations 1 and 2, with the assignment mod = i d . Excess functions M and M are related by e

M

6

= M

E

+ /

3Ml<1

(3P

>*T,x

(39)

dP

P

which i s a special case of Equation 3. Unlike the ideal-gas case,^ no simple general closed-form expression can here be written for P . By d e f i n i t i o n , P i s i n this case the pressure for which the i d e a l solution has the same molar volume V as the real solution at the given temperature and composition. Since

t h i s pressure must be found as a solution to the equation Jx V (T,P*) - V(T,P,x) i 6

(40) E

C l e a r l y , numerical r e l a t i o n of M to M v i a Equations 39 and 40 requires equation-of-state information, both for the real pure com-

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

ABBOTT AND NASS

Table I.

Residual Functions from a Pressure-Explicit Equation of State

P

r

(11)

- pRT(Z-l)

„r Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

1 Classical Solution Thermodynamics

_ 2

P

RT

=

;

(

|Z

d^ (12)

H

r

- -RT

Z

^

+ RT(Z-l)

(13)

dp. p

A

r

G

r

- RT / (Z-l)

(14)

dp. (15)

= RT / ( Z - l ) ^

+ RT(Z-l)

(16)

C„ =

C

r p

= C

(17)

r v

-R

+

R[

Z

+

T(|f)

p ) X

]

2

[Z + p ( { f ) . ] T

_ 1

x

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(18)

8

EQUATO INS OF STATE: THEORIES AND APPLICATIONS Table I I .

Residual Functions from a Volume-Explicit Equation of State

(19)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

"

Rt2

Z

/ (H) x f " ^ ' »

(20)

i UTJP.X P

(21)

Pi

RT

R

T

+

" / [ (|f)p, ^

f

X

A*

=

(22)

RT / ( Z - l ) ! ^ - RT(Z-l)

(23)

RT / ( Z - r f

(24)

P

2

-RT ; [T(^)P. - (H)P,] f X

R .

C

+ R

R[Z

+

T (

||

) p ) x ]

(25)

x

2

[ z

.

p j!) j-i (

T j

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(26)

1.

9

ABBOTT AND NASS

Classical Solution Thermodynamics

]

Table I I I . Residual Functions from A

2

r

=Ρ

r

_

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

r

S

"

„Γ

G

r

(|4 r

2 ,3A , τ

(ϊΓΪρ,χ

|~3( Α ) 1 L 3τ Jp,x Γ

=

τ

l~3(pA )] r

.

Γ r H

=

r A

Γ

(fM

+ τ f-^-) + ρ 3 τ -'ρ,χ *-3ρ τ,χ ν

c{

-

-

,

μ

'

;

Μ

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

10

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

e

ponents and for the r e a l mixture. Hence the r e l a t i o n between M and M i s not as "clean" as that between M and M . Rough closed-form approximations to Equation 39, appropriate for applications to condensed phases, may be found however. We write Equation 39 as E

r

M

e

» M

+ (!|H

E

T)X

R

(P-P*)

(41)

where the derivative i s evaluated at the pressure Ρ of the r e a l mix ture. An expression for Ρ follows from Equation 41 by the assign ment M = V, because V = 0 i d e n t i c a l l y . Thus we f i n d that e

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

p

"

p

< 4 2 )

Τ ^ Λ

'

J ι ι i where

i s the isothermal compressibility of pure i : 1

3

V

i

A l l quantities on the right side of Equation 42 are evaluated at pressure P. According to Equation 42, the sign of V determines whether Ρ i s less than or greater than P. Combination of Equations 41 and 42 gives e

i which i s the required approximation to Equation 39. Particular cases of Equation 43 are generated on s p e c i f i c a t i o n of M and of the corresponding derivative (3Μ^^/3Ρ)χ . Table IV summarizes expressions f o r this derivative i n terms of the volumetric proper t i e s of the species composing the mixture. To demonstrate i t s application, l e t us take M = A. Then, by Equations 43 and 48 we f i n d that x

A

e

« A

E

G

E

- A

E

+ PVΕ ]

But +

PV

E

and hence we have rationalized the approximation A

e

« G

E

a result frequently used i n molecular modeling of the constant Τ,Ρ,χ excess Gibbs energy.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

A B B O T T A N D Ν ASS

11

Classical Solution Thermodynamics

. ,id 3M — x

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

Table IV.

Expressions for

;

^3P

V

T,x

V

" [¥i i

p

u

x

v

(44)

T

I i
H

x

v

( 4 5 )

i

v

( 4 6 )

I i i - ^ W i i

S

- Σχ.β^ i Ρ

Α G

Ι 1 ΐ ΐ χ

κ

ν

( 4 8 )

Yx.V. i 1

(49)

1

2 C

(47)

p

-

3 β

1 - Tjx ( -) V 1

i T

p

1

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(50)

12

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Relations between Excess Functions and Residual Functions. Excess functions and residual functions are related. The relationships are most easily established through M and M . By Equations 4 and 37, we have E

Ξ M

R

- (M

id

R

ig

- M )

(51)

But

Λα i

i

1

and x

3- Jx.£nx i

M

? i i

1

1

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

where 0

(M = U,H,C )

R

(M - S)

p

-RT

(52)

(M = A,G)

Hence Equation 51 becomes M

E

= M

R

8

- (|x M i

Ix^/ )

1

(53) i

1

1

E

e

R

Equation 53 i s a basis for relations connecting M or M to M or M . It provides a l i n k between excess functions and the equation of state and i t suggests how physical interpretations applying to par t i c u l a r residual functions are carried over to the corresponding excess functions. Experiment provides values of the constant - Τ,Ρ,χ excess func tions Μ , whereas the constant - T,V,x residual functions M , par t i c u l a r l y A , are most cleanly related to a pressure-explicit equation of state. By Equations 7 through 10, we have r

r

r

M

K

- M

r

+ 3- Jin Ζ

(54)

with 3- defined by Equation 52. Combination produces an expression r e l a t i n g M to M : E

of Equations 53 and 54

r

(55)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

ABBOTT AND Ν ASS

13 Classical Solution Thermodynamics

Table V summarizes formulas for the constant - Τ,Ρ,χ excess Ε Ε Ε Ε r functions Η , S , G , and C i n terms of A and i t s temperature p

derivatives. These recipes are useful for testing the a b i l i t i e s of a PVTx equation of state to represent liquid-mixture properties. Consider as an example the van der Waals equation of state, Equation 34, for which A i s given by Equation 33. Application of Equations 56 through 59 gives r

E

H (vdW)

S (vdW) E

(ap-^ a X i

(61) b

p

^ ~ i i i ^

E

(62)

E

G (vdW)

H (vdW) - TS (vdW)

cJ(vdW)

R < 11 -

E

(60)

) + PV^vdW)

Rlx.£n p

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

i i P i

(

1 - b

P

) 2

p -1

2a

ii RT

(l-b

i i P i

)

P i

(63)

E

Here, V (vdW) i s the excess volume implied by the equation of state. Unsubscripted quantities are mixture properties, and subscripted quantities refer to the pure f l u i d s . P a r t i a l Properties Rationale and D e f i n i t i o n s . The partial-property concept i s central to applied solution thermodynamics. F i r s t , i t represents a formal (but arbitrary) basis for apportioning a mixture molar property M amongst the constituents of a phase. Second, i t provides an elegant apparatus for describing i n f i n i t e l y - d i l u t e solutions. F i n a l l y , i t serves as a unifying concept i n formulating mixture equilibrium problems, because the chemical p o t e n t i a l and i t s r e l a t i v e s stand to the Gibbs energy and i t s r e l a t i v e s as p a r t i a l properties: see Table VI. _ The conventional d e f i n i t i o n of a p a r t i a l property is

where by implication temperature, pressure, and composition are favored independent variables. This choice i s e n t i r e l y appropriate i n the laboratory frame of reference, because temperature and pressure are the variables susceptible to precise measurement and control. There are instances however where i t i s useful to broaden the partial-property concept, to accommodate alternative choices of

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

14

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table

V.

Excess

H

=

Functions

(Α

Γ

- |χ

and

Γ

the

Residual

Helmholtz

Energy

)

ι Α ι

V,x

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

,Ε + PV

S" =

(56)

3T

V

4

(57)

+ Rjx Jn(V/V ) 1

(Α

Γ

1

- Jx.A?) i 1

1

RT^jmiv/v^

(58)

+ PV

3

Cp = -τ 3T

2

•I /v.x

\

r

-Σ*·ΗΓ) i MjT

2

V j

(59)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

15

Classical Solution Thermodynamics

A B B O T T A N D Ν ASS

independent variables. Foremost among these i s the application of pressure-explicit equations of state to the c a l c u l a t i o n of chemical and phase e q u i l i b r i a . Table VI.

Important P a r t i a l M

M. 1

G

(65)

"i in λ

G/RT

(66)

1

G /RT

Jin φ.

(67)

AG/RT

Jin a

(68)

in

(69)

R

±

E

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

Properties

G /RT

y

±

Reis (3_) discusses the generalization of the partial-property concept to other sets of independent variables, and Abbott (4^) has extended the d e f i n i t i o n of Equation 64 to higher-order derivatives with respect to mole numbers. A result common to both kinds of generalizations i s that the "summability feature" of the M-^, v i z . ,

M

= Yx.M^ i 1

(70)

1

i s i n fact possessed by a very large number of " p a r t i a l properties", i n addition to M-^. The generalization of Equations 64 and 70 for choices of inde pendent intensive variables other than Τ and Ρ i s quite simply rationalized. We outline a development here. Let the t o t a l prop erty M- = nM of a phase be a function of the set of mole numbers ηι,η2,···, and of two arbitrary intensive variables X and Y. Then the t o t a l d i f f e r e n t i a l d(nM) corresponding to an a r b i t r a r y change of state i s 1

d(nM) =

3(nM) dX 3X Y,n J

3(nM) dY 3Y X,n

i

3(nM) 3 ηi J

(71) Χ,Υ,η.

where subscript η denotes constancy of a l l mole numbers and nj denotes constancy of a l l mole numbers save n-^. The following iden t i t i e s apply to Equation 71: d(nM)

ndM + Mdn

dn, i

= ndx. + x^dn ι i

3(nM))1 _n 3X "JY,n -

/3M\

H,x

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

16

EQUATIONS O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

- • ΦΙ where subscript χ denotes constancy of a l l mole fractions. Additionally, l e t us define the generalized p a r t i a l property Η± as

(72)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

Combining the last s i x equations and c o l l e c t i n g c o e f f i c i e n t s of η and of dn, we obtain

+ [Μ-ΣΧΑ] dn

- 0

But quantities η and dn are independent and arbitrary; the two bracketed terms must therefore separately be zero, and we find that (73)

dM and

-

(74)

I*A

Equation 73 i s merely a special case of Equation 71, with n=l. Equation 74 i s "new" however;__it i s the required extension of the "summability feature" of the to other classes of p a r t i a l proper ties. Other analogs of the usual partial-property relations are found by straightforward mathematics. For example, according to Equation 74, the t o t a l d i f f e r e n t i a l dM i s dM = Yx.dM, + YM. dx, j i i L ι i But this expression must be equivalent to Equation 73; comparing the two gives immediately a generalized Gibbs-Duhem equation: dY χ which i s merely an extension of the f a m i l i a r

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(75)

1.

A B B O T T A N D Ν ASS

Classical Solution Thermodynamics

! Ά • (S) P,x«

+

17

(SL-

(76)

'Τ,χ

Relations among P a r t i a l Properties of Different Types. Sometimes i t i s necessary to convert one type of p a r t i a l property to another. Let us define, analogous to ÎL, a second p a r t i a l property M^:

(77) 3

L

n

i

J «,Y',n. X

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

where intensive variables X

1

and Y* may be different from intensive

variables^X and Y. We wish to relate p a r t i a l property Mj[ to p a r t i a l property M^. By Equation 71 we f i n d that

|~3(nM)1 L

9

n

fa (nM)l

m

i Jx',Y',n. "

L

9 X

J ,n Y

[3(nM)1 L

^Vx',Y',n

J3Y \

Jx,n

9 Y

/ 3X \

l3

+

Vx',Y',n.

J

~3(nM)"| L

9

n

i

Jx,Y,n

which becomes on s i m p l i f i c a t i o n

(78) +

n

(lf)

X ) X

(l^) , x

) Y ) ) n

_

Equivalent statements of Equation 78 are possible. derivative (3Χ/3η^)χΐ^yt may be written as

For example, the

>n

111]

. . (i2L]

9n

\ i>X',Y',n.

/ax 1^1 9n

fâlL] a

^ ' χ ' , χ \ Vx',X,n.

f

- - <x-

x)

i'x',Y',n.

The derivative ( 3 Y / 3 n ) i ι .. may be s i m i l a r l y rewritten. i

x

> γ

>n

Only two classes of p a r t i a l properties are of importance to us here: those defined at constant Τ and Ρ ("laboratory" p a r t i a l properties), and those defined at constant Τ and V ("equation-ofstate" p a r t i a l properties). Thus we define, as special cases of Equation 72,

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

a(nM) 3n.

(64) Τ,Ρ,η.

and

3(nM)1

M. ι

•

â

n

(79)

i JT.V.n.

Here and henceforth the t i l d e (~) i s used to distinguish a constant - T,V p a r t i a l property from the conventional constant - T,P

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

variety. Note that can equally well be considered a constant T,p p a r t i a l property, where p i s molar density. Both classes of p a r t i a l properties obey the summability r e l a t i o n of Equation 74: (70)

ί ι i and M =

(80)

Yx.M, i

1

1

Each class of p a r t i a l properties has i t s own Gibbs-Duhem equation:

+

sir, (if).

dP T.x

(76)

and

. m)

d TdT 3TA V,x

i

+ (if ι dV

(81)

F

T,x

F i n a l l y , by Equation 78, we f i n d the following relations between the two classes of p a r t i a l properties:

(82)

and (83) T,x^°

i T,V,n.

Equation 82 effects a conversion of laboratory p a r t i a l proper t i e s to equation-of-state p a r t i a l properties. Since

V

i

~

3(nV) 3n 4

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

19

Classical Solution Thermodynamics

A B B O T T A N D Ν ASS

we can rewrite i t as

K

±

- M.

(V - V.)

+

(If)

(84) Τ,χ

The conventional p a r t i a l molar volume i s thus a key quantity for this type of conversion. More common are applications requiring conversion of constant • T,V p a r t i a l properties to constant - T,P p a r t i a l properties. Here Equation 83 i s appropriate; the analog of Equation 84 i s

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

*i

=

\

+

( P

" V (If). Τ,χ

where 3(nP) 9, i jT,V,n n

j

An equivalent result involving compressibility factors i s — M

±

~ - M

±

, RT . ~ ν / 3M + - (Z - Z ) [ ±

(85)

π

Τ,χ

As an example of an application of this and e a r l i e r material, consider the following standard problem: to determine an expression for the component fugacity c o e f f i c i e n t φ^, convenient^for use with a pressure-explicit equation of state. We know that £ηφ^ Is a R

constant - T,P p a r t i a l property with respect to G /RT: see Equation 67, Table VI, Moreover, by Equations 10 and 16, we have the following recipe for G /RT: R

ê

·

P

z-i-iinz / (z-i)^ +

μ

ο

Since density appears e x p l i c i t l y i n this equation, an expression for R

Gj[ /RT i s readily found.

It i s

A l l that remains i s to relate G^/RT (= £ηψ ) to C^/RT. 1

R

apply Equation 85, with the assignment M = G /RT. derivative of G /RT i s

Here, we

The pressure

R

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

20

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Hence

and thus we f i n d that

ΐηφ

= Ζ - 1 - ί,ηΖ + f

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

±

(ζ Ί)

(86)

^

±

Equation 86 can of course be obtained by other procedures. In this example we have attempted to make f u l l use of material presented i n t h i s and preceding sections. Suppose that Ζ i s represented by the van der Waals equation of state, Equation 34: 1 ap 1-bp " RT

Z(vdW)

(87)

Then

(S -b)p

χ

Ζ (vdW)

-

1

1

Ρ

±

- .

T bf>

?

(1-bp)

2

T R T

and hence, by Equations 86 and 87,

ΐ,ρ JUcjj.CvdW) =

Here, quantities a^ and

(a + a.)p

-γζς^'

R T

- £n(l-b )Z p

(88)

are p a r t i a l equation-of-state parameters:

9

ϊ

=

Γ (na)1 L

1=

L

i-ΐτ,ρ,η

JT

η

E x p l i c i t expressions for ^ and ί>^ require e x p l i c i t expressions for the mixing rules for parameters a and b.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

ABBOTT AND NASS

21

Classical Solution Thermodynamics

Mixture Fugacity Behavior Fugacity Coefficients, A c t i v i t y C o e f f i c i e n t s , and Henry's Constants. Component fugacity c o e f f i c i e n t s are readily obtained from a PVTx equation of state. For developing and testing equations of state for phase-equilibrium applications, however, i t i s sometimes useful to deal d i r e c t l y with quantities conventionally used for description of the l i q u i d phase, e.g., a c t i v i t y c o e f f i c i e n t s and Henry's constants. We review i n the following paragraphs the connections among these measures of component fugacity behavior, and i l l u s t r a t e how they are determined from pressure-explicit equations of state. The fugacity c o e f f i c i e n t i s defined as

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

(89)

where f^ i s the fugacity of species i i n solution. coefficient is

ι

x t 1

The a c t i v i t y

1

where f° i s the standard-state fugacity. Two standard states are popularly employed: Lewis-Randall ("Raoult's-Law") standard stat< for which f? i s the fugacity of pure i at the mixture Τ and P,

f°

±

(91)

(LR) - f .

and Henry's-Law standard states, for which f J (HL) =

%

where Henry's "constant"

(92)

±

. i s defined as

(93)

the l i m i t being taken at the mixture Τ and P. If we write Equation 90 as

i

Ρ

Ρ

then we see that a l l that i s required to "convert" a fugacity coef f i c i e n t to an a c t i v i t y c o e f f i c i e n t i s an expression for the r a t i o

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

22

f°/P.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

For Lewis-Randall standard states, we have

where φ± i s the fugacity c o e f f i c i e n t of pure i . Thus the conven t i o n a l Lewis-Randall a c t i v i t y c o e f f i c i e n t i s

(94)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

For Henry*s-Law standard states, we have

1

±

1·

— Ρ

lim — x. χ +0 ι n

But

—

lim X

i*°

=

Ρ l i m φ.

X j L

X

±*°

1

where φ ^° i s the fugacity c o e f f i c i e n t at i n f i n i t e d i l u t i o n .

9f.

= ?l °°

Thus

(95)

±

1

and the Henry s-Law a c t i v i t y c o e f f i c i e n t

is

(96)

Conversion from fugacity c o e f f i c i e n t s to a c t i v i t y c o e f f i c i e n t s and Henry's constants i s thus straightforward. One needs i n addi tion to the component fugacity c o e f f i c i e n t one or another of i t s l i m i t i n g values, v i z . , φ

ί

φ

1

Ξ

Ξ

lim

φ

±

lim φ^^ x^+0

For a pressure-explicit equation of state, both of these are found as l i m i t s of Equation 86.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

ABBOTT AND NASS

Classical Solution Thermodynamics

23

Consider as an example the van der Waals equation of state, for which φ^ i s given by Equation 88.

For pure i , this equation yields

as a special case

b

^ i

(

v

d

W

)

i i

p

2

i

a

i i

p

i

IP

=T ^ T ii i

"

( 9 ? )

^ " " i l P i * !

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

H

where the doubly-subscripted parameters refer to pure i . Equations 94, 88, and 97 when combined produce an expression for the Lewis-Randall a c t i v i t y c o e f f i c i e n t implied by the van der Waals equation. Evaluation of φ ι " (and hence of or y± ) requires a l i t t l e more care, because the state of i n f i n i t e d i l u t i o n for a species i n a multicomponent mixture can i n p r i n c i p l e be defined i n many ways. The natural d e f i n i t i o n of this state i s as that state for which x^ approaches zero as the i - f r e e mole fractions x^ remain constant. (H. ere, χ'. Ξ X./^X,

J 88 yields

J

£ηφ

(vdW) =

±

, where j,k ψ i.)

By this d e f i n i t i o n , Equation

k

b V ^

(a'+â^V gj±

,

in
(98)

For the general multicomponent case, the primed quantities i n Equation 98 are i - f r e e mixture properties. Parameters a^°° and b are p a r t i a l equation-of-state parameters, evaluated at the same i - f r e e composition as the mixture properties. For the binary case, say of i n f i n i t e l y d i l u t e solute 1 i n solvent 2, Equation 98 reduces to the simple result

^ °°P 1

* <

(

v

d

W

)

"

+3

^22 Ί°°) 2

2

T=b b"i pr 2 2

-

ρ

RT

- *°(l-b P )Z 22

2

2

2

(99)

Here, the 1-free "mixture" i s just pure solvent 2. P a r t i a l Equation-of-State Parameters. Composition i s introduced into many a n a l y t i c a l engineering equations of state v i a "one-fluid theory", i n which an equation of state for a mixture i s assumed to have the same functional form as that for the pure species. Component mole fractions appear e x p l i c i t l y only i n mixing rules for the equation-of-state parameters. As i l l u s t r a t e d by the example just considered, evaluation of φ^, γ^,

or γ^* then requires

expressions for the p a r t i a l equation-of-state parameters. Letting π denote a generic equation-of-state parameter, we define, analo gously as for any p a r t i a l property,

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

24

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

=

(IOO) 3

. Π) (nw)1 L

3 n

ijT,

(101) p >

n ,3

Usually, parameter ir i s taken to depend at most upon temperature and composition; i n such cases the r e s t r i c t i o n s to constant Ρ or ρ are superfluous i n these d e f i n i t i o n s , and

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

-

_ ~

_ Γ3(ηπ)1

Note however that Equation 101 accommodates the currently-popular concept of "density-dependent mixing rules". Since pressuree x p l i c i t equations of state are favored for engineering applica tions, we henceforth consider only ϊ ^ . Development and testing of mixing rules i s a major area of research i n applied thermodynamics, and new formulations appear regularly: see other papers i n this volume. For concreteness, and to i l l u s t r a t e procedures, we treat here only the familiar "van der Waals prescription", according to which parameter π i s quadratic (or, as a special case, linear) i n mole f r a c t i o n : w =

g

W

( 1 0 2 )

*

Application of Equation 101 to Equation 102 yields on rearrangement the simple result

* i " JVki " * 2

( 1 0 3 )

k where π i s the mixture parameter, given by Equation 102. In deriving Equation 103, we assume that the parameters remain unchanged on permutation of subscripts: πι^ ~ π^· For pure i , Equation 103 yields the expected result:

lim x. + l ι

- „

w ±

1 ±

50

To evaluate π/ (for i n f i n i t e l y d i l u t e i ) , we f i r s t introduce i - f r e e mole fractions into Equations 103 and 102, obtaining

;

i

"

2

V i i

+

2 ( 1

)

^i J

, x

k

i r

ki"

1 f

k

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

A B B O T T A N D Ν ASS

Classical Solution Thermodynamics

25

where w =

+ 2

X

i

( l -

X

i

)

r

^

k

i

(Ι-χ/ΐΤΐ;»^ k ι

+

(104)

Combination of these equations gives

;

= ^ ) .

+ (i-x ) n'Y;(2. 2

u

t

k i

^

k i

)

<

105

>

Here, the primed mole fractions are i - f r e e mole fractions, and the primed sums s p e c i f i c a l l y exclude species i . Equation 105 i s e n t i r e l y equivalent to Equation 103; taking the l i m i t x^+0 we obtain d i r e c t l y an expression for

V- n W ""V ,x

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

at i n f i n i t e d i l u t i o n of i : (2,

(106)

Similarly, we f i n d from Equation 104 i n the l i m i t as x^+0 an expression for the i - f r e e mixture parameter π : 1

*·

( 1 0 7 )

- l'i'Wu

Equations 102, 103, 106, and 107 complete the apparatus required to evaluate φ^ and related quantities from a pressure-explicit equation of state, with mixing rules given by the van der Waals prescription. With respect to examples treated previously, they apply i n par t i c u l a r to the van der Waals equation of state, where π i s iden t i f i e d with parameters a and b. Numerical Examples We i l l u s t r a t e the use of preceding material with two numerical examples, both for the van der Waals equation of state. F i r s t , con sider the van der Waals excess functions, for which expressions are given by Equations 60 through 63. Calculation of numerical values of these quantities requires values for the equation-of-state para meters. For binary mixtures containing species 1 and 2, s i x parame ters are needed: a ^ , b ^ , &22> ^22> 12> * 1 2 The f i r s t four are found from information on the pure components; the interaction parameters a\2 and b^2 estimated from combining rules or, pre ferably, from mixture data. Since the application i s to the representation of excess func tions for l i q u i d mixtures, i t i s reasonable to determine the and b±± from data on pure l i q u i d s . Various combinations of properties are employed for this purpose; we choose here the liquid/vapor a

a

r

at

anc

b

e

e

saturation pressure P^ , the molar density

of the saturated

ον l i q u i d , and the molar heat of vaporization ΔΗ^ .

ον A molar property change of vaporization ΔΜ^ of pure f l u i d i i s defined as

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

26

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

AMj

Ε M^(T,P* ) - M*(T,P* )

V

at

at

(108)

where the terms on the right side denote molar properties of pure i as saturated vapor and as saturated l i q u i d . Since temperature and pressure are uniform, ΔΜ* Τ,Ρ,χ

i s just a difference between constant -

residual functions: ΔΜ

Aν

(109)

Here, the terms on the right are constant - Τ,Ρ,χ residual functions for pure i as saturated vapor and as saturated l i q u i d :

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

V

M*'

at

8

at

at

8

at

= M^(T,P* ) - M j ( T , P ^ )

M*>* = M*(T,P* ) - M j ( T , P ^ )

Residual functions are readily determined from a PVTx equation of state by procedures reviewed e a r l i e r . Hence, by Equation 109, so also are property changes of vaporization. By Equation 109, Consider the molar heat of vaporization Δ# £v i

Η

ΔΗ'

i

" i

and, by the d e f i n i t i o n of H, H* = U* + PvJ where, by Equation 7, ,R U i

= u;i

Thus, by the l a s t three equations,

(110)

Equation 110 expresses the molar heat of vaporization i n a form con venient for use with a pressure-explicit equation of state. Constant - T,V,x residual i n t e r n a l energies are found from Equation sat α ν i s related to p. and ρ by the equation of state 12, and Ρ itself. For the van der Waals equation of state, we have by Equations 110, 36, and 34 that 1

1

1

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

ABBOTT AND NASS

Classical Solution Thermodynamics

27

a ( p j " ρ*) + ?\ \-\ - -j-) Λ

V

AHj (vdW) =

(111)

u

Pi

Pi

where Ρ

1-b

a (pp

112

< >

tl

l l P l

and

;Rv T

Pi

1

b

a^Cpp"

(113)

N

v

1-K 1

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

, v 92

K

pSat

p

ii i

Given experimental values f o r P*

, p*, and ΔΗ~

at specified T, one

solves Equations 111, 112, and 113 f o r p^, a ^ , and b ^ .

The value

γ

of p^ so obtained i s merely an intermediary

quantity; because the

equal-fugacity requirement f o r liquid/vapor equilibrium i s not invoked, p^ i s not necessarily the "true" saturation vapor density implied by the equation of the state at the specified T. (The equal-fugacity requirement would provide a fourth equation; vapor pressure P^ would then be treated as an unknown, to be determined at

along with p^, a ^ , and b ^ . ) The above-described procedure when applied separately to pure 1 and to pure 2 provides values for ^H> 22> * ^22' ^° ^*- * a\2 and b j ^ assume the a v a i l a b i l i t y of data f o r H and V , each at a single composition. The working equations follow from Equations 60, 34, and 102, applied to the l i q u i d phase: a

w

a n c

e

n(

E

E

E

E

H (vdW) - - ( a p - x a p - x a p ) + PV (vdW) 1

PivdW) =

-

E

V (vdW) - ρ"

1

b =

x

b

i n

+

x

a

b

ap

- x^"

2 2 a - x ^ a ^ + 2 22 x

1 1

2 22

+

+

2 x

2 x

1

1

2

2 2

2

(34)

- χ ρ 2

x

Χ

(115)

2

a

l 2 12 x

(114)

2

(116)

b

i 2 i2

(117)

Here, l i q u i d molar densities p i , p2> and ρ are found as solutions to Equation 34, under appropriate assignments f o r the equation-of-state parameters. Agreement with experiment i s forced f o r H^ and V at the single states from which a^2 * 12 determined; calculations E

a n (

b

a

r

e

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

28

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

of H and V at other states, or of other excess functions at any states, constitute extrapolations. When compared with experiment, these extrapolations provide tests of the van der Waals mixing rules and (to a lesser degree) of the a b i l i t y of this equation of state to represent properties of the l i q u i d phase. The l i t e r a t u r e abounds with such comparisons and i t i s not our purpose to survey them here. Instead we show i n Figure 1 predicted values of the scaled excess functions V /xiX2» G /x^X2RT, H /x^X2RT, S /xiX2R> and Cp /x x R for the system argon(l)/krypton(2). Purecomponent parameters were estimated as described above from satura t i o n data compiled by Vargaftik (5_). Parameters a i 2 and b ^ ^ estimated from equimolar values of H and V at zero pressure, as given by the correlations of Lewis et a l . (6_) for H at 116.9 Κ and Da vies et a l . (7_) for V at 115.77 K. Both pressure and temperature effects are i l l u s t r a t e d i n Figure 1. P a r t i c u l a r l y to be noted are the e s s e n t i a l equivalence of the 0 bar and 1 bar isobars at 120 K. This j u s t i f i e s the frequent use of the zero-pressure l i q u i d state i n calculations of excess functions from equations of state. For example, one obtains for the van der Waals equation at zero pressure an e x p l i c i t expression for the l i q u i d density: E

E

E

E

E

1

2

w

E

r

e

e

E

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

e

P*(vdW;P=0) - ^ ( 1 +

7

f ^)

Moreover, the expression for the excess enthalpy s i m p l i f i e s to H (vdW;P=0) - - ( E

a

p

-

ï

x

a 1

p

i i i

)

Next we examine the composition dependence of Henry's constant ^1;2,3 f ° solute species 1 i n a mixed solvent containing species r

2 and 3.

Here, the solute-free mole fractions x^ and x^ are

appropriate measures of composition.

Subtleties of behavior are

n i c e l y displayed through the "excess" quantity ^ r r ^ . 2 3» defined as n

n

X

x

( 1 1 8

* ^ ; 2 , 3 -= *-«%Z,3 " 2 ^ 1 ; 2 " 3 ^ 1 ; 3 where'^^2

a n c

* ^1.3

solvents 2 and 3. T,P,

a

r

e

>

Henry's constants for species 1 i n pure

The comparison i n Equation 118 i s made at uniform

and composition, and £η^/ί „ i s i d e n t i c a l l y zero i f the three * ' Ε species form an i d e a l solution; however, £n^f. ^ i s not a true excess function as defined by Equation 37, because ζηΊψ^^ 3 ia mixture molar property. '' According to Equation 95, Henry's constant i s proportional to the fugacity c o e f f i c i e n t at i n f i n i t e d i l u t i o n . Combination of Equations 118 and 95 thus y i e l d s the general result 9

# 9

s

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

n

o

t

A B B O T T A N D Ν ASS

Classical Solution Thermodynamics

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

I.

Figure 1 (A,B). Scaled van der Waals excess functions f o r l i q u i d mixtures of argon( 1 ) and krypton(2). (The 10,000-bar isobar i s not shown i n A) .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

29

30

EQUATO INS OF STATE: THEORIES AND APPLICATIONS

G /x^x^RT v s . x^ P=0 bar_

a t T = 120 Κ

"p^Too"

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

P=1000

P=10000

0.0

0.2

0.4

0.6

0.8

1.0

(C)

Figure 1 (C,D). Scaled van der Waals excess functions f o r l i q u i d mixtures of argon( 1 ) and krypton(2).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

A B B O T T A N D ΝA S S

31

Classical Solution Thermodynamics

0.30

0.25

0.20

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

0.15

0.10 P=10000 0.05

H /x^x^RT v s . x^ at T = 120 Κ

0.00 0.0

0.2

0.4

0.6

0.8

1.0

(Ε) 0.20<

T=115K

0.16,

0.12

0.08

0.04

Τ=14θ"κ H*Vx^x^RT v s .

X

^·

i ^ ^ ^ - -

at P = 0 0.00 0.2 -0.04

0.4

0.6

0.8

(F)

Figure 1 (E,F). Scaled van der Waals excess functions f o r l i q u i d mixtures of argon( 1 ) and krypton( 2) .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

P=10000 bar

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

P=1000

a t T = 120 Κ -0.30 0.0

0.2

0.4

1.0

0. (G)

-0.15 T=115 Κ -0.20

-0.25

-0.30 T=140 -0.35 S^/x^x^R v s . -0.40

X

l

at Ρ = 0 -0.45 0.0

0.2

0.4

0.6

0.8

1.0

(H)

Figure 1 (G,Η). Scaled van der Waals excess functions f o r l i q u i d mixtures of argon( 1 ) and krypton( 2) .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

A B B O T T A N D Ν ASS

33

Classical Solution Thermodynamics

0.10·

o.oo. P=1000 bar -0.10. P=100 -0.20-

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

P=10

C /x-.x R v s . Ρ 1 2 n

at -0.60J\ 0.0

1

N

\

T = 120 Κ

1 0.2

0.4

0.6

0.8

1.0

(I)

T=115 Κ

0.8

1.0

Figure 1 ( I , J ) . Scaled van der Waals excess functions f o r l i q u i d mixture of argon (1) and krypton ( 2 ) .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

34

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

(119)

Αη

Γ;2,3 ~2^Γ;2 " 3 *Γ;3 Χ

n

* ^l;2,3

χ

For the van der Waals equation, £ηφ" i s given by Equation 98. With quadratic mixing rules f o r parameters a and b, we f i n d from Equations 98, 99, 106, and 107 the following expressions f o rthe fugacity c o e f f i c i e n t s : 2(xj>b *<;2,3

2 ( x

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

12

( v d W )

a

2 12

+ Φι Η>' 1-b* 3

*3*13W

+

,

md-b'p'jz

RT

(120)

1

Λ

» (2b -b )p £t^, (vdW) - — : r 22 2 j î î N

1 2

9 1 ; Z

2 2

b

2 a

2

p

p

12 2

—

£n(l-b p )Z 22

Λ

(2b

b

2

(121) 2

)

οο » 13" 33 P3 £ηφ, .(vdW) = — — ' 33 3 b

2a 13 3 — —

p

p

- £n(l-b p )Z 33

Solute-free mixture parameters a

1

and b

f

3

(122)

3

are given by

2 a* = ( x p a

2 2

2 + (x^) a

3 3

+ 2x£x^a

2 b' = ( x p b

2 2

2 + (x^) b

3 3

+ 2x^b

23

(123)

2 3

(124)

and l i q u i d densities and compressibility factors are found as solu tions to the equation of state. For the present exercise, ten parameters are needed: &22*22> 33> 33> 12> 12> 13> 13> 23> 23* Ideally, the purecomponent parameters would be estimated from liquid-phase data (e.g., as i n the excess-function example), and the interaction para meters from liquid/vapor e q u i l i b r i a and g a s - s o l u b i l i t y data. We adopt f o r this example a more straightforward approach. Parameters an * ii estimated from c r i t i c a l constants Τ ^ and V ^ v i a the c l a s s i c a l relations b

a

b

a n c

a

b

b

a

r

a

b

a

a

n

d

b

e

0

c

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

Classical Solution Thermodynamics

A B B O T T A N D Ν ASS

a,, = ii

*jT RT 8

35

V c.^

(125)

(126)

Interaction parameters are determined from the conventional com bining rules

a

ij

b

- "-'ijX'ii-j/' (1

)(b

+

2

( 1 2 7

2

>

(128)

ij - ^i ii V'

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

3

where parameters k^j and £jj are pure numbers, of absolute value less than unity. We wish to i l l u s t r a t e the effects of varying •Ε k . and £ on the "excess" quantity fcn^f-, ~, i . e . , the sens i t i v i t y of mixed-solvent Henry's constants to the numerical values of the interaction parameters. The parametric study i s done for a simulated ternary system at 300 Κ and 1 bar, i n which hydrogen(l) i s the solute, and n-heptane(2) and n-decane(3) compose the mixed solvent. Parameters for the pure f l u i d s are obtained from Equations 125 and 126 and parameters a23 and b23 are fixed once and for a l l by setting k23 = £23 0 Equations 127 and 128. Assignment of numerical values to 9

=

k^

3

and £ ^

then permits c a l c u l a t i o n of An^f-j^ 3 from Equations 119

through 124. Numerical results are displayed on Figure 2. Figures 2A and 2B i l l u s t r a t e the effects of independently varying the energy interac t i o n parameters; here, we have set £ 1 2 * &13 ~ 0· Figures 2C and 2D s i m i l a r l y show the effects of varying £ 1 2 and £ 1 3 , with k\2 ^13 0. The results confirm that mixed-solvent Henry's constants, l i k e excess functions for l i q u i d mixtures, can serve as probes for assessing mixing rules and combining rules for PVTx equations of state. =

=

Closure Connections between the PVTx equation-of-state formalism and the conventional apparatus of c l a s s i c a l solution thermodynamics are cleanly exposed through a few unifying concepts, e.g., generalized deviation functions, generalized p a r t i a l properties, and component fugacity c o e f f i c i e n t s . We have found the notion of p a r t i a l equation-of-state parameters to be p a r t i c u l a r l y helpful, because i t allows one to postpone questions r e l a t i n g to composition dependence u n t i l they r e a l l y need to be addressed. Much of the substance of this communication resides i n d e f i n i tions and generalizations, and i n the summaries of working formulas collected i n the tables. To keep the paper to a reasonable length, we have provided examples and i l l u s t r a t i o n s for but a single

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

Figure 2 (C,D). van der Waals Henry's constants at 300K and 1 bar, f o r hydrogen( 1 ) i n mixed solvents containing n-heptane(2) and n-decane(3).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

38

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

equation of state: the van der Waals equation. The p r i n c i p l e s and procedures are of course easily applied to other, more r e a l i s t i c equations of state. Exercises i n synthesis necessarily build on precedents, i n this case too diffuse and numerous to c i t e i n d e t a i l . We are however pleased to acknowledge as general sources of i n s p i r a t i o n the published researches of P.T. Eubank, K.R. H a l l , the late A. Kreglewski, M.L. McGlashan, K.N. Marsh, J. Mollerup, S.I. Sandler, R.L. Scott, K.E. Starling, and J. V i d a l . To these and to other equation-of-state enthusiasts we acknowledge our indebtedness.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

Acknowledgments This work was p a r t i a l l y supported by the National Science Foundation under Grant No. CPE-8311785. K.K.N, i s pleased to acknowledge support provided by an AAUW Educational Foundation Fellowship, i n the form of Conoco Engineering/Mobil Awards. Eugene Dorsi assisted with the calculations. Legend of Symbols a,b

=

parameters i n van der Waals equation of state

a^

=

a c t i v i t y of species i

A

-

molar Helmholtz energy

Cp,Cy

=

molar heat capacities

f^

=

fugacity of pure i

f^

-

fugacity of species i i n solution

f°

=

standard-state fugacity of species i

G

=

molar Gibbs energy

H

=

molar enthalpy

=

Henry's constant for species i

=

arbitrary molar, or intensive (e.g. Μ = Ρ ) , property

=

constant - T,V,x

=

constant - Τ,Ρ,χ deviation function

=

constant - T,V,x

=

constant - Τ,Ρ,χ excess function

M d M

deviation function

D M e M

excess function

Ε M Μ

Γ

M

s

constant - T,V,x constant - Τ,Ρ,χ

residual function residual function

M*"

=

t o t a l property Ξ nM

M^

=

generalized p a r t i a l property

M^

=

constant - T,P p a r t i a l property

»

constant - T,V p a r t i a l property

ΔΜ

-

constant - Τ,Ρ,χ

property change of mixing

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

A B B O T T A N D Ν ASS 0

γ

ΔΜ^

=

molar property change of vaporization of pure i

η

•

amount of substance ("mole number")

Ρ pSat

• _

pressure liquid/vapor saturation pressure of pure i

R

=

universal gas constant

S

=

molar entropy

Τ

-

absolute temperature

U V

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

Classical Solution Thermodynamics

= a

molar internal energy molar volume

χ

-

mole f r a c t i o n

X,Y

-

arbitrary intensive variables

Ζ

=

compressibility factor Ξ PV/RT

Greek Letters 3

=

γ κ

= =

π ρ

volume expansivity = V ^ a V / a T ) r ,x a c t i v i t y c o e f f i c i e n t of species i isothermal compressibility = -V (3V/3P) ι ,x

=

absolute a c t i v i t y of species i

=

chemical potential of species i

= s

arbitrary equation-of-state parameter molar density

τ

=

"coldness" = Τ *

=

fugacity c o e f f i c i e n t of pure i

φ^

-

fugacity c o e f f i c i e n t of species i i n solution

Superscripts id

=

denotes an i d e a l - s o l u t i o n property

ig

•

denotes an ideal-gas property

mod

=

denotes a model mixture property

«

=

denotes a property of a species at i n f i n i t e d i l u t i o n

Literature Cited 1. Van Ness, H.C.; Abbott, M.M. "Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria," Appendix C, McGraw-Hill, New York, 1982.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

39

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch001

40

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

2. Abbott, M.M.; Prausnitz, J.M. "Generalized van der Waals Theory: A Classical View," manuscript in preparation. 3. Reis, J.C.R. "Theory of Partial Molar Properties," J. Chem. Soc., Faraday Trans. II 1982, 78, 1595. 4. Abbott, M.M. "Higher-Order Partial Properties," seminar presented at the University of California, Berkeley, 21 Sept. 1983; unpublished notes. 5. Vargaftik, N.B. "Tables on the Thermophysical Properties of Liquids and Gases," 2nd Edition, Wiley, New York, 1975. 6. Lewis, K.L.; Lobo, L.Q.; Staveley, L.A.K. "The Thermodynamics of Liquid Mixtures of Argon + Krypton," J. Chem. Thermodynamics 1978, 10, 351. 7. Davies, R.H.; Duncan, A.G.; Saville, G.; Staveley, L.A.K. "Thermodynamics of Liquid Mixtures of Argon and Krypton," Trans. Far. Soc. 1967, 63, 855. RECEIVED November 5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2

The Collinearity of Isochores at Single- and Two-Phas Boundaries for Fluid Mixtures 1

2,3

2

2

John S. Rowlinson , Gunter J. Esper James C. Holste , Kenneth R. Hall , Maria A. Barrufet , and Philip T. Eubank 2

2

1

Physical Chemistry Laboratory, Oxford University, Oxford 0X1 3QZ, United Kingdom Department of Chemical Engineering, Texas A&M University, College Station, TX 77843

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

2

Fluid isochores for mixtures of fixed overall composition generally change slope on passing across the isopleth (dew-bubble point curve, DBC) from the homogeneous to heterogeneous phase region on a pressure/temperature diagram. A thermodynamic proof is given which shows the isochores to be always collinear at the cricondentherm (or any temperature extremum), rather than at the mixture critical point. The proof agrees with a different, more mathematical proof given earlier by Griffiths. These thermodynamic proofs are supported by our new, high-precision density data for the CH /CO and N /C0 equimolarbinariesin addition to previously published measurements for He/ He and for H /CH at Duke University and Rice University, respectively. Collinearity of the isentropes at the cricondenbar is also demonstrated and supported by recent calculations using our CH /CO data. Because fluid isochores for both pures and mixtures in the homogeneous region are approximately linear, most accurate equations of state (EOS) begin with this premise and then add correction terms for curvature. The present results contain thermodynamic constraints for such EOS. 4

2

2

2

3

4

2

4

4

2

Figure 1 i l l u s t r a t e s a t y p i c a l dew-bubble point curve (DBC) or isop l e t h f o r a binary mixture of f i x e d o v e r a l l mol f r a c t i o n , Zy The mixture c r i t i c a l point (CP) i s shown t o l i e between the point of maximum pressure, the cricondenbar (CB), and the point of maximum temperature, the cricondentherm (CT). However, i f the slope of the c r i t i c a l locus, (dP /dT ), i s positive for a particular z^ (1 = more v o l a t i l e component?), £hen the CP w i l l l i e o u t s i d e the CB/CT gap. This occurrence i s usual for z^ > 0.8 causing a CP to the l e f t of the •Current address: Institut fur Thermo- und Fluid-dynamik, Ruhr-Universitàt, D-4630 Bochum 1, Federal Republic of Germany. 0097-6156/86/0300-0042S06.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986. two

critical components.

(T)

a

point

for

the

as

the

CB a n d for mixture

respectively,

the are well

as

mixture

CT

binary

showing typical

diagram for

equimolar).

curve (here,

cricondentherm,

composition

(isopleth)

(P)/temperature

DBC

the

pure

CP d e n o t e s

of

the

and

whereas

cricondenbar

constant

and

A pressure

locus

that

the

mixture

of

1.

c r i t i c a l

Figure

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

44

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

CB

but

below

also the

The the DBC.

density

Figure

shown

two-phase

to

(dP / d T ) L

when

for

some

have

l i m i t i n g

are

from

CT t o

the the

slope

of

fixed

on

at

phase

location

mixtures

CB.

slopes

identical single

of

P

> 0 c a u s i n g a CP

P

°

m o n o t o n i c a l l y as

CP t o

a steeper

the

of

increases

2 from

sides

slope

independent

BASIC

The

the

at

the

DBC f r o m

CT.

is

we

Above

steeper.

CP, which

traverse

density

two-phase

the

the

side

low

iso-

side

of

the

single

the

the

CT,

These

indeed

the

results

f a i l s

to

composition.

IDENTITIES

Imagine

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

in

are

isochoric are

< 0.1

°

We p r o v e t h a t

exist

ζ

for

saturation

DBC a s

chores and

happens

CT

a

fixed

gas

of

total

dV*

= V* (dT)

volume

cell

g

volume

V .

containing

liquid

of

total

volume V

and

Then,

+ V * (dP)

+

(dn^

+

(1)

(dn ) 2

where

V* Τ

=

(3V*73T)_ I I P,n ,n 1

,

vjΡ

2

(3V*73P)

I I T,n ,n

T

1

2

0

and

dn.

=

(V* + VÎjp

AV .

between

dy

=

t"ne

variables 3

addition

+ (Vp + V*)

dT

_ rr

-0

(V? -

V.)

-

diî :

The

= dyS

two

1

[2].

, η

= AV

dP

is

1

the

phases.

( Τ , Ρ, η

f = - sf

Two

(dT)

+ vj

equation

for

dV°

the

expression

for

1

y^

is for

of

the

fraction

( d T ) - AV

equation

2

to

of

2

partial

equations

from

the

(2)

(dn )

molar

volumes

connecting

equilibrium

the

four

conditions

(dP)

+

(3μ /3η,) 1

1

0

Τ,Ρ,η*

(dn,)

of

+

1

(3)

9

sum

mol

+ AV

difference

) follow

Τ,Ρ,η*

the

(dn,,)

(dn )

μ ι

where

similar

First

(3 /3η_)

x^

a

further

1

where

of

1 provides

Equation

where

cr

dn..

dy^ to

*

last in

two

terms

the

( d P ) = (3u /3y )

the the

mol less

(|

fraction volatile

T p

in

also

phase.

3 results

Equation

1

is

liquid

O i ^ / S x ^

1

the

a

with

similar

in

(dy ) - Ο ν / ^ Τ , Ρ

component

(dx^)

Equatinfe

vapor

phase.

(dx ) 1

The

(4)

analogous

is

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2.

$

•trn M

!^

It

Μ

/J %

"i

1

l

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

/

Ι I

1 ***

1 •

î ϋ t-t

f

/

Fy /

i

-

I

è

ψ

+

-

1

•Λ ,·f • Hit

1

—_|_ t

1

!

-Γ TTT 1

su

y i> k

1

:—_μ__ ! /

... ! /

&

;

"f

trif • ! 1,

!

ι !

1

ψ * \i I

y* ψ*

F

J*

I

/

• •J

... !

n.i

l 1

?"f

- -

"\

:::.|·:::

f y r

-

1

! 1

Ρ

4.. I

rrrrt .r-

/

Γ

"1

,,

FTi l "

ipi

i

45

Collinearity of Isochores

ROWLINSON ET A L .

rrrt-...

/

/

i

ι

lit I

•:.:|::::

-•·(·-·

—ή—

~ ; τ*

-41

itt*

4,

""*r

&é

~^

4.+U .ttr

M

3

1 Ι

1

p?

-a,

*H n

tH rfft til! tît.

i

.

if.l

+•<++

tM f n

u

"«

t.

fftf

:

ii'4

ϋ

jtt*

ft

4

1

4

tttt

;FTÎ

.rrr îttt:

tttt

τ Figure chores a

binary

the

mixture

isochores

isentropes Both

A qualitative

2.

and i s e n t r o p e s

the

choric

fixed

(lines

(lines

density

subscripted

of of

of

constant

constant of

exhibiting

the single

composition.

and entropy

i n order

inflection

diagram

i n both

Note

density,

entropy,

the p)

iso for

collinearity at

tracing

magnitude.

of

regions

the

S) a r e c o l l i n e a r

a r e monotonie

increasing

the nature

and two-phase

CT. at

of

The

the CB.

t h e DBC a n d a r e IIL is

the i s o -

locus.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

46

AS

(dT) - AV

2

(dP) = Ο μ / 3 γ )

2

2

1

τ p

(dy ) -

(δμ^χ^^ρ

1

(dx^)

(5)

A p p l i c a t i o n of the Gibbs-Duhem equation to each phase y i e l d s x Oyyax^p =- x Ομ /8χ ) and y ( 9 μ / dy ) ^ = 1

T

-y

(3y /3y )

2

2

(5) by x

1

2

p τ

2

·

2

1

ρ

T

1

M u l t i p l i c a t i o n of Equation

followed by addition

1

1

(H) by x

p

T

and Equation

1

yields

<x.AV.> (dP) - <x.AS.> (dT) - U x / y ) 2

Oy-^y^p

^y-j)

T

(6)

where <x..AV.> s X-.AV + x^àV and Δ χ a ( y - χ ). L i k e w i s e , m u l t i p l i c a t i o n or* Equation (4) by y and Equation (5) by y f o l l o w e d by addition provides 2

1

1

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

2

(dP) - (dT) = (Δχ/χ > Ο μ /9x ) Ί

2

1

(dx^

pT

(7)

Before combining Equations (2), (6) and (7), we modify Equation (2) by f i r s t noting that x

l

9

2

(dn/dT).. = x, (dn_/dT)„ + n 1 ν,η^ ,n 1 2 V,n^ ,n n

n

2

2

(dx?7dT) 1

v

n

(8)

n

ν,η^ ,n

2

and that a s i m i l a r equation can be w r i t t e n f o r the^gas phase. The reader i s reminded that dn^ = dn whereas n^ = n^ + n . M u l t i p l i c a t i o n of Equation (8) by y and of the gas phase equation by x followed by subtraction yields g

1

(y

- χ ) (dn /dT) = n*y

1

g

(dx /dT) + n x

1

where a l l t o t a l derivatives indicate a constancy n^ and n^. Likewise, (y

-

]

δ

X(|

) (dn /dT) = η χ 2

2

(dy^dT) + n'y

(9)

(dy /dT)

of t o t a l volume V,

(dx^dT)

2

(10)

Equation (2) together with Equations (9) and (10) f o r ( d ^ / d T ) and (dn /dT), respectively, becomes 2

il

g

1

(dP/dT) -

g

n (dx /dT)+n (dy /dT)<x.AV.>-(V?;+V )(Ax) — J — 1 (V£ + V ) (Δχ) 1

(11)

g

Equation (6) shows that (dy.j/dT) - [<x.AV >(dP/dT) - <x.AS.>] y /(Ax) Ο μ /3y ) i

2

1

1

pT

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(12)

2.

Collinearity of Isochores

ROWLINSON ET A L .

47

where

Oy^By^p

τ

= <χ.Δν.Χ3Ρ/3γ

)

1

(y^àx)

τ

=- <χ.Δ§.>( Β Τ ^

)

ρ

(y /Ax) 2

(13)

Substitution of Equations ( 1 2 ) and ( 1 3 ) into Equation ( 1 1 ) along with analogous equations for (dx^dT) and O y ^ B x ^ p ^ , results i n (dP/dT) = 3x

9x

^ ^ [ ( ^ • ( ^

dP

_

3y

9y

dP

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

g

(Vp+V )(Ax) (1*0 δ

T h i s equation i s s o l v e d f o r (dP/dT) w i t h the q u a l i t y , q Ξ. [ η / ( η A

n )]

1

= (vl/n ),

A

f

introduced as well as v* s O v / 3 T )

p

δ

ν ,

δ

+

v£ and

P" -(dP/dT) = g

[(1-q)v^+qv ](Ax)+(1-q)Ox /3T)p+q<x.AV >Oy /3T)p i

i

2

i

2

(15) g

[(1-q)Vp+qv ](Ax) + (1-q)(8x /3P) +q<x AV >(8y /3P) i

Because

(3Ρ/3Τ)

i

2

=- (3x /3T)

χ

2

T

i

(3P/3x )

p

2

i

2

T

, an alternate form of

T

Equation (15) i s g

[(1-q)v%v ](Ax)-(3P/3T) 1

-(dP/dT) =

where η

Α

I

v

· n*-(3P/3T) · η

_ 1_ [(1-q)v£+qv*]Ux)+n + η

= ( 1-qXy.AV.X3x /3P) 2

T

and η

δ

Yp

Xp

δ

^ —

(16)

δ

= q^.AV.X3y /3P) 2

T

PURE COMPONENT CHECK Both η and r\ are everywhere zero f o r a pure compound causing Equation (16) to reduce to -(dP/dT) - [(1-q) v* + qv^

- (Δν) (dq/dT) ]/[ 1-q)v£ + qv^]

(17)

For either pure components or mixtures of fixed o v e r a l l composition, q u a l i t y l i n e s form a f a m i l y of curves i n the heterogeneous r e g i o n i s s u i n g from the c r i t i c a l p o i n t . Any continuous path through the

American Chemical Society, Library

1155

18th St., N.W.

Washington, DX. 2003S In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

48

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

heterogeneous CP

via

CP,

Equation

f -(v

T

region

ending

line

tangency.

straight

(17)

p

= (3P/3T)

c

v

,

p

° f l u i d ) ,

=

the

collinearity

of

was

van

known t o

BINARY

fLxed

v ^ a n d vZ T r homogeneous

in

(dq/dT)

value

of

•+ vj* s o

= 0 at

the

q nearing

the

that

(dP°/dT)

=

r

^

phase

side

derivaties

v

T

are

slope the

of

vapor

der

both

the

divergent

c r i t i c a l

pressure

but

t h e i r

r a t i o

isochore

at

with

critical

curve

the

the

CP.

is This

isochore

[3].

Waals

M I X T U R E AT T H E CRICONDENTHERM ( C T )

the

CT,

(3T/3y ) 2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

(f

identically

At

-

I Here the

.

CP r e s u l t s any

P

f and

the For

contains

f /v )

at

(3P/3T)

and

p

Equation

and

<x AS > i

(13).

i

With

(3P/3y

are

)

finite

q unity,

n

g

are

so

infinite

that

= 0 and

<x AV > i

i

(3P/3T)

·

CT

while

Δχ,

zero

from

is S

n

=-

q

<χ.Δν\>· 1

yρ

1

(3y /3T) = (1). (0)-(f) =0, where f = finite (nonzero). Equation (16) then reduces to -(dP/dT) = (v^/v ) =- ( 3 P / 3 T ) or c o l l i n e a r i t y i s o b t a i n e d f o r any e x t r e m u m (maximum o r m i n i m u m ) i n t h e 2

p

g

temperature dew-point point

on

the

curve

curve

DBC.

but

the

isochores

including

a

the

diagram

for

a

diagram

for

here

case

are

the

mixture

critical

C0 /N 2

As

phase

first

"gas",

increases again

it

is

termed

"gas".

Both

are

collinear.

respectively,

the

a

GRIFFITHS

Levelt

an

earlier

Sengers

appeared and

as

[4]

an

Meyer

[5 ],

measurements. leads

us

to

believe

of

terse,

b r i l l i a n t

two

different

(ψ,ω) this

dC

their

He/

discussions

with

that

proof

to

whose

plane curve

this

We r e p e a t to

call

the

proofs

lead

to

the

same

the

derivatives

are I

is

DBC. II.

a

limited

of

of a

1

this wider

proof

pick

is

because

conclusion. ξ(ψ,ω)

discontinuous and

to

chemists

version

his

which

density

and

Griffiths

take

MT,

Behringer

isochoric

employed

Let

and

Griffiths

attention

can

a

finally,

CT

Doiron,

mostly

earlier,

argument

regions

the

DBC

analogous

and,

engineers

known

given

to

to of

He

a of

homogeneous

at

a backwards

The r e a d e r

across

separating

is

here it

proof

mathematical

quantities

function

contains

the

the

ρ

due

article

is

classical

"liquid"

proof

important

and m a t h e m a t i c a l .

standard

various

the

2

behavior

no CP n o r C B b u t of

and

previous

proof

Compared

concise A

in

physicists.

audience. these

noted

which Our

number

more

has

appendix

then ρ

OF

has

density

isochores

I S O C H O R I C C O L L I N E A R I T Y PROOF

3

the

bubble-

Figure

is

tne

on

on the

qualitative

which

(MT).

CT l i e s

is

Figure

2

CT a n d a m i n i m u m t e m p e r a t u r e once

the

it

exhibiting

point.

equimolar

that

where

i d e n t i c a l .

showing

binary

vapor/liquid

qualitative

assumed

unusual

conclusions

pressure/temperature fluid

We h a v e

in

The

thrice be

along

a

to

relate

continuous

a curve

differential

of

in ξ

the

along

is

= (3ξ/3ψ)

1

ω

dij, +

Ι

]

( 3 ξ / 3 ω ) * do) = ( 3 ξ / 3 ψ ) ^ ψ ψ ω

+

(3ξ/3ω)^ ψ

1

άω

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(18)

Collinearity of Isochores

ROWLINSON ET AL.

49

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

2.

3·

Figure with

no

A qualitative c r i t i c a l

diagram

point

(e.g.,

difference

between

isochoric

tracing

DBC f r o m

low

the

to

representing equimolar

slopes

high

on

a

binary

CO^/N^).

either

side

mixture

Note of

the

densities.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

the DBC

50

E Q U A T I O N S O F STATE: T H E O R I E S

where Ο ξ / δ ψ ) r e f e r s to the d e r i v a t i v e at the DBC side of r e g i o n ^ , etc. Then 1

[(3ξ/3ψ)* -

Οξ/3ψ)"

]

=-

[Οξ/3ω)ψ -

AND

APPLICATIONS

taken from

Ο ξ / 3 ω ) ^ ] du>

the

(19)

or δ Οξ/3ψ)

= - (da>/di|i) ·δ Οξ/3ω)

ω

(20)

where δ indicates the difference, Ι-ΙΙ. We are s p e c i a l l y interested i n δ(3Ρ/3Τ) , for which Equation (20) reads Ρ» 1 z

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

δ (3P/3p)

where

„

T

· δ (3P/3T)

(dT/dp)

the t o t a l derivative (dT/dp)

„

(21)

i s taken along

the

DBC.

This

derivative i s positive at low densities passing through zero at the CT to become negative at higher densities (see Figure 1). G r i f f i t h s proves that the left-hand side of Equation (21) i s always nonnegative so that δ(3Ρ/3Τ) must be negative below the CT, positive above Ρ» 1 the CT and thus zero i t s e l f at the CT. To prove that δ(3Ρ/3ρ) £ 0, Equation (20) i s again applied 1 but with ξ s ρ, ψ = Ρ and ω = ζ^ at constant temperature: Z

τ

,Z

δ Ορ/3Ρ) The

ζ

= " (dz /dP)

> τ

1

derivative

· δΟρ/Sz^p^

T

(22)

(3ρ/3Ρ)

> 0 for mechanical 1 ' A Maxwell r e l a t i o n ,

stability

(see

Ref.

z

[2], p. 18). Ορ/3ζ ) Ί

Ρ ί Τ

= - ρ

where Δ = δ(3ρ/3Ρ)

ζ

2

(3Δ/3Ρ)

(23)

> τ

ζ

- μ^, i s then used to obtain -

^

ρ

2

(dz /dP) 1

T

· δ(3Δ/3Ρ)

^

ζ

(24)

Equation (20) i s applied a t h i r d time with ξ = Δ, ψ = Ρ and ω s ζ^ to replace δ(3Δ/3Ρ) with δΟΔ/Βζ,. ) : D

Ζ^,Ι

δ(3ρ/3Ρ)

Now

ζ

ρ

> χ

(3Δ/3ζ )

2

τ

Ι Γ , Ι

(dz^dP)

2

· δ (3Δ/3ζ ) Ί

ρ

χ

(25)

i s zero i n the two-phase region (due to a phase rule

I r, 1

constraint) and i s nonnegative i n the homogeneous region f o r material s t a b i l i t y or (3 G /3z.,) > 0, where G i s the molar Gibbs energy m ι ρ 11 πι

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2.

ROWL1NSON ET AL.

Collinearity of Isochores

(see Réf. [2], p. 115).

51

Hence, δ(3ρ/3Ρ)

< 0 or δ(3Ρ/3ρ) > 0 -j » i i » * £ 0 for mechanical s t a b i l i t y . This completes the z

because (3p/3P)

z

Ζ

Γ proof of G r i f f i t h s . EXPERIMENTAL RESULTS ο

h

The 80.5? He/19.5? He mixture of Ref. 5 from Duke University showed collinear isochores at the CT for a system with the CP to the l e f t of the CB on the P/T diagram. F i g u r e 7 of Ref. 5 a l s o showed the slope (dP/dT) to i n c r e a s e but weakly w i t h temperature and w i t h o v e r a l l density i n the heterogeneous region. Later, isochoric measurements for a 20.05? H /79.95? CH^ mixture at R i c e U n i v e r s i t y by Kobayashi and coworkers [6] i l l u s t r a t e d c o l l i n e a r i t y at the CT for a system without a C B — a t least not within 30 degrees of the CT temperature. At the same time, measurements for C0 binaries i n our labora tories at Texas A&M University showed isochoric c o l l i n e a r i t y for a nearly equimolar mixture of CO /CH^ as seen i n Figure 4. The CP for t h i s mixture i s not known exactly but has been estimated using a BACK equation of s t a t e (EOS). L a t e r , we measured a n e a r l y equimolar mixture of CO^N^ r e s u l t i n g i n the quantitative Figure 5, a dramatic i l l u s t r a t i o n of isochoric c o l l i n e a r i t y at the CT. As discussed i n a previous s e c t i o n , t h i s mixture has no CP nor CB but a CT and a minimum temperature (MT) as shown by the qualitative Figure 3» based partly on necessarily less precise data at the higher densities.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

2

p

COLLINEARITY OF ISENTROPES In the fundamental equations of thermodynamics for pure components the v a r i a b l e s are Ρ, T, S and V. Ρ and Τ are analogous p o t e n t i a l functions of zero degree of mathematical homogeneity whereas S and V are analogous f u n c t i o n s of f i r s t degree. The i s e n t r o p i c slope, (3P/3T) , i s c o l l i n e a r w i t h the vapor p r e s s u r e curve at the CP f o r pure components as i s the isochoric slope. For binary mixtures the r o l e of S and V can be r e v e r s e d i n any of the proofs given above w i t h the r e s u l t t h a t i s e n t r o p e s are collinear at the CB. The qualitative nature of binary isentropes i s i l l u s t r a t e d i n F i g u r e 1. For mixtures, i t may be s a i d t h a t "volume i s p r e j u d i c e d i n favor of temperature whereas entropy f a v o r s pressure" . To confirm experimentally the c o l l i n e a r i t y of isentropes at the CB, we have taken the equimolar CO^CH^ data of F i g u r e 4 and c a l c u lated entropy increments on isotherms v i a the identity S

p

-2

US) m

Ί.

= T

/ I

OP/9T) ρ

(dp/p ) 2

(26)

p

1

To traverse temperature i t i s convenient to use the entropy residual f u n c t i o n , S (p,T): r

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

52

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

210

230

250 TEMPERATURE

270

(K)

XI = 0.476 X2 = 0.540 X1 - CO2 and X2 - CH4 Figure

4.

carbon

dioxide

dentherm. imental

but

Binary

mixture

data

illustrating

The c r i t i c a l estimated

point

from

for

52.40

isochoric a

(CP) o f

BACK

mol

% methane/47.60

collinearity this

equation

diagram of

state

at

the

is

not

[10].

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

mol %

criconexper-

2.

Collinearity of Isochores

ROWLINSON ET AL.

Ρ (s£/R)

1

= Τ

'

!

53

Ρ 1

OZ /3T

)

m

(dp/p)

ρ

I

+

0

1

^ V

)

/

p

]

T

d

p

(

2

7

)

"o

where Ζ (= P / p R T ) i s t h e c o m p r e s s i b i l i t y f a c t o r a n d » S (T,p) m m m S * ( T , p ) w i t h S* the p e r f e c t gas m i x t u r e v a l u e b a s e d upon a r e f e r e n c e p r e s s u r e and t e m p e r a t u r e , Ρ and Τ , r e s p e c t i v e l y . Like Equation (26),

Equation

therm

crosses

we

use

Equation

examine. point or

pressure

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

(3P/3T) phase

(27)

or

ρ

bubble suffers

a

p

D

,

and second

and

when

a

(3)

at

at

a

must to

β

isotherm

by t h e

easier at

the

a

to

dew-

pressure

derivatives, in

(2)

the

across

a higher

is

of

is

integrated

to

upper

function

of

be

ρ

(26)

iso

Although

bubble-point

D

ρ

When t h e

taken.

DBC f i r s t

P p»

from

the

continuous

discontinuity

(27)

density

simply replaced

is

the

discontinuities

zero

ρ

be

Equation

crosses

, Equation

to

p

must

calculations,

of

frèm

integration.

precautions

isotherm ρ

liquid,

is

S

)

(1)

from

point

isotherm,

the

Because / 3T

compressed

isothermal

our

density,

.

steps:

region

for

that

a n d (3Z

separate

assumes

DBC, s p e c i a l

Imagine

density,

the

(27) the

three two-

density

s u p e r c r i t i c a l ,

dew

point.

density

Along

but

(3S

m

in the the

/3p)

T

t h e DBC.

Values of S c a l c u l a t e d from E q u a t i o n (27), see T a b l e I, were t h e n g r a p h e d as S versus pressure along isotherms for the CO^CH^ b i n a r y . T h e s e i s o t h e r m s m u s t show a p o s i t i v e i s o b a r i c h e a t capacity, C = T (3S / 3 T ) , for thermal s t a b i l i t y . The c o u n t e r derivP,z m » 2 n

p

z

2

2

ative,

(3S/3P) ^ - ~p~ · (3p/3P)_ · (3P/3T) „ , is usually m ι , Z p ι,Zp P' 2 n e g a t i v e but (3P/3T7 may be n e g a t i v e i n t h e t w o - p h a s e r e g i o n n e a r Ρ» p T

Z

the

CP.

With

results

were

these

made o n t h e

high-preci.sion, differentiation the

case

points

in

the

in

a

S/P

this

graph,

supports We Figures (3T/3V

(i.e.,

the

4

)_ m P,z

7.

6,

withstand

and 7. .

of

Equation

region.

Here

of

crossplot

isentropic

data

positive Both

pure

so

that

the

methane

collinear

with

the

always

fewer

data

phases

are

be

of

at

shown

pure C 0

at

two

that

the

of

on at

Figure exhibit

vapor

pressure

lower

Figure

7

each

(T,P)

of

is

(ρΟ /Τ)

2

slopes 2

γ

2

(3 P/3T )

2

on

a P/T

t h e CB.

p,z these

isotherms made

desirable

2

isentropes

and

our

well

temperatures),

> (3P/3T)

n

can

our was

2

erally

While not

have

of

are

lower

collinearity

(3P/3T) S,z

it

is

when two

isentropes

and

The d i f f e r e n c e

Further,

f i r s t

massaging of

more

pressures

that

we

such

equilibrium

judicious

a

(27),

two-phase

6.

mixing capability.

Although

checked

wards. (3P/3T)

usually

without

higher

notion

also

measurements

slight,

Figure

diagram—Figure entropies

Figure

concerned about

cell

our

diagram,

integration

are

some s m o o t h i n g o f

S/P

two-phase

blind

Following the

and

i n mind,

sensitive

P(p,T)

and s e c o n d

present

criteria

7 are this curve

is

C

S,z

·

gen-

2

concave

up

behavior

with

at

Z

c' 2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

the

CP.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Figure 5. Binary mixture data for 44.70 m o l % carbon d i o x i d e / 5 5 . 3 0 mol % nitrogen i l l u s t r a t i n g isochoric c o l l i n e a r i t y at the c r i c o n dentherm for a system known to exhibit the qualitative behavior of Figure 3 at the higher pressures and densities.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

(73

δ

η

r

ΊΟ

>

α

>

GO

m

Ξ

m Ο

Η 3C

m

>!

en Ο m

ζ

δ

c

m Ο

4^

ROWLINSON

ET

AL.

Collinearity of Isochores

T a b l e I: T h e c a l c u l a t i o n o f E n t r o p i e s f o r 5 2 . 4 0 m o l % C H Reference State: P

0

55 a n d 47.60 m o l % C 0

4

= 0 . 1 0 1 8 2 5 M P a (1 A t m . ) , T

0

2

= 278.15 Κ

The entries are: pressure (Roman), ( M P a ) , and
Density ( m o l / m T(K)

820.871 0.7817

1246.85 2.8496

8074.69 6.2404

4799.85

7589.85

8.8193

12.267

10509.1 16.098

) 11850.4 18.246

16560.7 32.090

18619.5 48.550

800.00 2. 080

S. 550

4.657

5.270

5.980

6.569

6.807

7.599

7.050

0.7511

2.7189

5.8710

8.1789

11.103

14.219

15.930

27.664

42.286

288.75 2.207

S. 679

4.787

5.4 08

6.112

6.702

6. 939

7.720

8.085

0.7085

2.5362

5.3479

7.2619

9.4650

11.644

12.823

21.687

33.787

278.20

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

2. S 86

-8.857

0.6724

2.3802

2.5S9

-4.012

0.6628

2.3386

4.967

5. 589

6.297

6.890

7.129

7.020

4.8953

6.4597

8.0573

9.4941

10.306

16.774

-£.275 26.764

260.08 -5.128

7.042

7.280

5.745

6.451

4.7738

6.2429

7.6807

8.9282

9.6523

5.164

5. 786

6.492

7.17£

7.488

4.7630

6.2236

7.6516

8.8780

9.5946

7.162

7.874

7.6266

8.8349

9.5450

000 15.493

-8.42S 24.927

256.61 2.580 0.6620

-4.058 2.3350

-8.205 15.380

500 24.764

256.80 -2.584 0.6612

-4.057 2.3318

5.167

5. 789

4.7537

6.2069

8.643

«S. 0 5 7

15.282

24.625

256.08 2.588 0.6606

-4.060 2.3290

5.170

5. 792

7. 886

8.141

8.405

4.7455

6.1879

7.6044

8.7966

9.5010

15.196

5.169

5.823

7.853

8.190

8.454

9.220

9.510

4.7096

6.1055

7.5084

8.6310

9.3108

14.823

23.964

-9.177

-9.471 24.501

255.80 2.590 0.6578

-4.062 2.3167

254.78 2.602 0.6562

-4-074 2.3098

5.182

5.907

7.-^1

8.378

8.652

0.^57

0.72£

4.6892

6.0584

7.4535

8.5365

9.2023

14.610

23.659

-8.725

254.20 2.609 0.6521

-4.081 2.2921

5.189

6.066

7.505

8.447

4.6372

5.9392

7.3147

8.2974

5.207

6.462

-& 072

4.6150

5.8884

7.2554

8.2297

8.8123

8.946

-9.212

7.1988

8.1650

8.7012

8.9287

-0.568 14.072

9.864 22.886

252.78 2.621 0.6504

-4.099 2.2845

-8.938

9.226

-IO.O4 13.842

10.5^ 22.556

252.10 2.685

-4-106

5.214

6". 530

4.5938

5.8398

-IO.O4

10.5./

0.6487

2.2773

2.642

•4-114

5.221

6.596

£.155

8.953

9.233

10.07

10.87

0.6414

2.2454

4.4790

5.6251

6.9487

7.8793

8.2954

12.681

20.856

13.624

22.242

251.50

248.84 2.67S

4.144

5.816

6.90S

8.421

9.189

9.363

0.6321

2.2049

4.2861

5.3540

6.6329

7.5183

7.9222

-10.08 11.451

-10.5£ 19.114

245.50 2.714

-4-184

0.6237

2.1686

2.750

-4.22I

0.6164

2.1367

-5.602 4.1134

7.105 5.1112

-8.518 6.3499

-9.216 7.1950

9.370 7.5878

-10.09 10.374

-10.46 17.561

242.50 -5.821 3.9621

-9.223

7.246 4.8986

6.1022

6.9119

9.373 7.2951

- 1 0 . OS 9.4364

-10.51 16.206

289.87 -2.782 0.6003

-4.258 2.0662

-5.999 3.6283

-7.366 4.4292

8.625

9.235

9.878

10.07

5.5553

6.2871

6.6490

7.3829

8.548

9.838

9.588

-10.55 13.236

284.08 -2.853

-4.828

-6.180

-7.323

-10.44

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

-10.80

56

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

-2

DBC^

Ο PU

H

ω

-10

10 PRESSURE .

.

0

0

•

•

Δ

Δ

DBC 2 6

30

20 (MPa) 239.438

•

*

*

234.076

2 4 5.5

•

·

·

248 . 849

252.1

2 54

•

·

·

2 5 5.8

256.608

2 73.2

2 88.752

2

0.081

•

•

•

Figure 6. The calculated entropies for equimolar m ethane/ car bo η dioxide mixture (see caption of Figure M). The slope of isotherms i s (-pr/v ) whereas the isobaric heat capacity, C , i s roughly the average temperature m u l t i p l i e d by the isobaric ^entropy increment divided by the temperature increment.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ROWLINSON ET A L .

Collinearity of Isochores

57

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

2.

Figure 7. A pressure/temperature diagram i l l u s t r a t i n g isentropes (dashed l i n e s ) f o r the n e a r l y equimolar methane/carbon dioxide mixture (see c a p t i o n of F i g u r e 4). S c a t t e r i n the c a l c u l a t e d entropies prevents a d e f i n i t i v e conclusion from these data alone of i s e n t r o p i c c o l l i n e a r i t y at the cricondenbar. However, these data are s u p p o r t a t i v e of that proof and show that c o l l i n e a r i t y occurs on the upper part of the DBC between 237-255K.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

58

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

THE ISOCHORIC INFLECTION LOCUS (IIL) For pure components, i t appears that the IIL always i n t e r s e c t s the vapor pressure curve at the CP and again, f o r some compounds, at a l i q u i d density roughly twice c r i t i c a l . Recent density data at Rice U n i v e r s i t y (20.05 m o l % H /79.95 m o l % CH.) and Texas A&M U n i v e r s i t y (52.40 m o l % CH.A7.60 m o l % CO and 44.70 m o l % C0 /55.30 m o l % N ) suggest that the IIL may i n t e r s e c t the DBC at the CT f o r b i n a r y mixtures. We know of no thermodynamic proof for t h i s behavior whether for pure components or mixtures of f i x e d composition. A n o n c l a s s i c a l proof might be required for pure compounds but i f the mixture inter s e c t i o n i s indeed the CT r a t h e r than the CP, then a c l a s s i c a l proof should be p o s s i b l e . C o n s i d e r a b l e importance i s attached to the IIL f o r not o n l y i s i t the locus of the extremum of the heat capacity C but also i t s different qualitative behavior at v,z d e n s i t i e s above c r i t i c a l for pure compounds has been r e l a t e d to molecular effects, such as polarity [7_]. 2

2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

2

DISCUSSION Although we have presented proofs for (1) isochoric c o l l i n e a r i t y at the CT and (2) i s e n t r o p i c c o l l i n e a r i t y at the CB assuming a b i n a r y mixture, i t i s obvious that these r e s u l t s apply a l s o to m u l t i component mixtures because the d e r i v a t i v e s are at constant o v e r a l l composition. From another viewpoint, any multicomponent system may be considered a "pseudo-binary" and the above derivations repeated. The derivative (3P/3T) i s important because a number of ρ, ζ 2 thermophysical property measurements are performed at constant volume and composition. For example, Nowak and Chan [8] have extended the "adiabatic non-flow saturation calorimeter method" to mixtures i n the heterogeneous r e g i o n . T h i s method f o r measuring l a t e n t heats of vaporization and heat capacities was made famous by Osborne, Stimson and Ginnings (OSG) working p r i m a r i l y on water at NBS i n the 1 930 s. The approximate working equations of Ref. 8 r e l a t e the apparent s p e c i f i c heat of the c a l o r i m e t e r to the d e s i r e d i s o b a r i c heat capacity, C , f o r both the l i q u i d and vapor at s a t u r a t i o n . Ρ Because mixtures are d i f f i c u l t to d u p l i c a t e , data are most con v e n i e n t l y taken along a s e r i e s of i s o c h o r e s . In the homogeneous phase region, the density i s lowered to the next isochore by simply exhausting part of the f l u i d or by expanding i t i n t o a second volume—as i n the Burnett-isochoric method for the density measure ments of Table I. F i n a l l y , the present work has an important bearing upon mixture equations of state (EOS). Many physical EOS for both pure components and mixtures begin w i t h s t r a i g h t i s o c h o r e s (van der Waals EOS) and then add c o r r e c t i o n terms to account f o r c u r v a t u r e [ 9 ] . I t i s an experimental fact that isochores are nearly linear i n the homogeneous region on a pressure/temperature diagram for both pures and mixtures. The c o l l i n e a r i t y of i s o c h o r e s at the CT for mixtures provides a sensitive constraint for EOS parameters. f

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2.

Collinearity of Isochores

ROWLINSON ET AL.

59

ACKNOWLEDGMENTS We w i s h

to

and

recalling

the

for

thank

National

Petroleum Research

D r . Anneke the

proof

Science

Research

edgment

goes

Project

financial

to

the

Fund

and

University

(Ref.

for for

the the

6 and

Sengers

for helpful

Griffiths.

(ACS),

Processors

Production

acknowledged.

Institute

data

in

and C 0 / N 2

the

Co.

2

data

and

and t o

GRI p r o v i d e d

references

support

taken

of

Association,

Special

(Chicago)

Thermodynamics. CO^/CH^

discussions

The f i n a n c i a l

Gas

Amoco

Co. are

Gas Research

Manager,

support

University

of

Foundation,

and Engineering

Little,

Levelt

Dr.

the at

cited

Exxon

acknowl Frank

primary

T e x a s A&M from

Rice

7).

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch002

LITERATURE CITED [1] King, M.B., "Phase Equilibrium in Mixtures," Pergamon Press, Oxford, 1969. [2] Rowlinson, J.S., and Swinton, F.L., "Liquids and Liquid Mixtures," Third Ed., Butterworths, London, 1982. [3] Keesom, W.H., Communs. Phys. Lab. Univ. Leiden, (1901), No. 75. [4] Levelt Sengers, J.M.H., on a visit to Texas A&M University to deliver the Lindsay Lecture "Dilute Near-Critical Mixtures", September 27, 1984. [5] Doiron, T., Behringer, R.P., and Meyer, H., J. Low Temp. Phys., (1976), 24, 345. [6] Magee, J.W. and Kobayashi, R., Adv. Cryog. Eng. (1 984), 29, 943. [7] Kobayashi, R., "Phase and Volumetric Studies for Methane/Synthesis Gas Separations: The Hydrogen-Carbon Monoxide-Methane System," Final Report to the Gas Research Institute (GRI-84/0084), Rice University, March 15, 1984; "Generalized Understanding Through Generalized Experiments," Lindsay Lecture (printed), Texas A&M University, Feb. 8, 1985. [8] Nowak, E.S., and Chan, J.S., Proc. 7th Sym. Thermo. Prop., (1977), 538, ASME, New York. [9] Nehzat, M.S., Hall, K.R., and Eubank, P.T., Adv. Chem. Series, (1979), 182, 109. [10] Kreglewski, Α., and Hall, K.R., Fluid Phase Equil.,(1983),15, 11. RECEIVED November 8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

3 The Equation of State of Tetrafluoromethane 1

2

J. C. G. Calado , R. G. Rubio , and W. B. Streett School of Chemical Engineering, Cornell University, Ithaca, NY 14853

An experimental study of the P-V-T properties of tetrafluoromethane (CF ) over wide ranges of temperature and pressure is used to test several semi-empirical equations of state and molecular theories. The experimental data have been correlated by the Strobridge equation, and comparisons are made with the Haar and Kohler, Deiters and BACK equations of state, as well as with the lattice gas model of Costas and Sanctuary, the variational inequality minimization (VIM) theory of Mansoori, and the perturbation theory of Gray and Gubbins.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

4

We have recently completed a detailed experimental study of the equation of state of tetrafluoromethane, C F , covering a wide range of temperature and pressure (1_). Tetrafluoromethane is an attractive substance from both the technological and theoretical points of view. It is widely used as a low-temperature refrigerant (Freon 14) and a gaseous insulator, and i t s molecules offer an interesting theoretical study of the underlying intermolecular forces. Whilst i t s grosser features may be considered to be representable by a quasi-spherical (or globular) molecule, i t s thermodynamic behavior, especially in mixtures, displays a nonideality that is symptomatic of the anisotropy inherent in i t s microscopic interactions. For instance, a whole variety of phase diagrams arise when tetrafluoromethane is mixed with hydrocarbons (2). Even the existing low-density studies lead to contradictory conclusions about the intermolecular potential of CFi*. While in some cases a simple (12,6) Lennard-Jones potential is able to describe the second v i r i a l c o e f f i c i e n t data (_3) > in other cases a spherical-shell model has been claimed to be necessary (4-6). 4

Current address: Complexo I, Instituto Superior Tecnico, 1096 Lisboa, Portugal. Current address: Departamento de Quimica Fisica, Facultad de Quimicas, Universidad Complutense, Madrid 28040, Spain.

1

2

0097-6156/86/0300-0060S06.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

3.

Equation of State of Tetrafluoromethane

C A L A D O ET AL.

61

Tetrafluoromethane seems to be an i d e a l l y placed molecule for t h i s kind of study: on the one hand, i t i s f a i r l y small, nonpolar, highly symmetric (tetrahedral) and thus suitable for the testing of a wide variety of s t a t i s t i c a l theories and semiempirical equations of state; on the other hand, since i t has a r e l a t i v e l y big octopole moment, i t exhibits enough anisotropy to serve as a discriminator against cruder theoretical approaches. As such, tetrafluoromethane seems to f a l l between methane (CH^) for which a (12, 6) Lennard-Jones potential seems to be adequate, and tetrachloromethane (CCliJ for which a spherical-shell or s i t e - s i t e model is necessary. Strong orientational c o r r e l a t i o n s , which persist over several molecular diameters, have been found for tetrahedral molecules (7,8), and interlocking effects have been detected in t e t r a c h l o r o methane molecules using B r i l l oui η scattering techniques (9). These phenomena suggest that the spherical reference system, frequently u t i l i z e d in perturbation and variational approaches (10) could f a i l for this kind of molecules. In addition, calculations carried out using the s i t e - s i t e d i s t r i b u t i o n function formalism show that the disagreement with results from computer simulation i s much larger than for diatomic molecules {11). Those interlocking and other effects should become more pronounced at higher d e n s i t i e s , hence the need to extend the pressure range for which P-V-T data are a v a i l a b l e . Powles et a K (12) have pointed out that pressure and configurational energy data over wide ranges of density and temper ature are necessary in order to improve intermolecular potential functions and to test theories. There have been several experimental studies of the P-V-T properties of CFu. but only up to about twice the c r i t i c a l density ( p = 7.1 mol dm" ). The most extensive and accurate measurements are those of Douslin et a U (4, 13) which cover the temperature range 273-623 Κ and pressures up to 400 bar. MacCormack and Schneider worked in the same temperature range but only with pres sures up to 55 bar (14), while Lange and Stein (6) and Martin and Bhada (15) extended the measurements to lower temperature (203 K) and pressures up to 80 and 100 bar, respectively. Staveley and his co-workers measured both the orthobaric densities (16) and the saturation vapor pressure {!]_) of tetrafluoromethane. Thermody namic properties of the saturated l i q u i d , from the t r i p l e - p o i n t 89.56 Κ to the c r i t i c a l point 227.5 K, have been calculated by Lobo and Staveley (18), while Harrison and Douslin (19) calculated them for the compressed gas (temperatures 273-623 Κ and densities 0.75 - 11.0 mol dm- ). There was obviously a need for more data, especially in the low temperature, high-pressure region. We studied t h i r t y - t h r e e isotherms in the temperature range 95-413 Κ and pressures up to 1100 bar, obtaining about one thousand and five hundred new P-V-T data points (_1). Figure 1 shows the Ρ - Τ regions for which data are now a v a i l a b l e . The P-V-T surface of tetrafluoromethane i s now thus well defined over a wide range of temperature and pressure, from the d i l u t e gas to the highly compressed l i q u i d . In this paper we examine the a b i l i t y of several types of equations of state and molecular theories to predict the P-V-T properties of CF^. Detailed tables of thermodynamic properties w i l l be published elsewhere (20). 3

c

3

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

62

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

1000

—

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

800

600

-

400

Γ

/

I

1

/

!

1 1

1

i

1

•

//|1"~Ί

0 0

Figure 1.

200

400

600

T/K

Pressure-temperature regions covered by different investigators : , t h i s work; , Douslin et _aK (4); - · - · - , Lange and Stein (6); , Martin and Bhada (15); - - - - , MacCormack and Schneider (14)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

3.

63

Equation of State of Tetrafluoromethane

C A L A D O ET A L .

Experimental Since our study covered both low and moderately high temperatures, low and high d e n s i t i e s , two d i f f e r e n t apparatuses were used. The measurements in the 95 to 333 Κ range were made in the gas-expan sion type apparatus which has been u t i l i z e d and described before (21). In this apparatus, the amount of substance contained in a calibrated 27.5 cm c e l l , kept at a measured pressure and tempera ture, is determined by expansion into a large volume, followed by suitable P-V-T calculations (the pressure in the expansion volume is kept under 1.5 bar, to avoid any uncertainties in the equation of state of the gas). For the runs above 330 Κ a modification of the direct-weighing apparatus described by Machado and Streett (22) was used. Here the amount of substance is measured d i r e c t l y by weighing a f u l l c e l l of approximately 100 cm capacity. Temperature control was achieved with a boiling l i q u i d - t y p e cryostat for the low-temperature apparatus (replaced by a simple water bath in the experiments above 270 K) and by a cascade-type oven in the higher temperature apparatus. Substances used in the boiling l i q u i d type cryostat were nitrogen (95 K), argon (100 120 K), methane (130 - 152 Κ ) , tetrafluoromethane (160 - 200 Κ ) , ethane (210 - 245 K) and monochlorodifluoromethane (252 - 263 K ) . The temperature was controlled to within ± 0.02 Κ in the cryostat, and to within ± 0.002 Κ in both the l i q u i d bath and the a i r oven. It was measured, in a l l cases, with a platinum resistance thermome ter and referred to the IPTS - 68. Pressures in the c e l l s were measured with a Ruska dead-weight gage, (model 2450), the absolute accuracy being 0.1% or better, and the precision being about 0.01%. The main source of error in both apparatuses l i e s in the imprecise knowledge of the volume of the systems. Details of the c a l i b r a t i n g procedures have been given in previous papers (21, 22). We estimate that the average absolute error in density is about 0.1% for ρ > 8 mol dm- , 0.3% for 2 < p/mol dm- < 7 and 0.4% for ρ < 2 mol dm" . The CFj4 used in this work was from Linde (maximum purity 99.7%). It was purified by fractionation in a low-temperature column with a reflux ratio of 19/20. The final purity is estimated to be better than 99.99%. 3

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

3

3

3

3

Results The over one thousand and f i v e hundred data points were correlated by the Strobridge equation in the following form (1) ρ = RTp + (A RT + A. + A /T + k j l + Ας/Τ *) ρ + (A RT + A ) p + Α Τ * + ( A / T + A / T + A / T ) e x p [ A p ] p + (A /T + A / T + Am/T^)exp[A p ] + A p 1

2

X

2

6

2

7

8

2

i 2

Ρ

3

9

1 0

3

1 3

2

3

3

16

1+

11

2

5

P

2

3

16

6

1 5

For the sake of completeness, the data of Douslin et al_. (4)_ and Lange and Stein (6_) were included in the c o r r e l a t i o n . Given their high q u a l i t y , a s t a t i s t i c a l weight of two was ascribed to the data of Douslin et a l .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

64

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

The parameters of Equation 1 have been obtained using a method based on the Maximum Likelihood Principle as described by Anderson et al_. (23). Values of the parameters are recorded in Table I, as well as t h e i r estimated uncertainties. Figure 2 shows the differences between experimental and calculated densities for several isotherms. Deviation plots for the other isotherms follow this same general trend: Equation 1 is able to represent the experimental results within t h e i r estimated e r r o r s , except for the low density region of a few isotherms. It is gratifying to note the good agreement obtained for the 227.53 Κ isotherm, which i s very close to the c r i t i c a l one. Table II shows the density values of CFi* calculated at round values of pressure and temperature from Equation 1. In Figure 3 we compare the data of Douslin et a l . (£) with the values generated by Equation 1. The agreement i s , in general, within the combined errors of experiment and f i t t i n g . Martin and Bhada (15) have also reported large differences for the 273 Κ i s o therm, from the equation they proposed. The agreement between the different sets of data can be examined in Figure 4 where we plot the function B , defined by v

B

v

= (Z - l)/p

(2)

where Ζ is the compressibility f a c t o r , against pressure. As i t has been pointed out by Douslin et a]_. (4J this plot provides an excel lent test of the quality of the compressibility values, and the extrapolation to Ρ = ο gives the second v i r i a l c o e f f i c i e n t , B. In the temperature range in which the three sets of results overlap, the agreement is very good. This is very encouraging since our experimental techniques were s p e c i a l l y designed for high d e n s i t i e s , whereas the other investigators were primarily concerned with the lower density region. Equations of state and perturbation theory With sixteen parameters Equation 1 is f l e x i b l e enough to correlate PVT data over wide ranges of temperature and density, even in regions where (ap/3p)j is r e l a t i v e l y large. It lacks sound theo r e t i c a l foundation, although the format is that of an empirical modification of the v i r i a l equation of state. Looking at equations with a firmer theoretical basis is not only an i n t e l l e c t u a l l y rewarding exercise but also a serious attempt to develop more r e l i able and universal equations and improve on our present predictive a b i l i t i e s . The remainder of this paper will be devoted to an analysis and comparison of some of the most successful semiempirical equations and theories proposed in the last few years. The old Cartesian approach of dividing a complex problem into simpler, more manageable parts is of relevance here. Perturbation theory does just that. The reference or unperturbed system is a simple model whose properties have been f a i r l y well understood. The real or complex system is recreated by adding successive layers of complexity (the perturbing terms) to the i n i t i a l reference system. The genius or insight l i e s in finding a reference system

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

-0.337±0.004

(4.00±0.05)10-

7

8

.2

K" 1

3

1

mol" *

2

2

3

16

15

14

13

12

11

10

9

R = 0.083144 bar dm K"

bar dm 12

9

1

bar dm m o l "

6

dm moV

6

bar dm Κ * m o l "

Estimated Variance of Fit = 15

5

2

(0.53±0.02)10-

6

8

(4.835±0.005)10

5

bar dm Κ

2

mol-

(1.347±0.002)10

4 6

6

2

bar dm Κ m o l -

-1947±3

3 5

bar dm m o l " 2

1.76±0.04 6

2

3

0.041±0.001

dm mol" 1

1

7

8

6

bar dm

bar dm

bar dm

mol"

1

-0.0040 ± 0.001

(4.749 ± 0.002)10-

5

dm mol " 6

bar dm

18

15

15

15

9

2

3

2

h

3

2

3

3

3

6

mol-

mol"

mol"

mol"

mol-

mol"

mol"

K

K

K

bar dm Κ

9

9

bar dm K

-(1.2011 ± 0.0004)10 bar dm

7970 ± 14

-28.6 ± 0.4

(6.566±0.007)10

-(1.102±1)10

(5846±5)10

- Coefficients Αχ to A ^ , t h e i r Estimated Error, and Gas Constant R for Equation 1 τ

1

1

Table I.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

5

5

5

66

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

ρ/bar Figure 2.

Density deviation plots Δ ρ = ( p - P ) / p χ 100 for several isotherms. p is the experi mental density and p i t s value calculated from Eq. (1). , 199.98 K; > 227.53 K; , 252.53 K; e

c

e

e

c

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

0.703

0.613

0.545

0.492

0.450

0.415

0.386

0.361

0.339

0.320

0.303

0.288

0.288

0.261

0.238

0.220

0.204

0.190

0.179

0.168

0.159

0.151

0.144

220

240

260

280

300

320

340

360

380

400

420

17.357

17.330

160

15.924

18.605

18.587

140

0.368

19.752

19.739

120

0.323

20.817

20.807

180

21.063

21.054

95

100

200

10

P/bar

0.433

0.456

0.483

0.513

0.547

0.587

0.635

0.693

0.765

0.858

0.987

1.180

15.970

17.385

18.623

19.765

20.826

21.072

15

0.725

0.767

0.813

0.868

0.931

1.007

1.100

1.218

1.377

1.607

2.003

14.332

16.058

17.438

18.659

19.790

20.845

21.090

25

13.746 11.812 9.346 7.092 5.679 4.801 4.207 3.774 3.442 3.177 2.960

5.778 3.541 2.839 2.436 2.162 1.958 1.798 1.668 1.559 1.466

17.796

17.565

12.576

18.908

18.745

16.606

19.970

19.852

15.287

20.981

20.892

16.260

21.218

21.134

14.716

100

50

5.660

6.084

6.592

7.206

7.953

8.800

9.938

11.158

12.445

13.716

14.927

16.069

17.152

18.190

19.198

20.188

21.148

21.376

200

21.446 20.567 19.684 18.810 17.943 17.081 16.221 15.366 14.520 13.691 12.891 12.131 11.419 10.762 10.161 9.165 9.121

20.385 19.454 18.521 17.583 16.632 15.667 14.687 13.700 12.724 11.781 10.897 10.090 9.367 8.728 8.166 7.674

400

21.302

300

10.205

10.682

11.200

11.760

12.363

13.006

13.687

14.399

15.138

15.897

16.671

17.458

18.256

19.067

19.893

20.735

500

11.057

11.515

12.006

12.532

13.092

13.685

14.309

14.962

15.638

16.336

17.052

17.785

18.533

19.299

20.085

20.892

600

11.755

12.194

12.661

13.159

13.685

14.240

14.823

15.430

16.061

16.713

17.384

18.074

18.783

19.512

20.264

21.039

700

12.343

12.766

13.213

13.687

14.187

14.712

15.261

15.834

16.429

17.045

17.680

18.336

19.011

19.709

20.431

21.179

800

12.852

13.259

13.690

14.144

14.621

15.121

15.644

16.189

16.755

17.341

17.947

18.574

19.221

19.892

20.588

21.311

900

Density values of CF^ ( i n mol dm ) at round values of temperature and pressure, c a l c u l a t e d from Equation 1.

5

T/k

Table I I .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

13.299

13.694

14.110

14.547

15.005

15.485

15.985

16.507

17.049

17.610

18.192

18.793

19.417

20.063

20.736

21.437

1000

13.698

14.082

14.485

14.907

15.349

15.811

16.293

16.795

17.316

17.857

18.417

18.997

19.599

20.225

20.877

21.557

1100

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

68

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

2

4

6

θ

10

1

p (mol/Γ )

Figure 3.

Density deviation plots for Douslin's isotherms, Δρ = ( p - p ) / p χ 100. p is the experimental value and p the calculated value from Eq. (1): , 273.15 Κ , 298.15K; , 323.15K; , 348.15K; - · · - · · - , 373.15K. Horizontal bars denote the precision achieved by our results in the d i f f e r e n t density regions. e

c

e

e

c

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

CALADO ET AL.

Equation of State of Tetrafluoromethane

-20

-40

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

-60

ο Ε

ω -100

/

-120

-140

-160-

J 10

Figure 4.

50

30

70 p/bar

90

Values of B calculated from Equation 2: χ , t h i s work; · , Douslin et a K (4) ; α , Lange and Stein (6). The points at Ρ = ο are the values of the second v i r i a l c o e f f i c i e n t recommended by Dymond and Smith (24). v

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

70

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

that is simple enough to allow i t s properties to be easily calcu l a t e d , and yet close enough to the real system to avoid the need of many perturbing terms. Most of the calculations have been made using a spherical model (hard-sphere) as the reference system. Obviously in this case a perturbation theory is feasible i f the molecules are not far from s p h e r i c a l . Recently Fischer ( 2 5 ) devel oped a perturbation theory which can e f f e c t i v e l y deal with aniso tropic molecules, namely those of the two-center type (later also t r i a n g u l a r , tetrahedral and octahedral molecules). The problem is that strongly anisotropic molecules are also usually polar, and the present theories are s t i l l unable to deal with both anisotropy and p o l a r i t y . CF^. is fortunately non-polar, but despite i t s high symmetry cannot be reasonably described as a spherical molecule. It is now well established that thermodynamic properties are markedly affected by the shape of the molecules. Some of the more interesting equations of state can be obtain ed from perturbation theory. Indeed, the e a r l i e s t of them, that due to van der Waals, can be derived from f i r s t - o r d e r perturbation theory with a reference system of hard-spheres (26). A common fea ture to many of these equations of state is i t s s p l i t t i n g , in true perturbational fashion, into two or more terms, accounting for d i f ferent levels and types of complexity. One of those terms is usually b u i l t around a convex hard body. We will concentrate our attention, however, on what may be called the second generation of van der Waals - type equations. These are equations which retain a good approximation for the repulsive part ( l i k e that given by the Carnahan-Starling or Boublik-Nezbeda equations) while trying to improve on the attractive part of the p o t e n t i a l . The Haar and Kohler equation of state A few years ago Haar et al_. proposed a new approach to calculating P-V-T- data ( 2 7 ) . The equation of state is s p l i t into two parts Ρ = P

B

+ P

(3)

R

where Ρβ is the so-called base equation (in the sense that i t is a physically based expression) incorporating the effects of molecular repulsion and a t t r a c t i o n , and PR is a sum of residual terms, usual ly a series expansion in terms of ρ and Τ or some empirical func t i o n . For globular molecules a modified Carnahan-Starling equa tion is often used as Ρβ, but as many as twenty-six adjustable parameters are sometimes needed for PR ( 2 8 ) . Kohler and Haar ( 2 9 ) showed that PR would be a universal func t i o n for nonpolar f l u i d s , provided that Pg takes into account the shape of the molecule. Recently, Moritz and Kohler came out with an empirical expression for PR (3(D), leading to the following f i n a l form for the compressibility factor Ζ

ζ Z

- z - | i^ h

n

,,,· ^

*» « p [ - J ! ^ ] , e

,„

i s the compressibility factor for a hard-convex body (a tetrahe-

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

3.

C A L A D O ET AL.

71

Equation of State of Tetrafluoromethane

dron for CFiJ l i k e that given by the Boublik-Nezbeda equation of state (31), d and d are universal constants (0.098 and 0.128, respectively, according to reference 30) and y = v*p is the reduc ed density, with v* = N/\V*, V* being the volume of the hard-body and N/\ Avogadro's number. y j is the reduced density of the r e c t i linear diameter p j (the arithmetic mean of the densities of the orthobaric l i q u i d and vapor) v

w

rc

rc

p

rd

"
+

P

G>

/ 2

T

< c

<>

T

5

For Τ > T , p | has a virtual value, obtained from the extrapola tion of the r e c t i l i n e a r diameter curve. Equation 4 has four parameters a , V , W and d (d/2 being the thickness of the smooth hard layer added to the hard body). For Τ < 0.6 T , V and W can be easily calculated from vapor-liquid e q u i l i b r i a using the second v i r i a l c o e f f i c i e n t ; thus, only d and a (or ao/v*) need to be f i t t e d at each temperature. The ratio a / v * for CFi* (-33 kj mol- ) is appreciably larger than for CH^ (-20 kJ mol" ), in q u a l i t a t i v e agreement with the corresponding values of the s o l u b i l i t y parameter. The hard-core volume v* is found to decrease with temperature, as expected, but following a d i f f e r e n t dependence law than that observed with the BACK equation (see l a t e r ) . Moritz and Kohler had reached a similar conclusion for methane (30). The observed density dependence for the van der Waals parameter a/v* c

rc

0

c

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

0

0

1

= RT ( Z

h

- Z)/y

(6)

i s also similar to that found for methane (30). Within the estimated uncertainties, we found that i t is possible to describe the temperature dependence of a l l four parameters in Kohler*s equation of state for C F , with simple functions incorporating only a few adjustable constants. Figure 5 compares the results obtained with Equations 4 and 1. It is obvious that Kohler's equation f i t s the data better than the Strobridge equation below T , but the quality of the f i t t i n g worsens dramatically above T . This suggests that there is s t i l l room for improvement in the density dependence of a/v*. 4

c

c

The Deiters equation of state Another of the new, physically based equations of state is that derived by Deiters (32) from a square-well potential model of depth (-ε) — (7)

RT

1 + c c

4y - 2y 0

(1

- y)

2

d

RbcT V w(y) z

exp

aw(y) ~~cT

-1

ii(y)

The equation has three adjustable parameters: the c h a r a c t e r i s t i c temperature a, the covolume b, and the correction factor for the number of density dependent degrees of freedom, c.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

72

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

100

Figure 5.

200

300

T/K

400

Mean standard deviations of density, σ, as a function of temperature. A , Strobridge equation; · , Kohler equation; o, Deiters equation.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

3.

C A L A D O ET AL.

73

Equation of State of Tetrafluoromethane a = ε/k b = N

(8) σ //2

(9)

3

A

c can also be interpreted as a shape parameter (c = 1 for spherical molecules); c = 0.6887 is a universal constant which accounts for the deviation of the real pair potential from the rigid-core model. Il(y) and w(y) are complicated functions of the density y and c, derived from s t a t i s t i c a l mechanics. w(y) may be described as an e f f i c i e n c y factor for the square well depth, and i t takes into account the fact that the structure of the f l u i d is more and more determined by repulsive forces as density increases. The repulsive part of the Carnahan-Starling equation ( 3 3 ) has been retained as a good approximation for r i g i d spheres. Equation 7 can be modified, to account for three-body forces, by intrQducing an additive cor r e c t i o n , λρ, to the reduced temperature, Τ = cT/a ( 3 2 ) . We have used Equation 7 as Pg in Equation 3 and allowed the three parameters a, b and c to be temperature dependent. Figure 5 compares the results obtained with the Deiters equation with those given by Equations 1 and 4 . For Τ < T the highest deviations occur in the low density region, where the isotherms are the steep est. The results usually f a l l between those obtained with the Strobridge and Kohler equations. Figures 6 and 7 show the f i t t e d parameters of Deiters* equation as a function of temperature. It is interesting to note that both (c) and (a) are constant within t h e i r estimated e r r o r s , although they take d i s t i n c t values below and above the c r i t i c a l temperature. Of course, the temperature dependence of any of the parameters is not a simple function in the c r i t i c a l region, so we cannot extrap olate over this region. The behavior found for parameter (b) is somewhat b i z a r r e . In the range Τ < T , (b) follows the usual trend, i . e . i t decreases with temperature ( 3 4 ) ; no explanation has been found for i t s apparent increase with temperature in the region Τ > T . It should be noted that similar behaviour of the param eters a, b and c has been found for trifluoromethane CHF , whose study is now under way. The calculations were repeated using a constant value for (c) throughout the entire temperature range (we used the more r e a l i s t i c 'high-temperature' value of Figure 6 , c = 1 . 0 8 ) , but l e t t i n g both (a) and (b) f l o a t . Under these conditions parameter (a) is found to decrease monotonically from a value of about 235 Κ at 100 Κ to about 175 Κ at 400 K. Parameter (b) retains, however, the peculiar behavior displayed in Figure 7 . The quality of the overall f i t t i n g also deteriorates when a constant value of (c) is used throughout, perhaps because packing effects become important at higher densities. Using the Deiters equation as Pg (with the "constant" values of a and c, and b f i t t e d to a Morse-like function) and an expres sion for PR l i k e that proposed by Haar et a l . ( 2 7 )

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

0

c

c

c

3

(10)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

74

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

200

300

400 T/K

Figure 6.

Parameter c in Deiters equation as a function of temperature (with error bars for estimated uncertainty).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

C A L A D O ET A L .

Equation of State of Tetrafluoromethane

100 Figure 7.

τ

1

200

300

r

T/K

400

Parameters a and b in Deiters equation as function of temperature (with error bars estimated uncertainty).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

76

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

We were unable, even with sixteen constants C j , to describe the whole set of P-V-T data with better accuracy tnan that provided by Equation 1. This could be due to the inadequacy of the residual terms used, but looking at Figure 5 suggests that an improvement in the low-temperature range can be obtained by substituting the Boublik-Nezbeda equation for the Carnahan-Starling equation in Deiters" model. This is confirmed by setting V = W = 0 in the Kohler equation (4) (which uses the Boublik-Nezbeda equation as a reference) and finding that i t leads to a better correlation of the experimental results than that obtained with the Deiters equation. Even more important, perhaps, that substitution would allow the separation of the contributions of external and internal degrees of freedom in the Deiters equation. One should then use data for molecules with vibrational degrees of freedom, instead of data on argon, as Deiters d i d . Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

n

The BACK equation of state One of the best equations of state based on the generalized van der Waals model is the so-called BACK (from Boublik-Alder-ChenKreglewski) equation (35). It is an augmented hard-core equation which combines the Boublik expression for the repulsive part of the compressibility factor, with the polynomial developed by Alder et AL- for the attractive part (36) p v

- ζ z +

RT Ζ

=

h

Z

a

*

1

(

3

Ύ

"

2

h

)

" =

y

*

(

3

γ

"

I l

η

m

mD

( nm

u

"

2

(l-y) 9

a 3

(11) *

Ύ

)

y

"

2

(12)

1 f 2 y 3

3

η V* ) (_) V

kT

1

m

(13)

Both the molecular hard core V* and the c h a r a c t e r i s t i c energy u are decreasing functions of temperature V* = V

[ 1 - C exp ( - 3 u ° / k T ) ]

0 0

u

ΊΓ

k

( 1 "

v

+

3

* ) kT '

(14)

(15)

Chen and Kreglewski (35) gave rules for calculating C and n, so that only three adjustable parameters remain in the BACK equation: V , γ and u /k. The f i t t i n g of our experimental data led to the following values: V =(0.0310 ± 0.0001)dm , γ = 1.099 ± 0.013 and u°/k = (225 ± 2)K. The results are plotted in Figure 8. The agreement is satisfactory in the c r i t i c a l region, but i t goes beyond the experimental error in the high density region. Besides, the value of γ does not c l e a r l y respect the tetrahedral geometry of the molecule, a fact which Moritζ and Kohler also found for methane (30). As mentioned before, the temperature dependence of the hard00

3

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

CALADO

ET

AL.

Equation of State of Tetrafluoromethane

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

3.

Figure 8.

Several isotherms for CF (in reduced variables). · , experimental data; , calculated from BACK equation; , calculated from l a t t i c e model; - · - · - , coexistence curve according to Lobo and Staveley (17). 4

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

11

78

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

core volume, as expressed by Equation 14, does not agree with the corresponding dependence in Kohler's equation. Use of the c o r r e l a tions proposed by Simnick et al_. (37) for the calculation of param eters V , γ and u°/k leads to even poorer agreement with the experimental r e s u l t s . Taking the entire P-V-T range covered in this work, the quality of the f i t t i n g obtained with the BACK equation is consider ably worse than that obtained with either the Kohler or Deiters equations, but this result should be set against the very small number of adjustable parameters u t i l i z e d in the BACK equation. 0 0

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

Cell and Lattice Models Although the c e l l and l a t t i c e models have been v i r t u a l l y abandoned for l i q u i d s , mainly because they over-estimate the degree of order, some of t h e i r modifications have been revived, with success, in the thermodynamic treatment of polymers. Recently Nies et a U (38) have shown that hole theories can be e f f e c t i v e in reproducing the P-V-T behavior of a r e l a t i v e l y simple f l u i d , ethylene. In the lowdensity region, they favor the lattice-gas model over the c e l l theory. Sanchez and Lacombe (39) developed a theory of r-mers based upon a l a t t i c e model for l i q u i d s . They used a simplified Ising model, and were led to a four-parameter equation of state which successfully predicted a liquid-vapor phase t r a n s i t i o n . The four parameters were the non-bonded mer-mer interaction energy, ε* (equivalent to a c h a r a c t e r i s t i c temperature), the close-packed -mer volume, v*, the l a t t i c e coordination number z, and the number of mers per r-mer, r. Later Costas and Sanctuary (40, 41J removed r as an adjustable parameter and set i t equal to the number of atoms, other than hydrogen, in the molecule. Their equation of state, in reduced variables, took the form [ρ(1-φ)/(1-φρ)]

2

+ ρ + f [ l n ( l - p ) - (ζ/2) Ι η ( Ι - φ ρ ) ] = 0

(16)

where Φ = (2/z) (1-1/r)

(17)

Since they found that the calculated quantities were r e l a t i v e l y insensitive to changes in ζ between the values 8 and 14, they chose to f i x ζ equal to 12. Equation 16 became then a two-parametric equation. In this work we used the Costas-Sanctuary equation of state in i t s three-parametric form. The adjusted values of the parameters were found to be p*/bar = 3338 ± 123, T*/K = 258 ± 2 and v * / l m o l " = 0.0476 ± 0.0002. The results are plotted in Figure 8. It can be observed that the BACK equation gives a better overall description of the P-V-T properties of CFi^ than the Costas-Sanctuary equation, especially near the c r i t i c a l region. At very high or very low densities and pressures the two models give almost identical r e s u l t s . For high densities and P < 0.8 the agreement between the experimental values and those calculated from the l a t t i c e model i s better than that obtained with the BACK equation, in particular

1

r

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

3.

C A L A D O ET AL.

79

Equation of State of Tetrafluoromethane

near the coexistence curve. In the range 0.3 < p < 1.8 the densities calculated with the BACK equation are too low, whereas those calculated from the Costas-Sanctuary equation are too high. In agreement with Costas and Sanctuary we have found that these conclusions s t i l l hold when the coordination number ζ changes from 8 to 12. r

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

The VIM theory In a previous paper we have already examined the a b i l i t y of both variational and perturbation (see next section) theories to describe the thermodynamic properties of CF^ (_1). Here we repro duce the main conclusions in order to assess how these theories fare in comparison with more empirical equations of state. The variational inequality minimization (VIM) theory as developed by Alem and Mansoori (42) led to a r e l a t i v e l y simple equation of state for non polar f l u i d s . They use a reference system of hard-sphere molecules whose diameter (σ) has been chosen in order to minimize the Helmholtz energy of the real system. The configurational entropy is then given by S

C

= -R y

H S

(4 - 3 y ) / ( l H $

- y

H S

)

(18)

2

Contrary to what Alem and Mansoori found for argon and methane, where σ was a linear function of density and inverse temperature, for tetrafluoromethane second order terms were needed. This r e f l e c t s , perhaps, the higher anisotropy of CF^. F i t t i n g of the P-V-T data led to the following values of the three adjustable parameters: ε/k = (220 ± 1) K; σ/nm = 0.414 ± 0.001 and 10 v/k = (8 ± 1) nm Κ " , ( ε , σ) are the usual param eters in a Lennard-Jones type potential and ν is the c o e f f i c i e n t of the Axil rod-Tel 1er correction for the three-body potential (42). The f i n a l results are compared with the experimental values τη Figure 9. The agreement is similar to that found by Alem and Mansoori for argon, but i t is worse than that for methane. The theory underestimates the temperature c o e f f i c i e n t of the density, leading to poor agreement in the low temperature region. 3

3

1

Perturbation theory The perturbation theory of Gray, Gubbins and coworkers (10) has been extensively applied to molecular liquids and their mixtures. Since i t uses a spherically-symmetric reference system, i t stands the best chance of success when applied to molecules which exhibit high symmetry. A previous study (43) proposed a model for CF^ which involved descriptions of the octopolar, anisotropic disper sion and charge overlap (shape) forces, in addition to a LennardJones (n,6) potential (19) (n,6) u = u + u (336) + u (303 + 033) + u (303 + 033) CF4/CF4

ο

ΩΩ

ov

disp.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

80

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

01 0 Figure 9.

1 200

ι 400

p/bor

ι 600

ι 800

1000

Comparison of experimental density values ( · ) with the predictions of the variational equation of state ( - · - · - ) and perturbation theory ( - - - - ) , as a function of pres sure, for different isotherms.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

3.

Equation of State of Tetrafluoromethane

C A L A D O ET A L .

81

The potential was found to be extremely hard, with a repulsive exponent η = 20, perhaps r e f l e c t i n g the high electronegativity of fluorine atoms. Best values of the intermolecular parameters were ε/k = 232 Κ and σ = 0.4257 nm. Use of the perturbation theory with the usual Padé approximant (I) led to the calculated densities shown in Figure 9. The agreement with experiment can be rated very good, the difference being, at most, 2% even at the highest pressures, where the reduced density is very close to the l i m i t of v a l i d i t y of the reference equation ( i . e . densities ρσ < 1.0). 3

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

Conclusions The usual pattern is to have a new approach, be i t an empirical equation of state or molecular theory, tested against a variety of systems, from pure f l u i d s to mixtures, from simple molecules to nasty ones (to borrow an expression from John Prausnitz). Here we have followed a different route, v i z . use the same body of experi mental data to assess the goodness of several semi-empirical equa tions and of some t h e o r e t i c a l l y based treatments. Obviously t h i s l a t t e r approach can only be of value i f the set of data is compre hensive enough to provide a s t r i c t test of the theory. In thermo dynamic terms this means data over wide ranges of temperature and density, so that f i r s t order properties (temperature and pressure c o e f f i c e n t s , for instance) can also be checked. The f i r s t attempt at a molecular understanding of the proper t i e s of a f l u i d was that of van der Waals, with his famous equa tion of state

-IL. . ^ _ RT

RTV

+

_L_

(20)

l-4y

Although the f i r s t (attractive) term was soon recognized as a good approximation for the attractive f i e l d , the second was thought, even by van der Waals, to be a poor representation of the repulsive forces. In his Nobel address of 1910 he was s t i l l wondering i f there was "a better way" of doing i t , adding that "this question continually obsesses me, I can never free myself from i t , i t is with me even in my dreams". The problem was only solved more than f i f t y years l a t e r , when good approximations for the equation of state of a system of hard spheres were proposed. In the meantime, many people were busy trying to improve on the van der Waals equa tion by retaining i t s bad term while trying to correct the good one. The Redlich - Kwong equation of state is perhaps the most celebrated outcome of this approach. PV R

T

(21) (V+bjRT 1

5

i-4y

No wonder i t has generated over one hundred modifications, despite i t s own successes. John Prausnitz has said that Equation 21 is to applied thermodynamics what Helen of Troy has been to l i t e r a t u r e " . . . the face that launched a thousand ships."

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

82

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

In the 60's and 70's more successful members were added to the family of van der Waals equations: these combined the attractive part of the original Equation 20 or even Equation 21 with a good hard-sphere equation of state. The Carnahan-Starling equation, mentioned in this paper, is a good example. In the last few years another type of equations of state has begun to emerge. They usually have a s t a t i s t i c a l mechanics basis and combine a good hard-sphere part (or some generalization to hard convex bodies) with a more f l e x i b l e and sophisticated representa tion of the attractive f i e l d than that provided by the van der Waals equation. The Deiters and Kohler equations are good specimens of this second generation of generalized van der Waals equations of state. Equations with a r e a l i s t i c molecular basis should be poten t i a l l y more universal in their a p p l i c a b i l i t y and thus better suited to sound predictive methods. There is obviously s t i l l room for progress. For instance, the Deiters equation could p r o f i t from a better description of shape e f f e c t s , while the Kohler equation is perhaps deficient in i t s depiction of the attractive f i e l d . Perturbation theory seems, at present, to be more promising than the variational approach, espe c i a l l y i f i t succeeds in combining the a b i l i t y to deal with polar i t y with the a b i l i t y to take shape into account. In any science, understanding must be synonymous with good quantitative agreement, and understanding really means the molecu l a r l e v e l , even in chemical engineering. We owe i t , after a l l , to van der Waals who, in his Nobel address, said: "It will be perfect ly clear that in a l l my studies I was quite convinced of the real existence of molecules, that I never regarded them as a figment of my imagination". The road is long and arduous, but van der Waals pointed the way. Glossary of Symbols

Latin Alphabet A-j (i = 1 - 16) a, a

= =

0

Β B b C C j c c d

v

n

0

d ,d U (y) k N/\ Ρ R v

w

= = = = = = = = = =

= = = =

c o e f f i c i e n t s of the Strobridge Equation (1) van der Waals parameters in Equation 4; charact i c temperature in Deiters Equation (7) second v i r i a l c o e f f i c i e n t function defined by Equation 2 covolume parameter in Equation 14 constants in Equation 10 shape parameter in Deiters Equation (7) universal constant in Deiters Equation (7) twice the thickness of hard layer added to hard convex core universal constants in Equation 4 a function of density in Deiters Equation (7) Boltzmann's constant Avogadro's constant pressure gas constant

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

3.

Equation of State of Tetrafluoromethane

C A L A D O ET A L .

Γ

S Τ υ u, u° V ν V ν* V* W w(y) 0 0

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

y ζ ζ

=

= = = = = = = = = = = = = =

number of mers per r-mer entropy absolute temperature total energy c h a r a c t e r i s t i c energy in Equations 13--15 volume volume parameter in Equation 14 parameter in Equation 4 volume of one mole of hard bodies volume of hard body parameter in Equation 4 e f f i c i e n c y factor for the square-well depth, Equation 7 reduced density compressibility factor = PV/RT l a t t i c e coordination number

Greek alphabet α γ ε

= = =

n ν Ρ σ

= = = =

Φ

=

c o e f f i c i e n t in Equation 10 shape parameter in Boublik Equation (12) intermolecular energy parameter (depth of potential) parameter in Equation 15 c o e f f i c i e n t in Axil rod-Teller t r i p l e potential molar density intermolecular energy parameter (molecular diameter); mean standard deviation parameter defined by Equation 17

Subscripts, superscripts a Β c e G h HS L R r rd Ω

= = = = = = = = = = = = =

*

=

attractive base (physically based) configurational; calculated experimental gaseous hard convex hard sphere Liquid Residual reduced (by c r i t i c a l parameters) r e c t i l i n e a r diameter octopole reduced (by c h a r a c t e r i s t i c parameters, other than c r i t i c a l ) characteristic

Literature Cited 1. 2.

Rubio, R.G; Calado, J.C.G.; Clancy, P.; Streett, W.B. J. Phys. Chem., in press. Paas, R.; Schneider, G.M. J. Chem. Thermodynamics, 1979, 11 (3) 267.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

83

84

3. 4. 5. 6. 7. 8. 9.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

27. 28. 29. 30.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

MacCormack, K.E.; Schneider, W.G. J. Chem. Phys. 1951, 19 (7), 849. Douslin, D.R.; Harrison, R.H.; Moore, R.T.; McCullough, J.P. J. Chem. Phys. 1961, 35 (4), 1357. Kalfoglou, N.K.; Miller, J.G. J. Phys. Chem. 1967, 71 (5) 1256. Lange Jr., H.B.; Stein, F.P. J. Chem. Eng. Data 1970, 15 (1), 56. Montague, D.G.; Chowdhury, M.R.; Dore, J.C.; Reed, J. Mol. Phys. 1983, 50 (1), 1. Narten, A.H. J. Chem. Phys. 1976, 65 (2) 573. Asenbaum, Α.; Wilhelm, E. Adv. Molec. Relax. Int. Proc. 1982, 22, 187. Gray, C.G.; Gubbins, K.E. "Theory of Molecular Liquids"; Oxford University Press: London, 1984. Monson, P.A. Mol. Phys. 1982, 47 (2), 435. Powles, J.G; Evans, W.A.; McGrath, E.; Gubbins, K.G.; Murad, S. Mol. Phys. 1979, 38 (3), 893. Douslin, D.R.; Harrison, R.H.; Moore, R.T. J. Phys. Chem. 1967, 71 (11), 3477. MacCormack, K.E.; Schneider, W.G. J. Chem. Phys. 1951, 19 (7) 845. Martin, J.J.; Bhada, R.K. AIChE J. 1971, 17 (3) 683. Terry, M.J.; Lynch, J.T.; Bunclark, M.; Mansell, K.R.; Staveley, L.A.K. J. Chem. Thermodynamics 1969 1 (4) 413. Lobo, L.Q.; Staveley, L.A.K. Cryogenics 1979, 19 (6) 335. Lobo, L.Q.; Staveley, L.A.K. J. Chem. Eng. Data 1981, 26 (4) 404. Harrison, R.H.; Douslin, D.R. J. Chem. Eng. Data 1966, 11 (3) 383. Rubio, R.G.; Streett, W.B. J. Chem. Eng. Data submitted for publication. Calado, J.C.G.; Clancy, P.; Heintz, Α.; Streett, W.B. J. Chem. Eng. Data 1982, 27 (4) 376. Machado, J.R.S.; Streett, W.B. J. Chem. Eng. Data 1983, 28 (2) 218. Anderson, T.F.; Abrams, D.S.; Grens, E.A. AIChE J. 1978, 24 (1) 20. Dymond, J.H.; Smith, E.B. "The Virial Coefficients of Pure Gases and Mixtures"; Clarendon Press: Oxford, 2nd. edition, 1980. Fischer, J. J. Chem. Phys. 1980, 72 (10) 5371. Henderson, D., in "Equations of State in Engineering and Research", Chao, K.C.; Robinson Jr., K.L. Eds.; ADVANCES IN CHEMISTRY SERIES No. 182, American Chemical Society, Washington, D.C., 1979. Haar, L.; Gallagher, J.S.; Kell, G.S., Natl. Bur. Stand. (US) Internal Report No 81 - 2253, 1981. Waxman, M. Gallagher, J.S J. Chem. Eng. Data 1983, 28 (2), 224. Kohler, F.; Haar, L. J. Chem. Phys. 1981, 75 (1) 388. Moritz, P.; Kohler, F. Ber. Bunsenges. Phys. Chem. 1984, 88 (8) 702.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

3. CALADO ET AL. Equation of State of Tetrafluoromethane

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch003

31. 32. 33. 34. 35.

85

Boublik, T.; Nezbeda, I. Chem. Phys. Lett. 1977, 46 (2) 315. Deiters, U. Chem. Eng. Sci. 1981, 36 (7), 1139. Carnahan, N.F; Starling, K.E. AIChE J. 1972, 18 (6) 1184. Aim K.; Nezbeda, I. Fluid Phase Equil. 1983, 12 (3) 235. Chen, S.S.; Kreglewski, A. Ber. Bunsenges. Phys. Chem. 1977, 81 (10) 1048. 36. Alder, B.J.; Young, D.A.; Mark, M.A. J. Chem. Phys. 1972, 56 (6) 3013. 37. Simnick, J.J.; Lin, H.M.; Chao, K.C. in "Equations of State in Engineering and Research", Chao, K.C.; Robinson Jr., R.L. Eds.; ADVANCES IN CHEMISTRY SERIES No. 182, American Chemical Society, Washington, D.C., 1979. 38. Nies, Ε.; Kleintjens, L.A.; Koningsveld, R.; Simha, R.; Jain, R.K. Fluid Phase Equil. 1983, 12 (1) 11. 39. Sanchez, I.C.; Lacombe, R.H. Nature (London) 1974, 252 (5482), 381. 40. Costas, M.; Sanctuary, B.C. J. Phys. Chem. 1981, 85 (21) 3153. 41. Costas, M.; Sanctuary, B.C. Fluid Phase Equil. 1984, 18 (1) 47. 42. Alem, A.H.; Mansoori, G.A. AIChE J. 1984, 30 (2) 468. 43. Lobo, L.Q.; McClure, D.; Staveley, L.A.K.; Clancy, P.; Gubbins, K.E.; Gray, C.G. J. Chem. Soc. Faraday Trans. 2, 1981, 77, 425. RECEIVED November 5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

4 Phase Equilibria for the Propane-Propadiene System from Total Pressure Measurements Andy F. Burcham, Mark D. Trampe, Bruce E. Poling, and David B. Manley

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

Department of Chemical Engineering, University of Missouri-Rolla, Rolla, MO 65401

Total pressure measurements were made on the propanepropadiene system from 253.15 to 353.15K. The data were reduced using the Soave-Redlich-Kwong Equation of state with modified mixing rules containing several parameters. Corrections were made for the effects of known chemical impurities in the experimental system, and vapor pressures and relative volatilities were calculated.

The phase equilibrium behavior of the propane-propadiene system was s t u d i e d u s i n g the t o t a l p r e s s u r e method. T h i s method has been applied to a number of systems of close b o i l i n g l i g h t hydrocarbons at the university of Missouri-Rolla (Steele e t a l . , 1976; Martinez-Ortiz and Manley, 1978; Flebbe e t a l . , 1982; Barclay e t . a l . , 1982). The t o t a l pressure method consists of experimentally measuring the vapor p r e s s u r e s and t o t a l volume o f two-phase mixtures under v a r y i n g c o n d i t i o n s o f temperature and o v e r a l l composition. These data were then reduced using a thermodynamically consistent set of equations to calculate phase compositions and densities. In past studies, i t has been necessary to use extremely pure m a t e r i a l s i n order to produce a c c u r a t e r e s u l t s . For p r a c t i c a l reasons t h i s has limited the method to investigations concerning the r e l a t i v e l y few chemicals which are e a s i l y p u r i f i e d . However, many chemicals present i n i n d u s t r i a l process mixtures cannot be studied i n pure form because they react with themselves. In p a r t i c u l a r , l i g h t hydrocarbon d i o l e f i n s and a c e t y l e n e s tend to d i m e r i z e at process temperatures. One o b j e c t i v e o f t h i s work was to extend the

0097-6156/86/0300-0086$06.50/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

4.

B U R C H A M ET A L .

Propane-Propadiene System: Phase Equilibria

87

p r e v i o u s l y developed technique to mixtures c o n t a i n i n g r e a c t i v e components. Propadiene was chosen because, among other reasons, i t i s d i f f i c u l t to obtain i n high purity; and i t i s moderately reactive. T h e r e f o r e , i t p r o v i d e s a good test case for the development of the necessary experimental and computational t o o l s . There are several ways to reduce the experimental t o t a l pressure data. In this work we chose to f i t the data with the Soave-RedlichKwong (SRK) e q u a t i o n o f s t a t e ( S o a v e , 1972) with a d j u s t a b l e i n t e r a c t i o n parameters. The SRK equation was chosen because i t has been used s u c c e s s f u l l y f o r c o r r e l a t i n g hydrocarbon p h y s i c a l properties and because i t was necessary to predict some properties i n a d d i t i o n to c o r r e l a t i n g the observed p r e s s u r e s . Multicomponent mixtures were prepared with propadiene which contained quantitatively known i m p u r i t i e s . The SRK i n t e r a c t i o n parameters f o r the minor constituents were estimated from the l i t e r a t u r e , but the parameters f o r the primary propane-propadiene binary were determined from the measured data. I t was necessary to add a d d i t i o n a l i n t e r a c t i o n parameters i n order to f i t the data within the estimated experimental error. To avoid errors i n calculated pure component vapor p r e s s u r e s and l i q u i d d e n s i t i e s , the SRK equation parameters were adjusted to y i e l d a c c u r a t e pure component v a p o r p r e s s u r e s . The C o s t a l d (Hankinson and Thompson, 1979) c o r r e l a t i o n was used to calculate the saturated l i q u i d d e n s i t i e s i n the volume balance e q u a t i o n . Pure propadiene vapor pressures were estimated from the experimental data s i m u l t a n e o u s l y with the i n t e r a c t i o n parameters i n an i t e r a t i v e procedure. Experiment The e q u i l i b r i u m c e l l which holds about 6 cc of sample i s shown i n Figure 1. The chemicals o f known composition were v o l u m e t r i c a l l y m e t e r e d i n t o t h e evacuated c e l l , and t h e i r exact amounts were determined (to within 0.0001 grams) on an a n a l y t i c a l balance. The c e l l was then p l a c e d i n a thermostat controlled to within 0.01 °C, and the pressure was determined by b a l a n c i n g the hydrocarbon vapor pressure on the bottom of the s t a i n l e s s s t e e l diaphragm with nitrogen on the top. The d i f f e r e n t i a l pressure n u l l was determined to w i t h i n 0.02 p s i a by the displacement transformer which had been previously calibrated. The nitrogen pressure was measured to w i t h i n 0.01 p s i a with a Ruska dead weight gauge and Princo barometer. The temperature was measured to within 0.025 °C by a platinum resistance thermometer and M u e l l e r b r i d g e . D e t a i l e d d e s c r i p t i o n s o f the equipment, o p e r a t i n g procedures, and e r r o r a n a l y s i s a r e g i v e n e l s e w h e r e (Barclay, 1980; Burcham, 1981, Barclay et a l . , 1982). Research grade propane was obtained from P h i l l i p s Petroleum Company, which s t a t e d t h a t i n f r a r e d and mass s p e c t r o m e t e r determinations showed the p u r i t y to be 99.99 mole p e r c e n t . After c a r e f u l degassing, the only impurity shown by gas chromatography was a trace amount of ethane. The propadiene was purchased from Columbia Inorganics Incorporated and had a stated purity of 99 weight percent. After careful degassing, a n a l y s i s by gas chromatography and mass spectrometry i d e n t i f i e d the impurities as l i s t e d i n Table I.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

TO NITROGEN

DISPLACEMENT TRANSDUCER

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

FERRITE CORE

4 - WIRE CABLE TO POWER SUPPLY AND VOLTMETER

0.002" STAINLESS DIAPHRAGM

COUPLING-

SAMPLE CAVITY

THERMOWELL

FILL LINE TO ATTACHED VALVE

Figure 1.

Sample c e l l and transducer

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

4.

BURCHAMETAL.

Propane-Propadiene System: Phase Equilibria

89

Table I. Propadiene Analysis

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

Compound Propadiene Propene Propyne Propane Ethene Cyclopropane

Mole % 96.84 2.11 0.89 0.088 0.047 0.031

Two c e l l s were used, and a r u n c o n s i s t e d o f f i l l i n g t h e e v a c u a t e d c e l l s 25 to 60 percent f u l l with degassed c h e m i c a l s , inserting them into the bath, and measuring the pressure at d i f f e r e n t temperatures. The f i r s t s e t o f measurements s t a r t e d at 80°C, proceeded down i n 25°C steps to -20°C; a second s e t o f measurements was then made from -20°C to 80°C. Comparison o f the d u p l i c a t e d measurements p r o v i d e d a check on p o s s i b l e e r r o r s due t o l e a k s , reactions, or non-equilibrium conditions. The experimental results a f t e r adjustment to exact temperatures are g i v e n i n T a b l e I I . The temperature adjustment procedure accounted f o r the difference between the actual bath temperature and the desired temperature. T h i s was l e s s than 0.15°C, and the correction (done with the Antoine equation) c o n t r i b u t e d n e g l i g i b l e error. Data Reduction In order to compare with l i t e r a t u r e data the pure propane vapor pressures were c o r r e l a t e d with the Goodwin equation (Goodwin, 1975). Ρ = P'EXPUX + BX

2

+ CX

3

+ DX(l-X)

1,5

)

(1)

X = (1 - T'/T)/(1-T'/T ) c

The constants given i n Table I I I , are regressed from the data of t h i s study; and a comparison with r e c e n t l i t e r a t u r e data i s shown i n Figure 2. The scatter i s t y p i c a l f o r l i g h t p a r a f f i n vapor p r e s s u r e data from d i f f e r e n t laboratories. The thermodynamic state of a closed system at fixed temperature and s p e c i f i c volume i s completely determined. Pressure can be measured, but not varied independently. Consequently, the p r e s s u r e measurements can be used to determine appropriate parameters i n the t h e o r e t i c a l d e s c r i p t i o n o f the system. T h i s was accomplished by f o r m i n g a s y s t e m o f e q u a t i o n s r e l a t i n g the known and unknown

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

6.19

5.91

5.91

6.19

6.19

6.19

5.91

0.1923

0.3045

0.5071

0.6293

0.7168

0.3782

0.9459

1.0000

5.91

Cell Volume cc

0.00088

Overall Mol Fraction Propane

1.7802

1.8030

1.4471

0.5660

0.46295

0.7216

0.4009

0.0000

Propane grams

Table II.

0.0926

0.2274

0.5199

0.3033

0.40935

1.4989

1.5314

1.4004

Propadiene (impure) grams

35.505 35.409 35.481 35.382

36.092 36.129

36.040 36.029

35.918 35.956

34.339 34.346

32.587 32.590

30.609 30.608

26.505 26.502

253.15 Κ Pressure Psia

79.971 79.881 79.935 79.856

80.805 80.778

80.824 80.800

80.327

77.865 77.871

73.897

70.813 70.800

63.163 63.169

278.15 Κ Pressure Psia

Temperature Corrected Pressure Data

156.627 156.533 156.472 156.395 156.522 156.442

157.158 157.649

157.632 157.846

157.005

155.711

153.263 153.264

146.643

141.393 141.437

128.864 128.897

303.15 Κ Pressure Psia

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

276.631 276.588 276.621 276.540 276.591 276.543

277.697 278.018

278.380 278.715

277.666

275.819

272.067 272.042

261.859

454.489 454.487 454.494 454.461 454.239 454.052

455.680 456.204

456.782 457.074

455.833

453.110

447.469 447.448

432.526

420.694 420.639

393.683 393.801

234.883 234.347 253.748 253.750

353.15 Κ Pressure Psia

328.15 Κ Pressure Psia

BURCHAM ETAL.

Propane-Propadiene System: Phase Equilibria

Table I I I . Goodwin Equation Constants f o r Propane and Propadiene ProDadiene -1.57086 9.00968 -3.35329 5.35825 14.696

ProDane 3.26655 0.65753 -0.18930

A Β

C D p', p s i a Τ', Κ T , Κ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

C

0.56757 14.696 231.1 369.82

280

238.7 393.0

300

320

TEMPERATURE, KELVIN

Figure 2 .

Comparison f o r propane vapor pressure

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

360

92

E Q U A T I O N S O F STATE: T H E O R I E S

AND

APPLICATIONS

v a r i a b l e s , and then s o l v i n g f o r the unknown v a r i a b l e s . components these equations are:

For M

M Component Mass Balances N

x

N

y

N

β

i ~ i x ~ i y

0;

i « l....,M

2

<>

1 Total Volume Balance

(3)

V--NV - Ν V =0 Τ xx y y 2 Summation Equations

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

l - ^ x

i

= 0 ; l - ^ y

i

= 0

(4)

2 Equations of State E(P,V ,T) = 0; E(P,V ,T) - 0 x y

(5)

M Equilibrium Constraints F (P,x ,T) - F (P,y ,T) = 0; i = 1,...,M i

i

i

i

(6)

These equations contain only parameters and p h y s i c a l l y s i g n i f i c a n t v a r i a b l e s . When V , T, and are known, and the parameters are T

specified; the equations can be solved f o r the 2M + 5 unknowns

x^

y , P , V , V , N , N . Comparison o f the measured and calculated i χ y χ y pressures provides v e r i f i c a t i o n of the specified parameters. The SRK equation of state i s : J

J

t

κ

Ρ = RT/(V-b) - a/(V(V+b))

(7)

a = Û bRTF/Q. a D

(8)

b = Û RT /P b c c

(9)

Q

= 0.42748

(10)

= 0.08664

(ID

where

U

n

a u

D

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

4.

BURCHAMETAL.

Propane-Propadiene System: Phase Equilibria

93

where F i s a parameter which f o r c e s e q u a t i o n 7 to reproduce pure component vapor pressures. Equations 7 through 11 correspond to the form described by Reid, e t . a l . (1977). I n i t i a l l y , the o r i g i n a l mixing rules with one i n t e r a c t i o n parameter (12)

(13)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

b

were a p p l i e d ; but, as d e s c r i b e d below, i t was n e c e s s a r y t o add a d d i t i o n a l parameters i n order to f i t the experimental data within i t s estimated uncertainty. Since the SRK e q u a t i o n p r e d i c t s l i q u i d s p e c i f i c volumes p o o r l y , the more a c c u r a t e C o s t a l d c o r r e l a t i o n (Hankinson and Thomson, 1979) was used i n the t o t a l volume balance equation only. B e c a u s e t h i s i s an e x t e n s i v e c o n s t r a i n t , thermodynamic consistency i s s t i l l s a t i s f i e d . At each temperature, pure component F parameters f o r equation (8) were determined so t h a t pure component vapor p r e s s u r e s were a c c u r a t e l y reproduced. For propane, the vapor pressure data of t h i s study were used. For the impurities propene, propane, cyclopropane, and ethene, l i t e r a t u r e vapor pressure data were used (Bender, 1975; Vohra e t . a l . , 1962; H e i s i g and Hurd, 1933; L i n e t . a l . , 1970; Douslin and Harrison, 1968). The parameters are given i n Table IV. I n t e r a c t i o n parameters, k ^ i n E q u a t i o n 12, f o r t h e minor impurities were considered r e l a t i v e l y unimportant and were assigned a value o f 0. The propene-propane parameter was determined from l i t e r a t u r e data (Manley and S w i f t , 1970) to be 0.0085, and the propene-propadiene parameter was estimated to be the same based on experience with hydrocarbon m i x t u r e s . The propane-propadiene parameter was estimated to be 0.028 from the e x p e r i m e n t a l mixture data of this study. Using the F parameters f o r a l l the components except propadiene, and the estimated i n t e r a c t i o n parameters; the t h e o r e t i c a l - m o d e l equations were applied to the impure propadiene vapor pressure d a t a . Pure propadiene F parameters and vapor pressures were calculated. The r e s u l t s are g i v e n i n Table V. Goodwin e q u a t i o n p a r a m e t e r s determined by a l e a s t squares f i t to the pure vapor pressures are given i n Table I I I , and a comparison with l i t e r a t u r e data i s given i n F i g u r e 3. The r e l a t i v e l y l a r g e scatter f o r these data i s probably due to the d i f f i c u l t y i n obtaining pure propadiene and i n keeping i t from d i m e r i z i n g . The e f f e c t of accounting f o r the 3.26 mole percent impurity present i n the propadiene used i n t h i s study was to reduce the estimated pure propadiene vapor pressure by about 1.5%. Next, the propane-propadiene i n t e r a c t i o n parameter, k , f o r 1 2

each data point was déterminée.

Each value of k

1 0

i s shown i n Figure

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

94

Table IV. SRK F Parameters, C r i t i c a l Properties, and Acentric Factors i n t h i s Study

253.15 278.15 303.15 328.15 353.15

Κ Κ Κ Κ Κ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

Τ ,K Ρ ,psia ω

Propane

Propene

Propvne

Cyclopropane

1.843929 1.596573 1.394985 1.227697 1.085501

Parameter F 1.798899 2.167369 1.558492 1.870922 1.362286 1.629764 1.199187 1.430623 1.059943 1.264037

i n Equation 2.032122 1.769849 1.554293 1.375080 1.223972 398.30

Ethene

Propadiene

8 1.189297 1.024429 0.890463 0.779921 0.687514 393.

282.35

369.82

364.90

402.39

616.41

699.06

816.27

809.24

731.28

793.58

0.218

0.132

0.086

0.1493

0.152

Table V.

0.148

Propadiene Properties (Duplicate Data Points)

253.15 Κ Impure Propadiene 26.505 Vapor Pressures 26.502

278.15 Κ 63.163 63.169

303.15 Κ 128.864 128.897

328.15 Κ 234.883 234.847

353.15 Κ 393.683 393.801

Pure Propadiene F Parameters

1.749740 1.749706

1.532509 1.532417

1.352641 1.352645

1.201258 1.201152

62.053 62.059

127.083 127.117

232.230 232.193

389.897 390.019

2.017308 2.017349

Pure Propadiene 25.882 Vapor Pressures 25.879

• Ο A Y X 260

280

300

THIS STUDY (PURE PfWPflOIENE) THIS STUDY (IMPURE PROPflOIENO RPI-4W (1969) HILL (1962) STULL (1947) HRKUTA (1969) 320

3«40

360

TEMPERATURE, KELVIN

Figure 3.

Comparison f o r propadiene vapor pressures

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

4.

Propane-Propadiene System: Phase Equilibria

BURCHAM ET AL.

4,

At a l l f i v e temperatures,

f r a c t i o n propane i n c r e a s e d .

k

1 2

95

i n c r e a s e d s t e a d i l y as the mole

At each temperature,

the value of

that minimized the sum of the squares of the pressure d e v i a t i o n s was also determined. A t y p i c a l pressure deviation plot (for 303.15K with 12 °· ) shown i n Figure 5. This figure demonstrates a nonrandom e r r o r p a t t e r n c o n s i s t e n t with the non-random composition dependence of k ^ - This indicated a need f o r additional parameter(s) k

=

0 2 7 9 2

i

s

i n the mixing rules for the SRK equation. SRK Equation with Additional Interaction Parameters

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

I n i t i a l l y , a second parameter (β) was added to the standard SRK mixing rule i n the form

k

12

=

k

12

+

P

X

1

/

V

(

1

4

)

This equation i n t r o d u c e s a l i n e a r composition dependence f o r the parameter k ^ . The inverse volume factor was incorporated to force 2

the mixing rule t o reduce t o a q u a d r a t i c form i n the low d e n s i t y limit. T h i s i s important i f the equation i s to be applied to the vapor as well as the l i q u i d phase. Equation 14 l e d t o s a t i s f a c t o r y r e s u l t s , but two q u e s t i o n s remained. How c o u l d e q u a t i o n 14 be extended to multicomponent mixtures, and what s i m i l a r i t i e s were there between e q u a t i o n 14 and other r e c e n t l y proposed mixing r u l e s ? Mathias and Copeman (1983) recently published a m o d i f i c a t i o n o f the Peng-Robinson e q u a t i o n (1976) with mixing r u l e s based on l o c a l composition concepts. When the SRK equation i s s u b s t i t u t e d f o r the Peng-Robinson equation, Mathias's l o g i c leads to ρ . V-b

_ —1 V(V+b)

(

_ ln(l+b/V) bRT

a

(15)

In equation 15, a and b are s t i l l given by equations 12 and 13. Two a d d i t i o n a l parameters per binary, d ^ and d ^ are introduced. For the system i n t h i s study values of d were to be i n t r o d u c e d f o r o n l y one of the binaries, namely, the propane-propadiene binary. When a l l d ^ values except d ^ and d are zero, equation 15 may be written as 2 1

V-b V(V+b) * (Xjbjgj+xjb^))

a

b

W

a

l V

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(16)

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

0.07

x MIXTURE DATA -H- HEIGHTS) AVERAGE 0.06 X X

0.05 X

ce

°T-

M X

0.04

X H X

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

•MX X X

260

WO

I î

X X

280

300

χ χ

χ

I

*

320

360

340

TEMPERATURE, KELVIN

Figure 4,

Propane-propadiene i n t e r a c t i o n parameters, k ij

values increase with increasing propane concentration.

0.5 0.4 0.3

— ι

1—

DEVIATION PLOT

K = 0.02792 |2

0.2 04

0.1

in W Pi

0

CM

I

-0.1

ο ο

-0.2 -0.3 -0.4 -0.5

_L 0.2

0.4

_L 0.6

OVERALL

MOLE

FRACTION

_L 0.8

1.0

PROPANE

Figure 5. Mixture pressure deviations at 303.15K f o r oneparameter SRK f i t .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Propane-Propadiene System: Phase Equilibria

BURCHAMETAL.

4.

where the two dimensionless parameters,

and g

97

are given by

2

2 ,2 β

1

(

1/2

1

7

)

and 2 2 _ c2 12 a

ff g

2

d

(a

(

1 (

1/2 ) b RT

R

)

8 )

1 / 2

i a 2

2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

Equation (16) written for a binary mixture i s „ P

RT V=b

=

1 , 2^ v7vTb) lW2

_

I a

l f i i

-V2>2

2^„, l 2

+ 2 ( a

a

.1/2 l 2

)

x

X

( 1

. . ln(l+b/V) - 12 l l l 2b k

_ X

g

b

2f^

In equations 15, 16, and 19, the g (or d) parameters p r i m a r i l y c h a r a c t e r i z e the mixing behavior i n the l i q u i d phase, while k^ has 2

an e f f e c t on both phases.

When g

and g

1

are zero, e q u a t i o n 16

2

reduces t o the o r i g i n a l SRK equation. In equations 15, 16, and 19, the term ln(l+b/V) plays the same role as 1/V i n e q u a t i o n 14. When g i s s e t equal to zero, equations 14 and 19 take on e s s e n t i a l l y the 2

same composition dependence. A g e n e r a l f u g a c i t y c o e f f i c i e n t expression can be obtained from equation 15. However, since the extra parameters were used o n l y f o r the propane-propadiene binary, i t was more convenient to use equation 16. The fugacity c o e f f i c i e n t expression that r e s u l t s from equation 16 i s 2a l

n

A

i

=

f

~"

fr^V

<-i>-m(z-

1 / 2

J "

vfu-Vlinii+b/v)

1 3

+

ln(l+b/V) .2-,-,

1/2 l

a

l X

l V

h_ l 2 V+b X

U

_ \ b

l n l

n

V±b V l

) ( x

M

X

g

g

l

b D

l

+

x X

) 2 2V

g

b

g

D nl

in ψ

•

1

Q. l n * f }

+

(20)

For components 1 and 2, °1

=

x

x

g

b

l 2 l l

/

2

+

X

(

X

2 V l

) g

b

2 2

/ 2

(

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1

)

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

98

and °2

=

x

x

g

b

l 2 2 2

/ 2

+

VVVW

2

<22)

For a l l other components, CL = 0. Results

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

The e x p e r i m e n t a l data i n Table II were recorrelated using equations 16 and 20 with g^^ set equal to zero. Values of the parameters t h a t were obtained i n t h i s two parameter f i t are l i s t e d i n Table VII. Figure 6 shows the deviation plot for 303.15 K. The extra parameter has e s s e n t i a l l y removed t h e non- random d e v i a t i o n p a t t e r n demonstrated by Figure 5. The parameters i n Table VII were used to c a l c u l a t e smoothed phase c o m p o s i t i o n s , volumes, and r e l a t i v e volatilities, α = (y^Xj)/(y /x ), Propane (1) i n propadiene ( 2 ) . o

2

f

2

These a r e shown i n T a b l e s V I I - X I and F i g u r e 7. The r e l a t i v e v o l a t i l i t y curves are not l i n e a r with mole f r a c t i o n . This non-linear behavior has been observed p r e v i o u s l y i n c l o s e - b o i l i n g hydrocarbon m i x t u r e s ( F l e b b e e t a l . , 1982; B a r c l a y et a l . , 1982); c h a r a c t e r i z a t i o n o f t h i s behavior i s important i n the d e s i g n of s e p a r a t i o n p r o c e s s e s . Hakuta et a l . (1969) r e p o r t e d r e l a t i v e v o l a t i l i t i e s f o r the propane-propadiene system and comparisons to t h e i r r e s u l t s shown i n F i g u r e s 8 and 9. A comparison to r e s u l t s reported by H i l l et a l . (1961) i s shown i n Table XII. As discussed elsewhere (Barclay, 1980; Burcham, 1981; Barclay et a l . , 1982) e s t i m a t e d probable e r r o r s i n the c a l c u l a t e d l i q u i d compositions and r e l a t i v e v o l a t i l i t i e s are +0.0005 mole f r a c t i o n and +0.004 units respectively assuming that the pure material analysis i s good and that calculated vapor and l i q u i d d e n s i t i e s are r e a s o n a b l y a c c u r a t e . A comparison of the calculated saturated vapor and l i q u i d compressibility f a c t o r s f o r propane with p u b l i s h e d data (Das and v

L

Eubank, 1973) shows an error of +1% i n Z - Z at 253.15°K and +4% at 353.°K. Since these e r r o r s generate e q u i v a l e n t percent e r r o r s i n l n o , t h e c a l c u l a t e d r e l a t i v e v o l a t i l i t i e s may be o f f a maximum additional 0.0010 units at the pure propane c o n d i t i o n s and 0.0100 u n i t s at the pure propadiene c o n d i t i o n s . Experimental propadiene densities are not a v a i l a b l e and no e f f o r t was made t o reduce t h i s contribution to the t o t a l error. As can be seen from e q u a t i o n 19, the b i n a r y i n t e r a c t i o n term when g i s zero i s given by 1

! _

k K

_ 12

χ x

g

b g

D

2 2 2

ln
In t h i s study, values of k Values of the term, * 8 *> 2

2

2

1 2

were always p o s i t i v e and l e s s than ln( 1+b/V)/2b, were always p o s i t i v e

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

0.04. and

BURCHAM ET AL. Propane-Propadiene System: Phase Equilibria Table VI.

Binary Interaction Parameters f o r the Propane-Propadiene Binary

τ,κ

k

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

253.15 278.15 303.15 328.15 353.15

g

12

0.1564 0.1147 0.0532 0.0440 0.0356

0.03329 0.02956 0.02775 0.02694 0.02599

0.5 0.4

I

I

2

Γ

K =0.02775

DEVIATION PLOT

|2

G * 0.0532 2

0.3 0.20.1-

-0.1-0.2o

ο

-0.3 -0.4-

_L

-0.5,

Ό

0.2

0.4

I

0.6

0.8

10

OVERALL MOLE FRACTO IN PROPANE

Figure 6. Mixture pressure deviations at 303.15K f o r twoparameter SRK f i t .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Table V I I .

Calculated Propane(l)-Propadiene(2) VLE at 253.15 K. LIQ V from C c s t a l d , Other Values from SRK Equation V Y

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

±

0.0 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000 0.5500 0.6000 0.6500 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000

0.0 0.1184 0.2019 0.2658 0.3180 0.3631 0.4038 0.4420 0.4788 0.5151 0.5516 0.5889 0.6272 0.6669 0.7082 0.7514 0.7967 0.8441 0.8938 0.9457 1.0000

P,psia 25.88 27.92 29.43 30.58 31.50 32.26 32.91 33.46 33.95 34.38 34.75 35.07 35.33 35.54 35.70 35.80 35.84 35.82 35.75 35.62 35.44

L I Q

,

3

cm /g-mol 63.81 64.56 65.31 66.07 66.83 67.59 68.36 69.13 69.90 70.68 71.47 72.25 73.04 73.83 74.63 75.43 76.24 77.05 77.86 78.67 79.49

Relative VAP Ζ 0.9593 0.9558 0.9532 0.9511 0.9494 0.9479 0.9467 0.9455 0.9445 0.9436 0.9427 0.9419 0.9412 0.9406 0.9400 0.9395 0.9390 0.9386 0.9383 0.9380 0.9378

Volatility 2.892 2.552 2.276 2.051 1.865 1.710 1.581 1.471 1.378 1.298 1.230 1.172 1.121 1.078 1.040 1.008 0.980 0.956 0.935 0.917 0.902

Table V I I I . Calculated Propane(l)-Propadiene(2) VLE at 278.15 K. LIQ V from Costald, Other Values f o r SRK Equation V

L I Q

,

3

Χ

P,psia

χ

0.0 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000 0.5500 0.6000 0.6500 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000

0.0 0.0932 0.1673 0.2293 0.2833 0.3319 0.3769 0.4197 0.4611 0.5017 0.5422 0.5830 0.6243 0.6664 0.7097 0.7541 0.8000 0.8475 0.8965 0.9474 1.0000

62.06 65.25 67.80 69.90 71.66 73.16 74.46 75.59 76.58 77.44 78.19 78.83 79.36 79.79 80.11 80.33 80.45 80.46 80.38 80.19 79,91

cm /g-mol 67.21 68.03 68.86 69.68 70.52 71.36 72.20 73.05 73.90 74.76 75.62 76.48 77.36 78.23 79.11 80.00 80.89 81.78 82.69 83.59 84.50

Relative VAP Z ¥

0.9218 0.9172 0.9135 0.9103 0.9076 0.9051 0.9030 0.9010 0.8992 0.8976 0.8961 0.8947 0.8934 0.8922 0.8911 0.8902 0.8893 0.8885 0.8878 0.8872 0.8867

Volatility 2.121 1.952 1.808 1.686 1.581 1.490 1.412 1.343 1.283 1.231 1.185 1.144 1.108 1.076 1.048 1.022 1.000 0.980 0.963 0.947 0,934

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Propane-Propadiene System: Phase Equilibria

B U R C H A M ET A L .

Table IX. Calculated Propane(l)-Propadiene(2) VLE at 303.15 K. LIQ V from Costald, Other Values from SRK Equation L I Q

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

Χ

Υ

χ

0.0 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000 0.5500 0.6000 0.6500 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000

Ρ,psia

χ

0.0 0.0771 0.1441 0.2040 0.2585 0.3093 0.3573 0.4033 0.4480 0.4919 0.5353 0.5785 0.6220 0.6658 0.7103 0.7556 0.8019 0.8493 0.8980 0.9482 1.0000

127.10 131.59 135.44 138.77 141.69 144.25 146.50 148.49 150.25 151.78 153.12 154.26 155.21 155.99 156.58 157.00 157.25 157.32 157.22 156.95 156.50

V , 3 cm /g-mol 71.42 72.33 73.26 74.19 75.12 76.07 77.02 77.97 78.94 79.91 80.89 81.87 82.86 83.86 84.86 85.87 86.89 87.92 88.95 89.99 91 03 t

Relative VAP Ζ 0.8676 0.8620 0.8571 0.8527 0.8487 0.8450 0.8416 0.8385 0.8357 0.8330 0.8306 0.8283 0.8262 0.8242 0.8224 0.8207 0.8192 0.8179 0.8166 0.8155 0.8146

Volatility 1.667 1.587 1.515 1.452 1.395 1.343 1.297 1.256 1.218 1.183 1.152 1.123 1.097 1.073 1.051 1.031 1.012 1.995 0.979 0.964 0.950

Table X. Calculated Propane(l)-Propadiene(2) VLE at 328.15 K. LIQ V from Costald, Other Values from SRK Equation L I Q

Χ

Υ

χ

χ

0.0 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000 0.5500 0.6000 0.6500 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000

0.0 0.0690 0.1314 0.1887 0.2422 0.2930 0.3417 0.3889 0.4350 0.4805 0.5256 0.5706 0.6158 0.6612 0.7071 0.7537 0.8010 0.8491 0.8983 0.9485 1.0000

P,psia

V , 3 an /g-mol

232.21 238.75 244.47 249.53 254.00 257.97 261.50 264.62 267.37 269.77 271.85 273.62 275.09 276.26 277.14 277.74 278.06 278.10 277.86 277.36 276 ?9

76.93 78.00 79.08 80.17 81.27 82.38 83.51 84.64 85.79 86.94 88.11 89.29 90.48 91.69 92.91 94.14 95.38 96.63 97.90 99.19 100.48

t

Relative VAP Z ¥

r

0.7939 0.7866 0.7799 0.7737 0.7679 0.7626 0.7577 0.7531 0.7488 0.7447 0.7410 0.7375 0.7342 0.7311 0.7283 0.7257 0.7234 0.7212 0.7192 0.7174 0.7158

Volatility 1.462 1.409 1.361 1.318 1.279 1.243 1.211 1.182 1.155 1.130 1.108 1.087 1.068 1.051 1.035 1.020 1.006 0.993 0.981 0.970 0.960

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

102

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

TABLE XI. Calculated Propane(l)-Propadiene(2) VLE at 353.15 K. LIQ V from Costald, Other Values from SRK Equation L I Q

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

Χ

Υ

α

0.0 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500 0.5000 0.5500 0.6000 0.6500 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000

χ

0.0 0.0626 0.1208 0.1757 0.2281 0.2785 0.3276 0.3757 0.4230 0.4700 0.5167 0.5634 0.6102 0.6572 0.7046 0.7524 0.8007 0.8496 0.8990 0.9492 1.0000

0

Ρ,psia

V , 3 cm /g-mol

389.96 398.94 406.97 414.17 420.64 426.44 431.63 436.26 440.36 443.97 447.09 449.74 451.94 453.71 455.04 455.94 456.43 456.51 456.18 455.47 454,37

84.99 86.35 87.73 89.14 90.57 92.03 93.51 95.03 96.57 98.14 99.74 101.37 103.04 104.75 106.50 108.28 110.12 111.99 113.92 115.90 117.94

0.2 MOLE

Figure 7.

04 FRACTION

0.6

Relative VAP Ζ

Volatility

0.6949 0.6849 0.6755 0.6667 0.6583 0.6503 0.6427 0.6355 0.6287 0.6221 0.6159 0.6100 0.6044 0.5992 0.5943 0.5896 0.5853 0.5814 0.5777 0.5744 0.5713

1.303 1.268 1.237 1.208 1.812 1.158 1.137 1.117 1.100 1.084 1.069 1.056 1.044 1.032 1.022 1.013 1.005 1.997 0.990 0.983 0.977

08

1.0

PROPANE

Relative v o l a t i l i t y of propane propadiene.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

B U R C H A M ET A L .

Propane-Propadiene System: Phase Equilibria

Figure 8. Comparison f o r r e l a t i v e v o l a t i l i t i e s at 273.15K. Line i s calculated from results of t h i s study, points are Hakuta's data (1969).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

104

EQUATIONS OF STATE. THEORIES AND APPLICATIONS

0

0.2

0.4 MOLE

0.6

FRACTION

0.8

1.0

PROPANE

Figure 9. Comparison f o r r e l a t i v e v o l a t i l i t i e s at 293.25K. Line i s calculated from r e s u l t s of t h i s study, points are Hakuta's data (1969).

TABLE XII.

Comparison of Experimental and Calculated Relative V o l a t i l i t i e s of Propane i n Propadiene

Liquid Phase Mole f r a c t i o n s

Relative V o l a t i l i t i e s Experimental Propane Propadiene Propene Propvne Temperature ( H i l l . 1961) Calculated 0.967 0.976 305.37 0.013 0.912 0.058 0.017 0.965 0.973 305.37 0.015 0.924 0.047 0.014 1.584 1.441 305.37 0.012 0.026 0.027 0.935 1.595 1.464 305.37 0.012 0.933 0.018 0.037 1.601 1.464 305.37 0.010 0.036 0.013 0.941 .974 0.971 333.15 0.012 0.055 0.919 0.014 .973 0.978 333.15 0.013 0.011 0.929 0.047 1.401 1.287 333.15 0.016 0.011 0,959 0.014

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Propane-Propadiene System: Phase Equilibria

BURCHAMETAL.

4.

105

l e s s than 0.045 (the value for pure propadiene at 253.15K). At the four higher temperatures, t h i s l a s t term was never more than 0.026. The data i n t h i s study were f i t to various forms o f equation 16, and none was judged to be as s a t i s f a c t o r y as the form just r e p o r t e d , i . e . with g^ = 0. For the case when a l l three parameters were f i t , the three parameters proved to be highly c o r r e l a t e d , the parameters d i d not vary smoothly with temperature, and the f i t was improved by an i n s i g n i f i c a n t amount. When other two-parameter combinations were tested, i . e . , k^ = 0, or g = 0, the f i t to the data was better than 2

2

with the one parameter model, but always worse than f o r the two parameter case when g^ = 0. When

i s zero, and g

2

i s a p o s i t i v e number, the p r e d i c t e d

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

r e l a t i v e v o l a t i l i t y i s increased at low propane mole f r a c t i o n s . When g^ i s a p o s i t i v e number, the e f f e c t i s t o l o w e r t h e r e l a t i v e v o l a t i l i t y predictions at high propane mole f r a c t i o n s . Summary New experimental t o t a l p r e s s u r e data f o r the propane-propadiene system with s e v e r a l known minor i m p u r i t i e s a r e p r e s e n t e d . A c o m p u t a t i o n a l method f o r r e d u c i n g t h e d a t a i s developed and demonstrated. A modification of the SRK e q u a t i o n o f s t a t e with an a d d i t i o n a l i n t e r a c t i o n parameter i s a p p l i e d . C a l c u l a t e d purepropadiene vapor pressures and r e l a t i v e v o l a t i l i t i e s for the propanepropadiene system are given. Acknowledgment The authors acknowledge the u n i v e r s i t y o f Missouri-Rolla computer center for computer time. Andy Burcham acknowledges the Chemical Engineering Department at UMR for f i n a n c i a l support. Legend o f Symbols mixture constants i n SRK equation; a , b

a,b

±

i;

a . i s f o r pure i at i t s c r i t i c a l

±

are f o r pure

temperature

CI

A,B,C,D,P',T' Ε F,

constants i n Goodwin vapor pressure equation binary i t e r a c t i o n parameter (see equation 15) equation of state (see equation 5) fugacity of i (see equation 6) p u r e component equation 8

g

g

l' 2

k°.

i n SRK e q u a t i o n , s e e

b i n a r y i n t e r a c t i o n parameters (see equations 16-19). In t h i s study g

k. .

parameter

1

=0

binary interaction parameter constants i n equation 14

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

106 Ν

moles; Ν , moles i n the l i q u i d phase; Ν , moles i n the χ y vapor phase

Ρ

pressure, psia; Ρ , c r i t i c a l pressure, psia

Q

±

Τ V

V X χ

T

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

y Ζ Greek α β

û

term i n fugacity

c o e f f i c i e n t expression, see equations

19-21 temperature. Κ; Τ , c r i t i c a l temperature molar volume ; V i s molar volume o f l i q u i d , V i s χ y molar volume of vapor t o t a l volume temperature function i n equation 1 mole f r a c t i o n i n the l i q u i d phase i n equations 2-6, i n either phase i n equations thereafter mole f r a c t i o n i n the vapor phase ν L compressibility factor; Ζ for vapor, Ζ for l i q u i d relative v o l a t i l i t y constant i n equation 14 fugacity c o e f f i c i e n t of i a pure number, see equations 8-11

Literature Cited

1. Am. Petr. Inst. Proj. 44, Supplement 1974. 2. Am. Petr. Inst. Proj. 44, Supplement 1977. 3. Barclay, D. Α., Ph.D. Thesis, University of Missouri-Rolla 1980. 4. Barclay, D. A.; Flebbe, J. L; Manley, D. B., J. Chem. Eng. Data 1982, 27, 135. 5. Bender, E. Cryogenics 1975, 15, 667. 6. Burcham, A. F., M. S. Thesis, University of Missouri, Rolla 1981. 7. Das, T. R.; Bubank, P. T. Adv. Cryog. Eng. 1973, 18, 208. 8. Douslin, D. R.; Harrison, R. H. J . Chem. Thermodyn. 1976, 8, 301. 9. Flebbe, J. L.; Barclay, D. Α.; Manley, D. B. J. Chem. Eng. Data 1982, 27, 405. 10. Goodwin, R. D. J. Res. Nat. Bur. Stand. 1975, 79A, 71. 11. Hakuta, T.; Nagahama, K.; Hirata, M. Bull. Jap. Petr. Inst. 1969, 11, 10. 12. Hankinson, R. W.; Thomson, G. H. AIChE J. 1979, 25, 653. 13. Heisig, G. B; Hurd, C. D. J. Amer. Chem. Soc. 1933, 55, 3485. 14. Hill, A. B.; McCormick, R. H.; Barton, P.; Fenske, M. R. AIChE J. 1961, 8, 681. 15. Lin, D. C.; Silberberg, I. H.; McKetta, J. J. J. Chem. Eng. Data 1970, 15, 483. 16. Manley, D. B.; Swift, G. W. J. Chem. Eng. Data 1971, 16, 301. 17. Manley, D. B. Hydro Proc. 1972, Jan., 113. 18. Martinez-Ortiz, J. Α.; Manley, D. B. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 346.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

4. BURCHAM ETAL.

Propane-Propadiene System: Phase Equilibria

107

19. Mathias, P. M.; Copeman, T. W. Fluid Phase Equil. 1983, 13, 91. 20. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. 21. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. "The Properties of Gases and Liquids", 3rd ed., McGraw-Ηill: New York 1977, 40. 22. Soave, G. Chem. Eng. Sci. 1972, 27, 1197. 23. Steele, K.; Poling, Β. E.; Manley, D. B. J . Chem. Eng. Data 1976, 21 399. 24. Stull, D. R. Ind. Eng. Chem. 1947, 39, 517. 25. Vohra, S. P.; Kang, T. L.; Kobe, Κ. Α.; McKetta, J. J . J . Chem. Eng. Data, 1962, 7, 150. December 12, 1985

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch004

RECEIVED

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

5 Nonclassical Description of (Dilute) Near-Critical Mixtures J. M. H. Levelt Sengers, R. F. Chang, and G. Morrison

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

Thermophysics Division, Center for Chemical Engineering, National Bureau of Standards, Gaithersburg, MD 20899

The organizers of this symposium have asked us to review the nonclassical equations of state and their applications in near-critical fluids and fluid mixtures. In this contribution, the emphasis will be on fluid mixtures. The behavior of fluid mixtures near the gas-liquid critical line has undergone a revival of interest for a variety of reasons. One is the appearance of a number of reports of large anomalies in properties such as apparent heats of dilution (1), apparent molar volumes (2) and apparent molar specific heats (3) of dilute salt solutions in near-critical steam, and of extraordinarily large enthalpies of mixing at supercritical pressures in mixtures with components of very different critical temperature (4). Another reason is the strong push for exploration of the supercritical regime in separation processes, a promising alternative to liquid extraction (5). Supercritical solubility is governed, in part, by the partial molar properties of the solute, which have been reported to behave anomalously. It seems therefore useful at this point to take stock and review: (1) what are the differences in prediction between classical and nonclassical equations for fluids and fluid mixtures; (2) how the nonclassical behavior is going to be meshed with classical behavior further away and (3) what types of nonclassical equations are available for fluid mixtures and what is their range of validity. The anomalous properties of dilute near-critical mixtures, which are of interest in custody transfer and in supercritical solubility, are discussed in the last part of this contribution. C r i t i c a l Behavior of Pure Fluids This section defines the terminology and summarizes the differences between c l a s s i c a l and nonclassical behavior of pure f l u i d s i n the This chapter not subject to U.S. copyright. Published 1986, American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

5.

LEVELT SENGERS ET A L .

Near-Critical Mixtures

111

b r i e f e s t of terms. The material presented here i s in standard notation and has been reviewed elsewhere ( 6 ^ 7j_ 8 ) . C r i t i c a l behavior i s called c l a s s i c a l (mean-field, or van der Waals-like) i f the Helmholtz free energy a ( V , T) can be expanded at the c r i t i c a l point i n terms of 6V = V - V and δΤ = Τ - T V being q

c>

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

the volume, Τ the temperature and the subscript c denoting the c r i t i c a l value. If such an expansion exists, thermodynamic properties vary i n accordance with simple power laws along given paths asymptotically near the c r i t i c a l point. The power laws and exponent values are summarized in Table 1 and the paths are indicated in Figure 1. Real f l u i d s do not behave c l a s s i c a l l y . Their coexistence curves and c r i t i c a l P - V isotherms are f l a t t e r than predicted by c l a s s i c a l equations, and their s p e c i f i c heat Cy shows a weak divergence (Table 1 ) . Renormalization-group theory explains how nonclassical exponent values result from the correlation of the c r i t i c a l fluctuations, and the theoretical exponent values for the universality class of 3D Ising-like systems have been confirmed in a number of delicate experiments, see, for instance ( 9 ) , ( 1_0). The scaling laws (6 - 8 , U_) that describe the thermodynamics of n e a r - c r i t i c a l f l u i d s can be viewed as a compact "packaging" of the power laws of Table 1. They imply the exponent equalities 2 - α - 0(6

+ 1)

; Ύ - β(δ - 1)

(1)

so that only two exponents are independent; these are, however, the same for a l l systems in the 3D Ising-like universality class. Likewise, only two amplitudes in the power-law expressions can be chosen independently; these are not universal. We w i l l c a l l the c r i t i c a l behavior shown by real f l u i d s and predicted by renormalization-group theory nonclassical. It i s important to note the difference between strongly (Ύ-like) and weakly (α-like) diverging properties i n the nonclassical case (j_2). Properties which are the same i n coexisting phases are called " f i e l d s " . Examples are pressure P, chemical potential μ and temperature T. Thus Ρ (μ, Τ ) , with Ρ = Ρ/Τ, μ = μ/Τ and Τ = 1/Τ i s an example of a thermodynamic potential i n which a l l variables are f i e l d s . First derivatives, the density ρ = (3Ρ/9μ) (3P/9T)

τ

and the energy density Up =

, are generally different in coexisting phases.

Such f i r s t

derivatives are called "densities". Other examples of densities include volume V , enthalpy H, entropy S, and composition x. The strong divergences are obtained ( c l a s s i c a l l y and nonclassically) i f a derivative i s taken of a density with respect to a f i e l d while another f i e l d i s kept constant, that i s , i n a d i r e c t i o n intersecting the coexistence curve in Figure 1 b . I f , however, a density i s kept constant at d i f f e r e n t i a t i o n , one obtains the nonclassical weak divergences; a constant - density direction i s p a r a l l e l to the coexistence curve in Figure 1 b . (J_2). Examples of strong and weak divergences in one-component f l u i d s are l i s t e d in Table I I .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

112

EQUATIONS O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table I C r i t i c a l e x p o n e n t d e f i n i t i o n s and v a l u e s The p a t h s a r e i n d i c a t e d i n F i g u r e 1 Property

Specific

D e f i n i t i o n of c r i t i c a l exponent

heat

Coexistence

Curve

c

v

- |T-T f

a

c

|p-p l " I T - T ι 0

Compressibility

κ

Critical

|P-P I

isotherms

τ

β

Value for real fluids

α =0

α =0 1

3 - 0.5

3 = 0 33

Ύ = 1

Ύ = 1

g

- |T-T f

Y

c

C

Classical Value

"

|P-P I

6

δ

=

3

C

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

δ

21

- 4 .8

5.

LEVELT SENGERS ET AL.

Near-Critical Mixtures

113

For d e t a i l s of the scaled form of the potential Ρ (μ, Τ) we refer to the l i t e r a t u r e (6 - 8). Conceptually, t h i s form i s construed as follows. The potential i s s p l i t into a part that i s analytic i n μ and Τ and a part that i s scaled.

The scaling

variables are chosen as a reduced temperature difference τ = Δ Τ i n the weak direction along the coexistence curve and a

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

reduced chemical potential difference h = Δμ i n the strong d i r e c t i o n intersecting this curve (Figure 2). The scaled part i s written i n algebraically closed form by means of Schofield's parametric variables, a distance-from-critical variable r and a contour variable θ. Two refinements are made. One goes by the name of "revised s c a l i n g " : the gas-liquid asymmetry c h a r a c t e r i s t i c of r e a l f l u i d s i s obtained by replacing the weak scaling variable by a linear combination of ΔΤ and Δμ (7, 1_3) which i s equivalent with " t i l t i n g " the h axis i n Figure 2. The other refinement i s c a l l e d "corrections to s c a l i n g " or "extended s c a l i n g " . These corrections, introduced by Wegner, account for the difference between the c r i t i c a l Hamiltonian of the r e a l f l u i d and the Hamiltonian characterizing the fixed point for 3D I s i n g - l i k e systems [7]. PVT and thermal data for a number of pure f l u i d s , including steam, ethylene and isobutane have been f i t t e d accurately i n the range of -0.01 to +0.1 i n reduced temperature around the c r i t i c a l point by means of revised and extended scaling with one Wegner correction term (8). Crossover from C l a s s i c a l to Scaled Behavior Universal scaling behavior r e s u l t s when the behavior of a system i s dominated by one length scale, that of the c o r r e l a t i o n length, which diverges at the c r i t i c a l point. When the system moves away from the c r i t i c a l point, other length scales begin to become important and the system gradually assumes the nonuniversal species-specific behavior that i s described by c l a s s i c a l equations. Only very recently, progress has been made with devising functions that scale near the c r i t i c a l point and are c l a s s i c a l at some distance from i t . There have been a number of attempts at empirically connecting " c r i t i c a l " and c l a s s i c a l " regimes. Switch functions have been introduced to smoothly connect the two regimes (14, 15)Î great d i f f i c u l t y i s , however, experienced with derived properties, because of anomalous contributions from higher derivatives of the switch function. Only i f c l a s s i c a l and scaled free energy are exceedingly close to each other i n the range of the switch can these anomalous variations be suppressed. H i l l reports good success with a switch function for steam (16). A true crossover function i s a function that behaves c l a s s i c a l l y far from and i s scaled close to the c r i t i c a l point. Fox transformed the variables i n a c l a s s i c a l thermodynamic potential i n such a way that nonclassical behavior was obtained near the c r i t i c a l point (j_7). His method gives good r e s u l t s for PVT and coexistence curve data but has not been t r i e d for higher derivatives such as C... White (18) has reported success with a

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

114

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table I I .

Strong and Weak Divergences

Strong (Ύ-type)

weak (a-type)

other (nondivergent)

second derivative in direction intersecting coexistence surface in field space

second derivative taken in coexistence, but not in critical surface in field space

second derivative along critical surface in field space

one-component τ

\dp)

V

K T

s --7

(lf)

N.A.

s

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

(")P

(")v

two-component, Δ=μ ~μ 1

τΔ

v \spy, ι

Tx

-

1

2

(*A

¥1

/<sv\

Px °ΡΔ Ρ Δ Ε

*

·

V

^ T J

P

c„ = τ Vx

I

Τ

(3χ/8Δ) , ρ τ

osmotic

(«)ρΔ

susceptibility Οχ/3Ρ)

Τ μ

, supercritical

Ox/3P),

solubility

Figure 2. Directions of the strong (h) and weak (τ) scaling variables i n the space of independent variables μ, Τ f o r a one-component f l u i d . In simple scaling, the h axis i s v e r t i c a l ; i n revised scaling, i t i s not.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

5.

Near-Critical Mixtures

LEVELT SENGERS ET A L .

115

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

systematic averaging procedure for fluctuations on various length scales for the van der Waals equation. Crossover from scaled to mean-field behavior was solved for the Landau-Ginzburg Hamiltonian for 3D I s i n g - l i k e systems (J_9, 20) for both the symmetric and the asymmetric case, by N i c o l l and Albright. The l a t t e r has constructed a crossover function that crosses from the scaled equation to the van der Waals equation i n a range around the c r i t i c a l isochore (21). C l a s s i c a l C r i t i c a l Behavior of F l u i d Mixtures By c l a s s i c a l behavior we mean again that the Helmholtz free energy a(V, Τ, χ ) , with χ the mole f r a c t i o n , can be expanded at the c r i t i c a l l i n e i n terms of i t s independent variables. The ideal-mixing term RT[x in χ + (1-x) Jin (1-x)] must be treated separately because i t i s not analytic at x=0 and x=1. The implications of c l a s s i c a l behavior can be visualized by assuming, with van der Waals, that the mixture, at constant composition x, i s in corresponding states with i t components. In Figure 3a, b, we draw the equivalent of Figure 1a, b for the mixture, i n reduced coordinates Ρ Ρ

f

p

c

and T

q

= Ρ/Ρ , V = V/V and Τ = T/T . For the mixture, c c c represent the pseudocritical parameters of the point

at which the mixture would become mechanically unstable i f i t would remain homogeneous. In r e a l i t y , the mixture becomes materially unstable before i t reaches the pure-fluid coexistence curve. The curve of material i n s t a b i l i t y , or dew-bubble curve, i s indicated i n Figure 3a. Inside this curve, the mixture s p l i t s into two phases of compositions other than x^, and does not follow the pure-fluid isotherms. In a constant - χ representation the c r i t i c a l point i s not at the top and there are no t i e l i n e s i n the p l o t . Figure 3a makes clear that the compressibility Κ , and Τ χ

therefore also α

ρ χ

and C

p x >

are f i n i t e at the c r i t i c a l point.

Thus, these mixture properties are not analogous to their strongly diverging counterparts in one-component f l u i d s . Mixtures do show strong divergences; and example i s the osmotic s u s c e p t i b i l i t y (3χ/3Δ)ρ , with Δ = μ which diverges because of the Τ

1

c r i t i c a l i t y conditions: (3Δ/3χ)

2

ρτ

2

= (3 G/3x )

2

p T

2

- 0 ; 0 Δ/3χ )

3

ρ τ

3

= (3 G/8x )

p T

= 0

(2)

In Figure 3b we show the equivalent of Figure 1b for a mixture of constant composition; also indicated are the isobar that passes through the maximum on the phase boundary i n Figure 3a and b, and the isotherm tangential to the phases boundary. In this example, the c r i t i c a l point i s located between the two extrema i n Figure 3b. This i s the most common, but not the only case. In Figure 3b we also draw a few isochores. Note that the c r i t i c a l isochore changes slope at the phase boundary while the isochore passing through the temperature extremum does not change slope (22). By means of the law of corresponding states, van der Waals was able to predict

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

116

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

almost a l l of the often-complex phase behavior of binary mixtures that has l a t e r been found experimentally. Van Konijnenburg and Scott ( 2 3 ) generated by computer a l l binary-mixture phase diagrams that follow from the van der Waals equation through the law of corresponding states. Nonclassical C r i t i c a l Behavior of F l u i d Mixtures The defects of a c l a s s i c a l description are as obvious i n f l u i d mixtures as i n pure f l u i d s . C l a s s i c a l models for the coexistence curves of mixtures are never f l a t enough at the top, since binary gas-liquid and l i q u i d - l i q u i d coexistence curves are cubic (£4, 25). G r i f f i t h s and Wheeler (J_2) generalized the scaling laws for pure f l u i d s to f l u i d mixtures by means of the concept of c r i t i c a l - p o i n t u n i v e r s a l i t y . Although, as we have seen, a one-component f l u i d and a mixture of constant composition do not have the same two-phase behavior, G r i f f i t h s and Wheeler postulated that they do i f the mixture i s considered at constant f i e l d . This p r i n c i p l e i s indicated schematically i n Figure 4. In this figure the independent f i e l d s , f o r the scaled potential developed by Leung and G r i f f i t h s (26), are the weighted average a c t i v i t y a =

+ a

2

and the normalized a c t i v i t y of the second component: ζ = c Here c^, c

2

2

a

2

/ (c

a

1

+ c

2

a ) 2

(3)

are two constants related to the choice of zeropoints

of chemical potentials, a^^ - exp (yVRT), and ζ runs from 0 (pure component 1) to 1 (pure component 2). The variable ζ, rather than the customary composition variable x, connects the a-T diagrams of the two pure components. Whereas i f χ i s the variable, a coexistence surface consists of two sheets, a dew and a bubble surface, i t i s single valued i f ζ i s the variable, since ζ i s a f i e l d . A point on the c r i t i c a l l i n e can now be approached i n one of three ways. If two f i e l d s are kept constant, f o r instance ζ and the temperature, a path intersecting the coexistence curve i s obtained. We expect this d i r e c t i o n , indicated by h i n Figure 4, to be "strong" just as i n the pure f l u i d , Figure 2, and corresponding second derivatives to be strongly divergent. Several examples are given i n Table I I . Note that the osmotic susceptibility Ο χ / 3 Δ ) and the s u p e r c r i t i c a l s o l u b i l i t y ρ τ

Ox/3P)

T

^

constant μ

i n the presence of a s o l i d or l i q u i d phase of (almost) 2

are i n this category of strong, Ύ-like divergences.

If one density i s kept constant i n the d i f f e r e n t i a t i o n , f o r instance the volume or the entropy, then the c r i t i c a l point i s approached along the coexistence surface, the weak direction indicated by τ i n Figure 4, and second derivatives diverge weakly. Note, i n Table 2, that Κ , C and α are i n this category. Τ χ

p x

ρ χ

F i n a l l y , when two densities are held constant, the path to a point on the c r i t i c a l l i n e becomes asymptotically p a r a l l e l to this l i n e . Since f i r s t derivatives such as volumes and entropies vary

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

LEVELT SENGERS ET AL.

Near-Critical Mixtures

(b)

P

1 x.

/

m

V^ A

ψ /

Ρ /

ι T

!m '''''À

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

T/T

c

Figure 3 a, b. They are the equivalents of Figure 1a, b for a mixture of constant composition x, by virtue of the law of corresponding states. · i s the pure-fluid c r i t i c a l point, a pseudocritical point for the mixture. The mixture phase-separates on the dew-bubble curve and does not attain the states — ~ inside this curve. Ο i s the real c r i t i c a l point of the mixture. Τ i s the maximum temperature, Ρ the m m maximum pressure, f o r the dew-bubble curve at x^ . In (b), several isochores are indicated. The c r i t i c a l isochore changes slope at the dew-bubble curve while the isochore through Τ does not. m

I

Figure 4. of

Coexistence surface and c r i t i c a l

independent

l i n e i n the space

variables a (average a c t i v i t y ) , c(=a /a) and T. 2

The scaling variables h (strong) and τ (weak) are chosen i n the plane of constant ζ. ζ i s the third variable chosen along the c r i t i c a l l i n e .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

118

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

smoothly

along

derivatives

critical

taken

in

the

In

to

construct

P(a, of

order

the

at

constant

pure

a regular

is

function

a

ζ

is

components

into

of

the

surface

and with

The

of

and the

the

pure

algebraically with

limited

adjustable for yet

been

with

corrections

simple

to

scaling

to

It

is

extent.

describe

one

of

v

mixtures

modifications, azeotropic Moldover, equation

it

He

order

region

describing

at

at

an angle

phenomenogical of

the

used

their

and

of

available It

is

and has

not

and

so-called

departure

e m p i r i c a l l y the

lines,

lines.

with

form,

scaling

nature,

number

fluids

the

variable.

in

equation

critical

ζ,

to

between

critical

large

scaled

near

split

parametric

interpolation

continuous

by

is

constant

interpolated

the

the that

Ρ(μ, Τ),

potential

l i q u i d - v a p o r asymmetry

(26,

from

the

corrections

properties,

27,

applied

as

to

such

have

the

true

to form

as

has

PVT a n d

of

minor

two-phase

modified been

and at is

of (29).

2

the

Leung-Griffiths

very

pressures

presently

in

successful

hydrocarbon mixtures,

model

states

and C 0 ~ e t h y l e n e

l i q u i d - v a p o r asymmetry

They have

curves

With

(28)

2

3]_).

Their

22). one-and

C0 ~ethane

in volatility,

pressure.

as

two-phase

describe

dew-bubble

differences

critical

been

(3£,

Τ ) , or

part,

one-component

and c o w o r k e r s

to

Pta^ the

for

postulated

The n o n u n i v e r s a l

therefore

a n d He

such

Rainwater

later.

il

J

has

mixtures

in

two-phase large

of

second of

The o r i g i n a l L e u n g - G r i f f i t h s

and

Q

C ,

(26)

Schofield s

only

which,

accommodate

a certain to

in

include

exponents

values,

to

used

ζ to

the

if

discussed

f

in

because

yet

result

exception

and h ,

fluid.

although

scaling

scaling.

critical

asymptotic

along,

with

power is

as

Again,

g a s - l i q u i d mixtures

generalized

apparent

been

of

τ,

restricted

parameters,

modeling

analogous

tedious,

be

background are

equation,

predictive

to

The s c a l e d

pure

components,

Leung-Griffiths

2).

written

analytic

1,

the

thermodynamic p o t e n t i a l

same f o r m

part.

the

with

and G r i f f i t h s

= 1 or

is

divergences

0 or

the

variables

analogy

those

-

Leung

of

(i

in

amplitudes

ζ

no

line,

a scaled

and a s c a l e d

coexistence complete

this

limits

gas-liquid mixture,

T)

line,

along

x-derivatives binary

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

the

are

even

down t o

limited

to

the in with

half

the

two-phase

states. The fluid

only

application in

other

mixtures

is

has

been

binary liquids,

Bartis

and H a l l

One o f liquid pure

us to

mostly

by

those

with

scaling

amplitudes

the

such

36)

model

that

lattice the

of

to

constructed

a

scaled the

Recently,

enthalpies

at

method,

analytic

of

potential

Johnson has

two

of

used

(3j0. (35).

for

variables

constant

the

boundaries

curves

equilibria

scaling

in

The

phase

binary of

pressure.

the In

nonuniversal

b a c k g r o u n d were

and volumes

applied

33).

coexistence gas-gas

a binary mixture

and the

been

(32,

description

a model

Leung-Griffiths

has

gas

closed-loop

a transformation of

pressure-dependent. excess

to

some w i t h

(L.S.,

analogy

correlate

decorated

applied

mixtures

fluid

nonclassical

the

made

this

potential

a binary liquid

to

mixture

(37). As

of

crossover importance urgent

this

date,

problem for of

no

attempt

fluid

has

mixtures.

supercritical fluid

been

made

to

solve

In

view

of

the

mixtures

this

the

growing

appears

to

be

task.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

an

5.

Near-Critical Mixtures

LEVELT SENGERS ET AL.

119

E x p e r i m e n t a l E v i d e n c e , o r Lack o f I t , For N o n c l a s s i c a l Behavior i n Mixtures For a l l g r o s s f e a t u r e s o f phase b e h a v i o r o f m i x t u r e s , c l a s s i c a l e q u a t i o n s do a r e a s o n a b l e j o b . There a r e n o t many m i x t u r e s f o r which we have t h e w e a l t h o f a c c u r a t e e x p e r i m e n t a l d e t a i l t h a t we have f o r some o f t h e pure f l u i d s , so t h e shortcomings o f c l a s s i c a l e q u a t i o n s a r e not as a c u t e l y f e l t . As we a l r e a d y mentioned, t h e most s t r i k i n g shortcoming o f c l a s s i c a l e q u a t i o n s i s i n t h e r e p r e s e n t a t i o n o f t h e c o e x i s t i n g phases near t h e c r i t i c a l p o i n t (24, 25) where n o n c l a s s i c a l r e p r e s e n t a t i o n s make a major improvement (V7, J_8, 25 - 3£ ). A l s o , t h e s t r o n g anomaly o f t h e osmotic s u s c e p t i b i l i t y ( 3 χ / 3 Δ ) has been determined from t h e

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

ρτ

i n t e n s i t y o f s c a t t e r e d l i g h t i n l i q u i d - l i q u i d (38) and i n g a s - l i q u i d (39) m i x t u r e s and t h e n o n c l a s s i c a l exponent v a l u e was found f o r Ύ. The s i t u a t i o n w i t h r e s p e c t t o t h e weak anomalies i s f a r l e s s c l e a r - c u t . Only t h e weak d i v e r g e n c e p r e d i c t e d f o r t h e s p e c i f i c heat C has been c o n v i n c i n g l y demonstrated i n l i q u i d - l i q u i d (40) p x

mixtures.

Those i n t h e expansion c o e f f i c i e n t

compressibility Κ

Τ χ

have not been seen y e t .

m i x t u r e s , t h e reason i s o b v i o u s . relations α

/ Κ

Ρχ Τχ "

d P / d T

l RL

:

C

C

α

ρ χ

and t h e

In binary l i q u i d

Since, according t o the Pippard

Px

/ V T a

Px *

d P / d T

l RL

where CRL denotes t h e c r i t i c a l l i n e , and s i n c e d P / d T | l a r g e i n b i n a r y m i x t u r e s , t h e anomaly i n 0 b e i n g v i s i b l e , f o l l o w e d by those i n α

ρ χ

ρ χ

W

C

C R L

i s very

has t h e b e s t chance o f

and i n Κ

Τ χ

, i n this

order.

I n g a s - l i q u i d m i x t u r e s , t h e L e u n g - G r i f f i t h s model has been used t o f i n d t h e d i s t a n c e from t h e c r i t i c a l l i n e where t h e weak 3 4 divergence of Κ can be seen. For He - He , t h i s d i s t a n c e was Τ χ

1 0

e s t i m a t e d t o be about 10 Κ ( 2 6 ) , which i s beyond e x p e r i m e n t a l accessibility. The p r e d i c t i o n o f a weakly d i v e r g i n g c o m p r e s s i b i l i t y l e a d s t o an apparent paradox (£l_) i n t h e case d e p i c t e d i n F i g u r e 3* because the c r i t i c a l i s o t h e r m has t o c r o s s t h e phase boundary a t t h e c r i t i c a l p o i n t w i t h a h o r i z o n t a l s l o p e ( F i g u r e 5 ) . From t h e relation d V / d P

l ,x a

=

(

9

v

/

9

p

)

+ T

x

(3V/3T)

p x

dT/dP|

(5)

o > x

where σ i s t h e phase boundary, we note t h a t i f ( 3 V / 3 P )

T x

i s to

d i v e r g e t o -«. w h i l e (dV/dP) and dT/dP| a r e f i n i t e and ° σχ σχ n e g a t i v e , then (3V/3T) has t o d i v e r g e t o -« . T h i s means t h a t i n 1

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

120

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

the one-phase region near the c r i t i c a l point, the expansion c o e f f i c i e n t must pass through zero and diverge to The negative expansion c o e f f i c i e n t makes i t appear that the isotherm enters the two-phase region while i t i s actually s t i l l i n one phase, and vice versa on the other side of the c r i t i c a l point. Some indications of negative expansion c o e f f i c i e n t s have been seen i n l i q u i d - l i q u i d mixtures near consolute points (42, 43) but never near gas-liquid c r i t i c a l l i n e s . By using the Leung-Griffiths model, Wheeler and two of us (RFC and GM) have found that i n 0.28 mole f r a c t i o n C0 i n 2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

-22 ethane, the expansion c o e f f i c i e n t changes sign at 10 i n reduced temperature from the c r i t i c a l l i n e ! (41). While several of the weak anomalies thus escape detection, some of the non-divergent properties actually appear to diverge.

Q

4 Thus the s p e c i f i c heat C of He - He behaves l i k e the weakly-divergent s p e c i f i c heat Cy of pure f l u i d s , while theory predicts i t to remain f i n i t e (j_2). The Leung-Griffiths model explains that the same mechanism, the small size of a weakly-diverging property, i s the cause of the apparent f i n i t e value of Κ« and the apparent divergence of C (26). A related theoretical prediction i s that the isochore-isopleth, a curve on which two densities, V and x are constant, should become confluent with the c r i t i c a l l i n e and therefore with the constant - χ phase boundary (Figure 5). The most detailed measurement of (9Ρ/3Τ) i s that of Doiron et a l . i n 3 4 He - He (22) i n which the c r i t i c a l point was approached to 1 mK but O P / 3 T ) remained f i n i t e . The Leung-Griffiths V x

Γχ

V x

t

νχ

Vx

model predicts 10~ approach dP/dT|

CRL

.

11

Κ for the region i n which (3Ρ/3Τ)

νχ

begins to

In the C0 - C Hg system, the isochoric slope 2

2

has to become negative (Figure 5) - according to the -22 Leung-Griffiths model, t h i s happens at 10 i n reduced temperature, the same point where the expansion c o e f f i c i e n t turns negative (41). It i s obvious that for a l l p r a c t i c a l purposes most of the predictions of nonclassical theory for the weak anomalies can be safely ignored. An exception i s the weak divergence i n 0 . The departures from c l a s s i c a l behavior f o r the strong divergences, however, are quite v i s i b l e and need to be taken into account i f accuracy i s desired. ρ χ

Dilute N e a r - C r i t i c a l Mixtures: C l a s s i c a l Analysis Dilute n e a r - c r i t i c a l mixtures show a p e c u l i a r i t y that at a f i r s t glance i s offensive to the thermodynamic1st; p a r t i a l molar properties of the solvent do not necessarily approach pure-solvent properties i n the l i m i t x-0 at the solvents c r i t i c a l point. This behavior was f i r s t experimentally studied by Krichevskii and coworkers (4_4, 45), and modeled by a c l a s s i c a l equation by Rozen (46). Wheeler (47) gave a nonclassical treatment based on the decorated l a t t i c e gas.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

5.

121

Near-Critical Mixtures

LEVELT SENGERS ET AL.

The reason for this abnormal behavior i s that the solvent c r i t i c a l point i s the intersection of the locus of i n f i n i t e dilution, (3Δ/3χ) + and of the c r i t i c a l l i n e , ( 3 Δ / 3 χ ) ρτ

0;

ρτ

as a consequence the l i m i t i n g values assumed by c e r t a i n p a r t i a l properties depend on the path of approach. This anomalous behavior i s r e a d i l y elucidated for the p a r t i a l molar volume V

- V - χ (3V/3x)

1

(6)

pT

since (3V/3x) In the l i m i t x+0,

Κ

pT

- VK

Tx

(3P/3x)

approaches the compressibility of the pure

Τ χ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

f l u i d which i s strongly divergent. by (44,

(7)

VT

The l i m i t of (3P/3x)

VT

i s given

48)

[Lim (3P/3x) ]° - dP/dx| VT

where dP/dx|°

are

dT/dx|^L

T

L/HL

- dP/dT|£

CRL

dT/dx|£

xc

(8)

RL

are l i m i t i n g slopes of the

critical

LKL

l i n e i n P-x and T-x space, respectively; the l i m i t i n (8), i n general, i s f i n i t e . Thus (3V/3x) diverges at the solvent's c r i t i c a l point. If the l i m i t x-0 i s taken f i r s t , i t diverges strongly, as the compressibility. As a consequence, the p a r t i a l molar volume of the solute pT

V

2

= V + (1 - x ) ( 3 V / 3 x )

(9)

pT

diverges strongly, with a sign equal to that of (3P/3x)

VT

in (8).

These diverging p a r t i a l molar volumes of the solute have been reported by Khazanova and Sominskaya for C0 (45), by Benson 2

et a l .

for NaCl i n steam (2) and by Eckert et a l .

i n ethylene

(49).

for naphthalene

Close to the solvent c r i t i c a l point,

and

V

2

can be predicted from the solvent properties and the i n i t i a l slope of the c r i t i c a l l i n e (48, 50, 51). In the c l a s s i c a l treatment of d i l u t e mixtures, the divergence of (3V/3x) along a path of choice i s obtained by expanding pT

(VK^)""

1

equals 0.

around the solvent's c r i t i c a l point, at which point i t We obtain (48);

V

K

ix

-

a

VVx

X

+

3

VVT

(

δ

Τ

)

+

a

VVVV

(

5

V

)

2

/

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

0

0

)

122

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

where subscripts V, χ, Τ indicate repeated p a r t i a l d i f f e r e n t i a t i o n of the Helmholtz free energy a(V, T, x) with respect to the variable indicated, and where the superscript c indicated that the derivatives are evaluated at the solvent's c r i t i c a l point. Note that (VK,^)

1

i s quadratic i n volume because the c r i t i c a l i t y c

conditions imply that a ^

c = 0, a^yy

v

• 0.

A t y p i c a l path i s the c r i t i c a l l i n e (Figure 5). ν κ

Here δΤ and

1

δν are linear i n x, so ( ) i s linear i n χ and (3V/3x) diverges as χ . The product x(3V/3x) i n (6) i s a f i n i t e Τ χ

pT

1

pT

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

constant and

does not approach V »

A second path i s the

c

2 coexistence curve at T=T (Figure 6). Here (δν) ~ χ and (3V/3x)p s t i l l diverges as 1/x, but with a different amplitude. On any 2 isothermal path on which (δν) varies more slowly than x, however, 2 the term i n (δν) w i l l dominate the c r i t i c a l behavior. One such path i s the c r i t i c a l isotherm-isobar (Figure 6) on which (δν) - χ, so that V- diverges as x*~ . On t h i s path, x(3V/3x) c

3

T

2/3

2

pT

approaches 0 and V^ •» V .

Since the derivative (3H/3x)

pT

is

asymptotically proportional to (3V/3x) : pT

Lim (H /V ) = Τ (dP/dT|£ x-»0 2

2

)

(11)

we expect the same anomalies i n p a r t i a l enthalpies as in p a r t i a l molar volumes (48, 50, 51_) as was also noted by Wheeler (47). Strongly increasing slopes i n the excess enthalpy curves of binary gas-liquid mixtures were reported by Christensen on isotherm-isobars near the c r i t i c a l points of each of the components (4, 50). The above analysis i s readily extended to higher-ordered derivatives such as the p a r t i a l molar s p e c i f i c heat C

p 2

- (3C /3x) . p

pT

This quantity i s predicted to diverge much more 2

strongly than the p a r t i a l molar volume, namely as (K ) T

i n the

2 l i m i t x+Q and as (1/x) strong divergence of C

along the c r i t i c a l l i n e . p 2

This extremely

was experimentally found by Wood et a l .

for NaCl i n steam (3). The c l a s s i c a l results are summarized i n Table 3. Dilute N e a r - C r i t i c a l Mixtures; Nonclassical Analysis We have investigated the nonclassical c r i t i c a l behavior of d i l u t e mixtures (48) by writing

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

LEVELT SENGERS ET AL.

Near-Critical Mixtures

X

X

1

1

1 Ρ /

2

/

\\ \\

\ Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

(a)

(b)

Figure 5. Nonclassical behavior near a mixture c r i t i c a l point. The compressibility i s weakly divergent, so the c r i t i c a l isotherm i s horizontal (a). The isochore-isopleth i s confluent with the phase boundary and the c r i t i c a l l i n e (b). If the slope of the phase boundary i s i s negative at the c r i t i c a l point, the expansion c o e f f i c i e n t diverges to -°° and the slope of the isochore-isopleth i s negative at the mixture's c r i t i c a l point.

Figure 6. The paths in V-x space at the c r i t i c a l temperature of the solvent along which we have calculated the divergence of V « 2

The c r i t i c a l l i n e i s shown i n projection.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

124

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

(aV/3x)

= θΥ/3Δ) /Οχ/3Δ)

pT

ρ τ

(12)

ρ τ

Both numerator and denominator, being second derivatives [of the potential μ (Δ, Ρ, Τ), (_1_2)] i n a d i r e c t i o n intersecting the 2

coexistence surface, diverge strongly at the c r i t i c a l l i n e . As a consequence, (3V73x) i s f i n i t e at the c r i t i c a l l i n e j u s t as i n pT

the c l a s s i c a l case, and not weakly divergent as stated i n r e f . (52). The osmotic s u s c e p t i b i l i t y i n the denominator, however, faces a dilemma when x+0 because i t i s zero at the pure-fluid axis and i n f i n i t e at the c r i t i c a l l i n e . By means of the Leung-Griffiths model we have been able to obtain the structure of this term. Schematically, i t can be given, to the leading order i n ζ and r , as

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

Οχ/3Δ)

* c(flnlte) + c (strong)

(13)

2

ρ τ

While the structure of the numerator i s θν/3Δ)

~ c(strong)

ρ τ

(14)

where ( f i n i t e ) denotes a thermodynamic derivative that remains f i n i t e , (strong) one that diverges strongly at the c r i t i c a l l i n e . Thus the structure of (3V73x) i s pT

(3V/3x) ax;

* .„ , (strong) ( f i n i t e ) + c(strong)

pT

(

}

Schofield's r denotes the distance from the c r i t i c a l l i n e at constant ζ, and the strong term can be written as proportional to -Ύ r . A path to the solvent c r i t i c a l point can be characterized by stating how r goes to zero as ζ shrinks, or by the r e l a t i o n r _

(16)

ε ζ

If ε i s small, ζ decreases much faster than r and the path i s close to the pure-fluid axis. I f ε i s large, ζ decreases slower than r and the path i s close to the c r i t i c a l l i n e . Intermediate paths are obtained by varying the value of ε. Thus we may indicate the behavior of (3V/3x) schematically by pT

(3V/3X)

~

il

-y

( f i n i t e ) + cr

-

^

£

Ύ

.

e

T

^

(

(finite) + ζ

Since for small ζ, ζ ~ χ, because the RTxfcnx term dominates the free energy, we obtain

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1

7

)

5.

Near-Critical Mixtures

LEVELT SENGERS ET AL.

(3V/3x)

"

125

L

— (finite)x

(18) e T

+ χ

When ζ goes to 0 at f i n i t e r, we retrieve, from (15), the strong divergence of (3V/3x) . If ζ and r both go to zero, there i s a pT

competition between the term i n χ ε i s large,

ε Ύ

and the term i n χ i n (18).

ε > 1/Ύ, εΎ > 1, ε > 0.81 then (3V/3x)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

V

1

pT

(19)

behaves l i k e 1/x and so does V

does not approach V .

2

(9).

In t h i s case

If ε i s small

ε < 1/Ύ, εΎ < 1, ε < 0.81 then (3V/3x)

pT

behaves l i k e 1/χ

does approach V .

εΎ

(20)

and so does \? . 2

On the c r i t i c a l l i n e (ε large) V

q

If

In t h i s case 2

diverges as

1/x. On the isothermal or isobaric coexistence curve h equals 0 and τ i s linear i n ζ to lowest order. Since τ i s linear i n r , we have ε=1 and (3V/3x) goes at 1/x. pT

On the c r i t i c a l isoterms-isobar, the pressure and temperature are to be kept constant and equal to those at the solvent's c r i t i c a l point. In this case both τ and h vary l i n e a r l y with ζ for ft Λ

small ζ leading to r (3V/3x)

and so does V « 2

pT

~ ζ or ε = 1/36. . 1/χ

Ύ / Β δ

1

« 1/χ "

The value of 1-1/6

I t follows that 1 / δ

= 0.79.

(21)

Therefore χ

(3V/3x)

0 and V\ V . The crossover from 1/x to slower-than 1/x 1 c behavior occurs at ε - 1/Ύ » 0.81 which i s between the isothermal or isobaric coexistence curve (ε*1) and the c r i t i c a l isotherm-isobar (ε=0.64). We have also analyzed the behavior of the p a r t i a l molar s p e c i f i c heat C . p2

The results of our analysis

are summarized i n Table I I I . The c l a s s i c a l and nonclassical treatments of d i l u t e mixtures show the same global features but d i f f e r i n d e t a i l . The differences are due to the different values assumed by the strong exponents 3, Ύ, δ and should therefore be experimentally detectable. Impure N e a r - C r i t i c a l Fluids In the custody transfer of f l u i d s , the density of the f l u i d i s often obtained from measurements of pure Ρ and temperature Τ and

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

pT

126

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

from the known equation of state. I t i s important to know what effect the presence of an impurity can have on the density at given Ρ and T. I f V indicates the molar volume, we may write V(P, T, x) = V(P, T, 0) + (3V/3x) x + . . . .

(22)

PT

Since (3V/8x)

pT

i s proportional to the solvent's compressibility,

impurity effects become larger the larger the compressibility. The expansion (23) i s not v a l i d at the solvent's c r i t i c a l point. At 1 /Λ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

T

c*

P

w

c*

e

h

a

v

e

χ

(<*

ν / 9 χ

)ρ

Τ

~

x

(21). So even an impurity on the

-5 l e v e l of 10 i n mole f r a c t i o n can s t i l l modify the molar volume on the 10? l e v e l , since 1/6 i s about 0.2 for f l u i d s . Impurity effects can be drastic near T ? because they may induce a phase t r a n s i t i o n . In Figure 7, we sketch how a nonvolatile impurity affects the molar volume of a n e a r - c r i t i c a l f l u i d . If i s obvious that impurity effects are highly nonlinear i n concentration. Impurity effects i n n e a r - c r i t i c a l f l u i d s can be calculated i n a consistent but inaccurate way by means of c l a s s i c a l corresponding-states treatments and their generalizations. They can be calculated consistently and accurately by means of the Leung-Griffiths model. In view of the complication of the l a t t e r approach i t i s tempting to model the pure f l u i d as accurately as possible with a multiparameter c l a s s i c a l or a nonclassical thermodynamic surface, and then to calculate the impurity effects by means of corresponding states (53). Unfortunately, t h i s idea i s incorrect as can be appreciated from Figure 3. The nonclassical treatment of the major component modifies the behavior around the pure-fluid c r i t i c a l point. This part of the graph, however, i s not reached by the mixture because i t separates at the curve of material i n s t a b i l i t y . The mixture therefore "sees" l i t t l e or nothing at a l l of the nonclassical behavior characterizing the major component. The effect of combining a nonclassical description of the major component with c l a s s i c a l corresponding states are more disturbing i n d i l u t e mixtures. To see t h i s , we need the c l a s s i c a l expressions for the i n i t i a l slopes of the c r i t i c a l l i n e , which are obtained from the c l a s s i c a l expansion of the Helmholtz free energy and from the conditions of c r i t i c a l i t y . They are q9

q9

2 dT c dx CRL

^ (a° ) -RT a ; x

x

^_

&

a

CRL RT a £

V T

'

d

c dT c VT dx CRL

+

4

(23)

x

the l a t t e r being the equivalent of (8). From (23) we derive for the i n i t i a l slope of the c r i t i c a l l i n e i n P-T space

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

5.

Near-Critical Mixtures

LEVELT SENGERS ET AL.

Table III C r i t i c a l Behavior in D i l u t e

path

V , H 2

K ,

2

C

T2

p 2

127

Mixtures

V

Lim x V

1

x->0

lim x-K)

0

AV

2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

Classical 1/x

isotherm-isochore

1/x

1/x

V c

5 / 3

2

RT / ( A - c . / A )

?V C

isobar-isochore

critical

1/x

line

1/x

1/x

1/x

2

AV c

I

RT / ( A - c / A )

fV

0

AV

C

C

c

RT /A c

AV

0

AV

c

RT /A c

AV

c

RT /A c

AV^ c

RT /A

0

?V

2

C

c

Λ

2

C

c

Nonclassical 1/χ ~ 2

isotherm-isochore

1/x

1/x

V

1 / δ

/V r

isobar-isochore

1/x

1/χ ~ 3

Ύ

/V r

critical

line

I

1/x

E

1

C

2

'

R T

c VVT

-

R

V V V T

x • x

a

?V

,

'CRL

C

—

'CXC

d T / d X

d T / d P

1

CRL

lcRL

lcXC

d

P

/

d

X

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

c

(ν /Κ 2

Τ χ

)

E Q U A T I O N S O F STATE. T H E O R I E S A N D

128

APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

V

χ

Figure 7. Impurity effects on extensive properties i n d i l u t e mixtures near the solvent's c r i t i c a l point are large and nonlinear i n concentration. The plot shows the r e l a t i o n between molar volume and impurity concentration at fixed T > 1

T

Q

and at four pressures: that of the mixture's c r i t i c a l point

Ρ , P > Ρ and two pressures Ρ , P„ at which the cm d cm m l actually induces a phase t r a n s i t i o n . 0

impurity

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

5.

Near-Critical Mixtures

LEVELT SENGERS ET AL.

129

where we have used a ^ - -dP/dT|£ , t h e l i m i t i n g s l o p e o f t h e T

xc

vapor p r e s s u r e c u r v e . I f t h e major component has been t r e a t e d n o n c l a s s i c a l l y , o r w i t h a very accurate c l a s s i c a l equation, the coefficient a ^ a

VVT *

9

(

ν

κ

χ

1

)

/

9

V T

Τ

w i l l be z e r o o r v e r y s m a l l because ~

Δ

τ

Ί

Μ

and Ύ > 1.

Thus, t h e i n i t i a l s l o p e o f

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

the c r i t i c a l l i n e i s f o r c e d t o be t h e e x t e n s i o n o f t h e vapor p r e s s u r e c u r v e , which c o n t r a d i c t s experiment. The combining o f a n o n c l a s s i c a l p u r e - f l u i d equation with corresponding s t a t e s f o r the m i x t u r e induces a s e r i o u s d i s t o r t i o n i n t h e phase diagram. Conclusions For one-component f l u i d s , a c c u r a t e n o n c l a s s i c a l f o r m u l a t i o n s o f t h e c r i t i c a l region are a v a i l a b l e , i n c o r p o r a t i n g the v a p o r - l i q u i d asymmetry as w e l l as c o r r e c t i o n s t o s c a l i n g . E m p i r i c a l t e c h n i q u e s f o r s w i t c h i n g from c l a s s i c a l t o c r i t i c a l b e h a v i o r have been worked o u t . The fundamental s o l u t i o n o f t h e c r o s s o v e r problem has advanced t o t h e p o i n t t h a t a p p l i c a t i o n t o one-component f l u i d s i s near. F o r b i n a r y m i x t u r e s a n o n c l a s s i c a l f o r m u l a t i o n i s a v a i l a b l e f o r t h e one-and-two-phase r e g i o n s m i x t u r e s w i t h c o n t i n u o u s c r i t i c a l l i n e s . A v a r i e t y o f n o n c l a s s i c a l models have been used t o r e p r e s e n t c o e x i s t i n g phases. No attempts have been made t o d e v i s e crossover functions. D i l u t e n e a r - c r i t i c a l m i x t u r e s p r e s e n t some s p e c i a l anomalies not o f t e n encountered i n thermodynamics. We d e s c r i b e these w i t h b o t h c l a s s i c a l and n o n c l a s s i c a l e q u a t i o n s . We argue t h a t a n o n c l a s s i c a l thermodynamic d e s c r i p t i o n o f t h e s o l v e n t can n o t be combined w i t h a c o r r e s p o n d i n g - s t a t e s treatment o f t h e d i l u t e mixture. For i n c o r p o r a t i o n of the n o n c l a s s i c a l e f f e c t s i n t o p r a c t i c a l thermodynamic s u r f a c e s o f f l u i d m i x t u r e s , t h e s o l u t i o n o f the c r o s s o v e r problem appears t o be t h e most urgent t a s k a t hand. Acknowledgment We have p r o f i t e d from d i s c u s s i o n s w i t h P.C. A l b r i g h t and J.C. Wheeler.

References 1. 2. 3.

4.

Busey, R.H.; Holmes, H.F.; Mesmer, RE. J. Chem. Thermodynamics 1984, 16, 343. Benson, S.W.; Copeland, C.S.; Pearson, D. J. Chem. Phys. 1953, 21, 2208. Copeland, C.S.; Silverman, J.; Benson, S.W. J. Chem. Phys. 1953, 21, 12. Smith-Magowan, D.; Wood, R.H. J. Chem. Thermodynamics 1981, 13, 1047. Wood, R.H.; Quint, J.R. J. Chem. Thermodynamics 1982, 14, 1069. Gates, J.Α.; Wood, R.H. J. Phys. Chem. 1982, 86, 4948. Christensen, J . J . ; Walker, T.A.C.; Schofield, R.S.; Faux, P.W.; Harding, P.R.; Izatt, R.M. J. Chem. Thermodynamics, 1984, 16, 445 and references therein. Wormald, C.J. Ber. Bunsenges. Phys. Chem. 1984, 88, 826 and references therein.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

130

5. 6. 7. 8.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

31. 32. 33.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

See for instance, "Chemical Engineering in Supercritical Fluid Conditions" (Paulaitis, M.E. et al., Eds). Ann Arbor Science Publishers, 1983. Sengers, J.V.; Levelt Sengers, J.M.H. in "Progress in Liquid Physics" (Croxton, C.A. Ed.); Wiley: Chichester, U.K., 1978; Ch. 4, p. 103. Levelt Sengers J.M.H.; Sengers, J.V. in "Perspectives in Statistical Physics" (Raveche, H.J. Ed.); North Holland Publ. Co: Amsterdam, Netherlands, 1981; Ch. 14, p. 239. Sengers, J.V.; Levelt Sengers, J.M.H. Int. J . Thermophysics 1984, 5, 195. Hocken, R.J.; Moldover, M.R. Phys. Rev. Letters 1976, 37, 29. Greer, S.C. Phys. Rev. 1976, A14, 1770. Widom, B. J. Chem. Phys. 1965, 43, 3898. Griffiths, R.B.; Wheeler, J.C. Phys. Rev. 1970, A2, 1047. Mermin, N.D.; Rehr, J.J. Phys. Rev. Letters 1971, 26, 1155. Phys. Rev. 1971, A4, 2408. Rehr, J . J . ; Mermin, N.D. Phys. Rev. 1973, A8, 472. Chapela, G.; Rowlinson, J.S. Faraday Trans. 1974, 70, 584. Woolley, H.W. Int. J. Thermophysics 1983, 4, 51. H i l l , P.G. Proc. 10th International Conf. on Properties on Steam" Moscow, USSR, 1984; and Int. J . Thermophysics; to be published. Fox, J.R. Fluid Phase Equilibria, 1983, 14, 45. White, J.Α.; Pustchi, M.E. Bull. Am. Phys. Soc., 1984, 30, 372; 1984, 29, 697; and references therein. Nicoll, J.E. Phys. Rev. 1981, A24, 2203. Nicoll, J.E.; Albright, P.C. Phys. Rev. 1985, B31, 4576. Albright, P.C. Ph.D. Thesis, U. of Maryland, 1985. Doiron, T.; Behringer, R.P.; Meyer, H. J. Low Temp. Phys. 1976, 24, 345. Van Konynenburg, P.H.; Scott, R.L. Phil. Trans. Roy. Soc., 1980, 298, 495. Kolasinska, G.; Moorwood, R.A.S.; Wenzel, H. Fluid Phase Equilibria 1983, 13, 121. B i j l , H.; de Loos, Th.W.; Lichtenthaler, R.N. Fluid Phase Equilibria 1983, 14, 157. Leung, S.S.; Griffiths, R.B. Phys. Rev. 1973, A8, 2670. Wallace Jr, B; Meyer, H. Phys. Rev., 1972, A5, 953. Chang, R.F.; Doiron, T. Int. J. Thermophysics 1983, 4, 337. Chang, R.F.; Levelt Sengers, J.M.H.; Doiron, T.; Jones, J . J . Chem. Phys. 1983, 79, 3058. D'Arrigio, G.; Mistura, L . ; Taraglia, P. Phys. Rev. 1975, A12, 2578. Moldover, M.R.; Gallagher, J.S. in "Phase Equilibria and Fluid Properties in the Chemical Industry" (Storvick, T.S. and Sandler, S.I., Eds.). ACS Symposium Series 60, 1977; Ch. 30, p. 498. Rainwater, J.C.; Moldover, M.R. in "Chemical Engineering in Supercritical Fluid Conditions" (Paulaitis, M.E., et al., Eds.), Ann Arbor Science Publishers, 1983; p. 199. Wheeler, J.C. Ann. Rev. Phys. Chem., 1977, 28, 411 and references therein. Wheeler, J.C. J. Chem. Phys. 1975, 62, 4332.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch005

5. LEVELT SENGERS ET AL. Near-Critical Mixtures

131

34. Andersen, G.R.; Wheeler, J.C. J. Chem. Phys., 1978, 69, 2082; J. Chem Phys., 1979, 70, 1326. 35. Bartis, J.T.; Hall, C.K. Physica, 1975, 78, 1. 36. Levelt Sengers, J.M.H. Pure and Applied Chemistry 1983, 55, 437. 37. Ewing, M.B.; Johnson, K.A.; McGlashan, J.L. J. Chem. Thermodynamics 1985, 17, 513; and to be published in the same Journal. 38. Chang, R.F.; Burstyn, H.; Sengers, J.V. Phys. Rev. 1979, A19, 866. 39. Giglio, M; Vendramini, A. Optics Comm. 1973, 9, 80. 40. Bloemen, E.; Thoen, J.; Van Dael, W. J . Chem. Phys. 1981, 75, 1488. 41. Wheeler, J.C.; Morrison, G.; Chang, R.F., accepted, J. Chem. Phys. 42. Morrison, G.; Knobler, C.M. J. Chem. Phys. 1976, 65, 5507. 43. Clerke, E.A.; Sengers, J.V.; Ferrell, R.A.; Battacharjee, J.K. Phys. Rev., 1983, 27, 2140. 44. Krichevskii, I.R. Russ. J. Phys. Chem. 1967, 41, 1332. 45. Khazanova, N.E.; Sominskaya, E.E. Russ. J. Phys. Chem. 1971, 45, 1485. 46. Rozen, A.M. Russ. J. Phys. Chem. 1976, 50, 837. 47. Wheeler, J.C. Ber. Bunsenges, Phys. Chem. 1972, 76, 308. 48. Chang, R.F.; Morrison, G.; Levelt Sengers, J.M.H. J . Phys. Chem. Letter 1984, 88, 3389. 49. Eckert, C.A.; Ziger, D.H.; Johnston, K.P.; Ellison, T.K. Fluid Phase Equilibria 1983, 14, 167. 50. Morrison, G.; Levelt Sengers, J.M.H.; Chang, R.F.; Christensen, J.J. in "Proceedings of the Symposium on Supercritical Fluids", (J. Penninger, Ed.). In press, Elsevier, Netherlands. 51. Levelt Sengers, J.M.H.; Morrison, G.; Chang, R.F. Fluid Phase Equilibria 1983, 14, 19. 52. Gitterman, M; Procaccia, I. J. Chem. Phys. 1983, 78, 2648. 53. Hastings, J.R.; Levelt Sengers, J.M.H.; Balfour, F.W. J. Chem. Thermodynamics 1980, 12, 1009. RECEIVED

November 8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

6

Prediction of Binary Critical Loci by Cubic Equations of State 1

2

2

Robert M. Palenchar , Dale D. Erickson , and Thomas W. Leland 1

Olefins Technology Division, Exxon Chemical Company, Raton Rouge, LA 70821 Department of Chemical Engineering, Rice University, Houston, TX 77251

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

2

This paper compares reduced forms of five different cubic equations of state of the van der Waals family as to their ability to predict critical loci of binary mixtures. For the calculation of binary critical loci, the constants for each equation were expressed as functions of composition by the same mixing rules which involve two unlike pair coefficients evaluated by fitting to the experimental critical temperature and critical pressure at the same single composition. Only the critical temperature and critical pressure loci of van Konynenburg and Scott's Type I systems are predicted quantitatively over their entire composition range by the equations examined in this way. Critical volumes in all systems and critical temperature and pressure loci in Type II systems are only qualitatively predicted. Unlike pair coefficients evaluated at one point on a critical line in a two-phase region do not produce satisfactory predictions in portions of a critical locus where a third phase appears. There are no major differences in accuracy among any of the equations in predicting Type I critical loci. However, the Teja equation gave best predictions of Type I critical temperatures and pressures. The Peng-Robinson equation gave best predictions of Type I critical volumes. In Class 1, Type II and in all Class 2 systems, the twoconstant Soave and Redlich-Kwong equations generally predict critical temperatures and critical pressures better than the more elaborate three-constant equations and two constant equations whose form has a larger deviation from that of the original van der Waals equation. The accurate prediction of the properties of equilibrium phases and vapor-liquid e q u i l i b r i a i n mixtures near their c r i t i c a l i s an 0097-6156/86/0300-0132$07.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

6.

PALENCHAR ET AL.

Prediction of Binary Critical Loci

133

important unsolved problem. The b a s i c d i f f i c u l t y i s t h a t t h e r e i s at present no accurate non-analytic equation of state for mixtures. The use of t y p i c a l e n g i n e e r i n g t y p e a n a l y t i c a l e q u a t i o n s of s t a t e w i t h e s s e n t i a l l y e m p i r i c a l m i x i n g r u l e s f a i l s i n the n e a r c r i t i c a l region. The n a t u r e of t h i s f a i l u r e i s w e l l u n d e r s t o o d . The d i v e r g e n c e of the p a r t i a l molar volume of a s o l u t e as i t becomes i n f i n i t e l y d i l u t e when the system approaches the s o l v e n t critical c o n d i t i o n s has r e c e n t l y been e x p l a i n e d c l e a r l y by Chang, M o r r i s o n , and L e v e l t Sengers (1). At f i n i t e c o n c e n t r a t i o n s the vapor and l i q u i d phase p r o p e r t i e s p r e d i c t e d by reduced a n a l y t i c a l e q u a t i o n s u s i n g p s e u d o - c r i t i c a l s approach each o t h e r too r a p i d l y as the system approaches i t s c r i t i c a l c o n d i t i o n s . When d e t e r m i n e d from reduced equations i n terms of p s e u d o - c r i t i c a l s t h e s e p r o p e r t i e s tend to predict critical conditions which l i e below the true critical locus. The t r a d i t i o n a l method of d e a l i n g w i t h t h i s problem has been t o s t o p phase e q u i l i b r i u m c a l c u l a t i o n s w e l l below the c r i t i c a l l o c u s and t o make an e m p i r i c a l e x t r a p o l a t i o n t o an e s t i m a t e of the t r u e c r i t i c a l p r o p e r t i e s of the s y s t e m . It may be possible to overcome these problems with c o r r e s p o n d i n g s t a t e s methods w h i c h use a p r o p e r l y s c a l e d r e f e r e n c e fluid equation. Pseudo-critical relations at present cannot a c c u r a t e l y p r e d i c t the second and t h i r d d e r i v a t i v e s o f the Gibbs f r e e energy w i t h r e s p e c t to c o m p o s i t i o n which are needed to d e f i n e the c r i t i c a l of a m i x t u r e . Furthermore, a d d i t i o n a l research is needed to d e t e r m i n e the b e s t way to map the c r i t i c a l r e g i o n of a m i x t u r e on t o the c r i t i c a l r e g i o n of a pure s c a l e d r e f e r e n c e w i t h o u t destroying the effective representation of multicomponent e q u i l i b r i u m phases which i s currently possible below the near c r i t i c a l region. A l t h o u g h r e s e a r c h w i t h t h e s e and o t h e r methods f o r i m p r o v i n g p r e d i c t i o n s i n the c r i t i c a l r e g i o n i s c o n t i n u i n g , a common need with any method under i n v e s t i g a t i o n is a procedure for e s t i m a t i n g the t r u e c r i t i c a l l o c u s of the m i x t u r e . F o r t h i s r e a s o n , t h i s paper examines a f e a t u r e of the c l a s s i c a l van d e r Waals f a m i l y of e q u a t i o n s which e n a b l e s them t o g i v e a v e r y good r e p r e s e n t a t i o n of the c r i t i c a l l o c i of s i m p l e m i x t u r e s by u s i n g properly assigned unlike pair interaction coefficients. This p r o c e d u r e may cause p r o p e r t i e s near the m i x t u r e c r i t i c a l t o be predicted poorly, but the critical locus itself is described remarkably w e l l . D u r i n g the l a s t twenty y e a r s a g r e a t d e a l of work has been done t o s t u d y the p r e d i c t i o n of b i n a r y c r i t i c a l l o c i i n t h i s manner ( 2 ]). U s i n g the o r i g i n a l van der Waals e q u a t i o n , van Konynenburg and Scott (8) q u a l i t a t i v e l y p r e d i c t e d n i n e d i f f e r e n t t y p e s o f b i n a r y critical loci. These d i f f e r e n t t y p e s a r e i d e n t i f i e d by the b e h a v i o r of c r i t i c a l p r o p e r t i e s i n a p r o j e c t i o n of the Ρ , Τ , χ s u r f a c e of the system on t o a p r e s s u r e - t e m p e r a t u r e p l a n e . These t y p e s a r e summarized i n T a b l e I . Thermodynamics of the C r i t i c a l P o i n t The thermodynamic c o n d i t i o n s f o r the e x i s t e n c e of a c r i t i c a l p o i n t were f i r s t d e r i v e d by Gibbs (9) more than a hundred y e a r s a g o . The d e f i n i n g e q u a t i o n s f o r b i n a r y system c r i t i c a l p o i n t s a r e as f o l l o w s :

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

134

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

(φ

0

Ρ,Τ =

<»

When working with pressure e x p l i c i t equations of state i t i s more convenient to express the above equations i n terms of their relationship to the Helmholtz free energy, which depend on the measured variables Ρ, V, T, and x. This relationship has been developed by Redlich and Kister ( 1 0 ) . The Redlich and Kister form for a binary system i s : 2 2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

d

1

Ρ,Τ d

X

X

V

X

1 2

d

l

X

V,T l RT(x -x )

3

%C

d

X

l

V,T

Ρ,Τ 3

9

- riL-i l d X

1

J

i v,

T

r

Ϊ

d2v

- ril-ï i P,

2

^Ρ,Τ

Ρ,Τ

ldX

r

J

a2p

l d V d X

T

ι J

i T

d x

l i V,T Equations of State Examined d

X

The c r i t i c a l l o c i c r i t e r i a i n Equations (3) and (4) can be evaluated with an equation of state relating the variables P,V,T and x. For this work five reduced equations of state of the van der Waals family were used. These equations contain two, and sometimes three, constants which are universal functions of the reduced temperature, c r i t i c a l constants, and the acentric factor, ω . Redlich - Kwong ( 1 1 ) : Ρ -

* a

R

T

-

V-b

* ϋ

K

Τ · i>V(V+b)

'

2>5

0.4278 R

2

/P„

Τ

b - 0.0867 R T / P c

c

Soave ( 1 2 ) : Ρ - RT _ a(T) V=b V(V+b) a(T)

f 6 K

= 0.42747 a(T) R

2

2

Τ /P

c

c

b - 0.08664 R T / P a(t) =* {l+m(l-TrO'5 2 c

c

)}

m

- 0.480 + 1 . 5 7 4 ω -

0.176ω

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

0

. )

6.

PALENCHAR

Prediction of Binary Critical Loci

ET AL.

135

Adachi (13): Y

RT l^b

'

a(T) V(V+c)

a(T) -

A

m w

5

2

Z {l+a(l-Tr°- )} R

Q

2

c

b = Β Z

c

R T /P

c

c = C Z

c

R T /P

c

c

c

2

T /P c

a - 0.479817 + 1.55553ω - 0.287787ω 3

A

0

= B (l+C) /(l-B)

c

2

3

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

Β = 0.260796 - 0.0682692ω - 0.0367338ω

c

2

[{(4/Β)-3}°·5-3]/2

C = Z

;

= [1/(1-B)]-A /(1+C) 0

Peng and Robinson (14): p — _RT _ a(T) " V-b V(V+b)+b(V-b)

( 8 )

r

a(T) = 0.45724 a(T) R b - 0.07780 R T / P c

o(T)

2

2

T /P c

c

{1+Κ(1-ΤΓΟ* )}

-

c

5

2

κ - 0.37464 + 1.54226ω - 0.26992ω

2

Teja U 5 ) : RT _ a(T) ~ V-b V(V+b)+c(V-b)

( 9 )

Y

a(T) = Ω α ( Τ ) a b = Ω, R b

Τ

R

2

'

2

T /Pc c

/P c e

c - Ω R Τ /Ρ c c e Ω = 3Z + 3(1-2Z )Ω + Ω a c C D D 2

2

+ 1-3Z^

Κ

c

Ω, equals the smallest positive root of: Ω J + (2-3Z )Ω + 3Z Ω - Ζ 3 - ο b c b c b c b

2

2

Κ

Ω

c

Ζ

= 1 - 3Z

c

c

= 0.329032 - 0.076799ω + 0.0211947ω

α(Τ)

= {1+F(l-Tr°« )}

F -

0.452413 + 1.30982ω - 0.295937ω

5

2

2

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

136

loci predicted by various equations using two unlike pair c o e f f i c i e n t s obtained from a single data point are i l l u s t r a t e d i n Figures 1-13. The errors reported i n Table II f o r a l l the c r i t i c a l properties are obtained by the procedure shown below f o r the c r i t i c a l temperature:

|( Average % Relative Absolute Error

)

T

-

- (χ )

S-^S k

Vexpt

3

I

c expt

w

1 Q Q )

3

The subscript j i n Equation (15) indicates represents the t o t a l number of data points.

(15)

a data

point and Ν

Unlike Pair Interaction Coefficients Table I I I shows the values of the unlike pair coefficients ζ and λ i n Equations (10) and (11) obtained by f i t t i n g to the experimentally measured c r i t i c a l temperature and c r i t i c a l pressure i n an equi-molar, or nearly equi-molar, mixture. Because of the empirical nature of the equations of state i n which these c o e f f i c i e n t s are used, their actual relationship to any e f f e c t i v e pair potential i s rather obscure. The name "unlike pair interaction c o e f f i c i e n t s " i s used i n an i d e a l sense only. Furthermore, i t i s important to note that these values apply only to the projected c r i t i c a l l o c i and bear l i t t l e resemblance to their optimal values obtained by f i t t i n g other portions of the P-T-x surface.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

1 2

1 2

Even when f i t t i n g c r i t i c a l l o c i alone, a s i g n i f i c a n t l y better correlation could obviously be obtained with any of the equations tested by optimizing either ζ^ when λχ *1·0> or by optimizing both ζ and X^ together, over the entire composition range. This, however, obscures the r e l a t i v e degree of composition dependence of these unlike pair c o e f f i c i e n t s i n the various equations of state and i s less revealing of the shape of the c r i t i c a l locus predicted by the a n a l y t i c a l form of any p a r t i c u l a r equation of state. However, among the equations with the same mixing rules with unlike pair c o e f f i c i e n t s evaluated i n the same way from the same single data point within the same group of data points, the equation which produces the lowest average absolute error f o r the entire group of these data points should produce the best c o r r e l a t i o n when i t s unlike pair c o e f f i c i e n t s are optimized f o r a l l data points i n the group. s

2

1 2

s

e

t

a

t

2

2

In a few cases i t was not possible to obtain an accurate prediction of both the c r i t i c a l pressure and the c r i t i c a l temperature with any combination of ζ χ ^ λχ2 values. These cases are indicated with a * i n Table I I I and the values presented are those giving the best prediction of T and P . a n

2

c

c

The policy of f i t t i n g the unlike pair c o e f f i c i e n t s at a single data point gives information regarding the r e l a t i v e degree of composition independence i n the values of these c o e f f i c i e n t s among the various equations of state tested. Furthermore, this policy also gives information about the shape of the c r i t i c a l loci predicted by the various equations. This i s i l l u s t r a t e d by Figures 10 and 11 where the c r i t i c a l temperatures and c r i t i c a l pressures of

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

6.

Prediction of Binary Critical Loci

PALENCHAR ET AL.

137

TABLE I van Konynenburg and Scott C l a s s i f i c a t i o n of Binary C r i t i c a l Loci Class 1 Description

Type I.

One continuous gas-liquid c r i t i c a l l i n e connecting the two pure component c r i t i c a l points CI and C2. CI represents the c r i t i c a l of the component with the lowest c r i t i c a l temperature and C2 i s the c r i t i c a l of the other component.

I- A. The same as I, with the addition of a negative azeotrope.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

II.

Two c r i t i c a l lines: One i s a vapor-liquid c r i t i c a l l i n e between the pure component c r i t i c a l s CI and C2. The second i s a l i q u i d - l i q u i d c r i t i c a l l i n e representing the merger of two liquids with limited m i s c i b i l i t y i n the presence of a s o l i d phase. Because of the limited m i s c i b i l i t y , this c r i t i c a l l i n e persists to Immeasurably high pressures so that i t s o r i g i n may be regarded as occurring at an i n f i n i t e l y large pressure. From this i n f i n i t e pressure this liquid-liquid critical line connects at lower pressure with an upper c r i t i c a l end point (UCEP), representing the high pressure termination point of a three-phase liquid-liquid-solid line. At this termination point the two l i q u i d phases become i d e n t i c a l . The o r i g i n of this l i q u i d - l i q u i d l i n e at an i n f i n i t e c r i t i c a l pressure i s designated with the symbol C by van Konynenburg and Scott (8). m

I I - A.

Type

The same azeotrope.

as

II,

but

with

the

addition

of

a

positive

Class 2 Description

III-HA. Two c r i t i c a l l i n e s , but no continuous c r i t i c a l l i n e runs between CI and C2. The f i r s t of the c r i t i c a l l i n e s i s a vaporliquid critical line from CI to the high temperature termination point (UCEP) of a liquid-liquid-gas three phase l i n e where the l i q u i d richest i n the component with the lowest c r i t i c a l temperature and the gas phase become i d e n t i c a l . The other critical line connects one of two possible low temperature origins to the c r i t i c a l C2 of the component with the higher c r i t i c a l temperature. The portion of this second c r i t i c a l l i n e near i t s low temperature o r i g i n i s a l i q u i d l i q u i d c r i t i c a l line which changes to a vapor-liquid c r i t i c a l l i n e before approaching C2. The two points of o r i g i n for this second c r i t i c a l l i n e identify two sub-types of Type III-HA defined as IIla-HA and IIIb-HA as follows: Continued on next page

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

138

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table I Continued IIIa-HA. In Type IIIa-HA the two liquids which merge along the low temperature portion of the second c r i t i c a l l i n e have limited m i s c i b i l i t y so that their merger requires increasingly higher c r i t i c a l pressures as the c r i t i c a l temperature i s lowered. The low temperature o r i g i n of this l i q u i d - l i q u i d c r i t i c a l l i n e i n this case may be regarded as occurring at an i n f i n i t e c r i t i c a l pressure and i s given the symbol C by van Konynenburg and Scott (8).

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

m

IIIb-HA. The second low temperature point of o r i g i n for the c r i t i c a l l i n e to C2 occurs at the high pressure termination point, or UCEP, of a three phase l i q u i d - l i q u i d - s o l i d l i n e at a point where the two l i q u i d phases become i d e n t i c a l i n the presence of the s o l i d . Type IIIb-HA was not considered by van Konynenburg and Scott because i t s o r i g i n at the UCEP of a three phase l i q u i d - l i q u i d - s o l i d l i n e cannot be represented by the o r i g i n a l van der Waals equation. In Type IIIa-HA or Type IIIb-HA the entire three phase l i q u i d liquid-gas l i n e l i e s at pressures above the vapor pressure curves of both of the two pure components. This condition represents what i s called "hetero-azeotropic" behavior and i s designated by the symbol "HA" i n the c l a s s i f i c a t i o n of this type of c r i t i c a l behavior. I I I . The same as III-HA except that the three phase l i q u i d - l i q u i d gas l i n e ending with the UCEP l i e s between the vapor pressure curves of the two pure components. Like Type III-HA, Type III may be subdivided into two sub-types I l i a and I l l b , depending upon the o r i g i n of the c r i t i c a l l i n e to C 2 . Type I l i a originates at and Type I l l b originates at the UCEP terminating a three phase l i q u i d - l i q u i d - s o l i d l i n e . Because the three phase liquid-liquid-gas l i n e does not l i e outside the two pure component vapor pressure curves, there i s no heteroazeotropic behavior i n Type I l i a or Type I l l b systems. IV.

Three c r i t i c a l l i n e s : A vapor-liquid c r i t i c a l l i n e runs from CI to the high temperature UCEP termination of a l i q u i d - l i q u i d gas three phase l i n e where a l i q u i d and the gas phase become identical. In this case, however, this liquid-liquid-gas line i s discontinuous. As the temperature i s lowered, i t ends at a LCEP where the two l i q u i d phases merge. At temperatures below this LCEP there i s a short range where the l i q u i d s are completely miscible and only a l i q u i d and gas phase are present. At s t i l l lower temperatures, there i s a second UCEP where two l i q u i d phases are again c r i t i c a l . Below this second, or low temperature, UCEP the liquid-liquid-gas three phase l i n e continues with decreasing temperature u n t i l i t ends with the formation of a s o l i d phase at a quadruple point. A second c r i t i c a l l i n e runs from the LCEP to C 2 . Near the LCEP i t i s a l i q u i d - l i q u i d c r i t i c a l l i n e and i n aproaching C2 i t changes to a vapor-liquid c r i t i c a l l i n e .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

6.

Prediction of Binary Critical Loci

PALENCHAR ET AL.

139

Table I Continued The third c r i t i c a l l i n e i s a l i q u i d - l i q u i d c r i t i c a l locus between the second or low temperature UCEP and a C point at an immeasurably high pressure. m

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

V.

Two c r i t i c a l l i n e s : In this system there i s a single three phase liquid-liquid-gas l i n e ending at a UCEP where one of the l i q u i d s and the gas phase become i d e n t i c a l . As the temperature i s lowered, this three phase l i n e stops at a LCEP where the two l i q u i d phases merge. One of the c r i t i c a l lines i s a gas-liquid c r i t i c a l locus from CI to the UCEP. The second c r i t i c a l l i n e runs from the LCEP to C2 and consists of a l i q u i d - l i q u i d c r i t i c a l l i n e near the LCEP which changes to a vapor-liquid c r i t i c a l l i n e when approaching C2.

V-A. The same as azeotrope.

Type

V

but with

the addition

of a negative

Class 3 The van Konynenburg and Scott c l a s s i f i c a t i o n includes a Class 3 behavior which i s exhibited by very complex mixtures with strong s p e c i f i c interactions, usually involving hydrogen bonds, between the components. These systems have LCSTs where there i s a minimum i n a T-x coexistence curve. Systems i n this class cannot be represented by equations of state of the van der Waals family.

y

-

\

~

\

/

\

ce Q. (I)

α.

\ »

TJ5J

EXPERIMENTAL TEJA EQUATION PEN6 AND ROBINSON

70·

TSO

Si

eSi

δϊ~

TEMPERATURE (R)

Figure 1.

C r i t i c a l l o c i of propane-hexane

system.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

\

/

\

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

tu oc V) 2CA ·

»

%T% ïSi

4··

EXPERIMENTAL TEJA EQUATION PEN6 ANO ROBINSON

Sî

sot

880

TEMPERATURE (R)

Figure 2. C r i t i c a l l o c i of m ethane-car bon dioxide system.

V en

en ο en S. tu m oc a.

»

"Soô

EXPERIMENTAL TEJA EQUATION AOACHI EQUATION 800

680

700

TEMPERATURE (R)

Figure 3.

C r i t i c a l l o c i of butane-carbon dioxide system.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

6.

Prediction of Binary Critical Loci

PALENCHAR ET AL.

» 0

*-*

141

EXPERIMENTAL (CURVE #1) EXPERIMENTAL (CURVE #2) SOAVE EQUATION

ο

α.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

3

CL

Ο Ο

CARBON DIOXIDE CRITICAL POINT

O*—

400

UCEP

S00

(expt.)

600

\\ Λ

\

DECANE CRITICAL P O I N T - ^ *

700

000

900

1000

1100

TEMPERATURE (R)

Figure 4. C r i t i c a l l o c i of de cane-car bon dioxide system.

9.29

9.49

9.99

9.99

N0LE FRACTION PROPANE

Figure 5.

C r i t i c a l l o c i of propane-hexane system.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

to Q. Œ ZD in in

*

t&i cr o.

X

» 0

/

EXPERIMENTAL (CURVE EXPERINENTAL (CURVE #2) REDLICH KW0N6 SOAVE EQUATION

BENZENE CRITICAL POINT

*^50

1000

10S0

1100

TEMPERATURE

1150

(R)

Figure 6· C r i t i c a l l o c i of benzene-water system.

EXP CURVE

#1

EXP CURVE

#2

SOAVE EQUATION

ο

—

II

WATER CRITICAL POINT

&—

β

^00

700

800

900

TEMPERATURE

Figure 7.

1000 (R)

C r i t i c a l l o c i of water-car bon dioxide system.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Prediction of Binary Critical Loci

PALENCHAR ET AL.

\

\

o.

\

ui

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

3

Q- S.

t

. φ

» 0

*^00

EXPERIMENTAL (CURVE #1) EXPERIMENTAL (CURVE #2) TEJA EQUATION SOAVE EQUATION

500

600

M

700

TEMPERATURE (R)

Figxare 8.

C r i t i c a l l o c i of methane-hexane system.

//«

\

\

if to ο.

.7/

to

HEPTANE

**l7~SOLID

CO

\

APPEARS

a. /i U C E F

/ • METHANE CRITICAL

» 0

-

EXPERIMENTAL (CURVE # EXPERIMENTAL (CURVE #2) TEJA EQUATION 2) SOAVE EQUATION

\ \

*

POINT

*^οο

soo

600

700

TEMPERATURE

Figure 9.

800

900

m

(R)

C r i t i c a l l o c i of methane-heptane

system.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

144

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

m: +

1

ι

Γ

τ

1

450 400 350

*

EXPERIMENTAL TEJA

EQUATION

00

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

TEMPE[RATURE

ο ο. in

.00

0.20

MOLE

0.40

0. 60

0.80

1.00

FRACTION METHANE

Figure 10. C r i t i c a l temperature of m ethane-car bon dioxide system.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

PALENCHAR ET AL.

UCEP

CO

145

Prediction of Binary Critical Loci

(PREDICTED)

QL

*

to

EXPERIMENTAL SOAVE

EQUATION

CO Œ CL

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

METHANE C R I T I C A L

*^40.0

342.0

344.0

346.0

POINT

348.0

TEMPERATURE

350.0

352.0

354.0

(R)

Figure 12. C r i t i c a l l o c i of m ethane-heptane system near methane c r i t i c a l point.

Figure 13. C r i t i c a l l o c i of methane-heptane system near methane c r i t i c a l point.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

146

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

TABLE I I Average Percent Relative Absolute Error i n Predicting Binary C r i t i c a l Loci by Cubic Equations of State

Redlich Kwong

System

Adachi

0.08 0.49 19.51

0.14 0.73 16.78

0.80 2.52 7.70

0.05 0.33 16.78

2.53 6.32

2.18 5.86

2.88 7.99

3.64 8.58

2.16 5.82

butane/ C2) co

CD

0.77 2.62 C3) 17.00

0.54 1.77 20.96

1.18 2.96 28.06

1.28 2.47 7.99

0.51 1.56 13.38

CD

1.67 9.63

2.81 16.72

3.63 27.92

2.33 29.72

.96 6.25

.77 7.81

.69 7.93

1.18 8.32

1.35 7.78

water/ ( 1.38 co
CD

1.14 16.62

6.61 224.55

1.28 35.84

2.02 27.70

CD

4.91 19.42

17.16 29.91

10.02 11.13

6.14 22.67

CD

2.43 3.39 18.89

9.36 24.31 23.50

5.01 5.72 8.65

4.90 8.30 22.23

CD

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

Peng and Robinson

Soave

propane 0.37 hexane (2) 1.09 C3) 18.31

CD

methane/ [2) co 2

2

decane/ { .81 co (C2) 10.23 2

CD

benzene/I water C2)

2

methane/{ 10.17 hexane 4C2) 12.44 methane/( 2.92 heptane C2) 11.49 C3) 12.31

(1) (2) (3)

Teja

percent r e l a t i v e absolute error i n c r i t i c a l temperature prediction percent r e l a t i v e absolute error i n c r i t i c a l pressure prediction percent r e l a t i v e absolute error i n c r i t i c a l volume prediction ( i f available)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

PALENCHAR ET AL.

6.

Prediction of Binary Critical Loci

147

TABLE I I I Values of f i t t e d parameters γ 12

a n

^

ζ 12 f o r each Cubic Equation of State

Redlich Kwong

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

system

Soave

Adachi

Peng and Robinson

Teja

propane/( 0·9244 hexane <[2) 0.9111

1.0097 0.9734

0.9094 0.8846

0.8320 0.8245

0.9897 0.9771

methane/(ID 0.9558 co ([2) 0.8765

1.0524 0.9222

1.0990 0.8871

0.7476 0.7893

1.0465 0.9250

butane/ (ID 0.9182 co <[2) 0.7567

0.9990 0.8390

1.2607 0.9713

0.6600 0.7475

0.9578 0.8294

decane/ I
1.3268 0.9352

-.2422 -1.0090

.4455 1.5599

1.400* .700

benzene/ ( [I) 0.8015 water <[2) 0.6556

0.8019 0.5980

.7257 .5536

.7210 .7694

.7562 .6110

water/ i[I) 1.0546 co <[2) 0.8828

1.0446 1.0407

1.000* 1.000

methane / [I) ( hexane {[2}

.9950 .9545

1.1310 1.1441

methane/([I) heptane <[2)

.9093 .7319

1.0237 0.9661

2

2

2

2

(1) (2) *

.5000* .6000 -.0096 -.2814

.6260 .95154

1.2307 1.2927

.6091 .7347

1.0682 0.9661

.3793 .6732

0.8861 0.7446

Calculated value for λ ι Calculated value for ς Values for \ and ς do not f i t the midpoint data exactly 2

1 2

i 2

1 2

American Chemical Society Library 1155 16th St., N.W. Washington, D.G. 20036 In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

148

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

The cone tante a, b, and c In the above equations are calculated e n t i r e l y from pure component properties* To extend the equations to mixtures, the following mixing rules were used: a - [ /α χ ]

b - [b

χ

l X ] L

+ /a x ] 2

1

2

+

+ b x ] + 2χ 2

c « c x + c x 1

2

2

2

2 x

x 1

( 2

1 ) y

^i2" l

ι Χ 2

( λ - 1)( 1 2

2 i

( 1 0 )

^2 )

(11) (12)

2

The constants i n these mixture equations can be evaluated e n t i r e l y from pure component properties alone by equating the parameters ς 12 and λ ι to unity. Equations (10) and (11) then become the mixing rules of the o r i g i n a l van der Waals equation. The ζ and λi2 parameters other than unity, must be determined by f i t t i n g the equation to actual experimental data for the mixture. 2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

1 2

Equations (3) and (4), together with an equation of state make up a system of non-linear equations i n three unknowns: T , P and V. For a given composition, the free energy equations were solved for the c r i t i c a l temperature and c r i t i c a l volume of the system. The c r i t i c a l pressure was then determined from the equation of state. U 8 e a < i n To calculate the values for λ ι " ζ 12» this work, the following functions were calculated at s 0.50: c

c

c

a n a

2

F

F

T

13

l - lexp " exp - ( c) lcl p

<> <>

p

14

ca

If experimental data at - 0.50 were unavailable, then the closest experimental value to Xi « 0.50 was used. Values of λ * ζ i2 calculated by using a damped Newton Raphson method, u n t i l the functions F i and F2 were less than 0.001. In most cases i n i t i a l starting values of λ ι • 1·0 and ζ 12 « 1.0 converged satisfactorily. In other cases much lower i n i t i a l t r i a l s were required. The values of λ ι and c obtained from data at X i « 0.5 were then used to calculate the entire c r i t i c a l locus from X i - 0.0 to x i » 1.0. a n a

w

e

r

e

1 2

2

2

i 2

Systems Selected for Study The P-T projections of the P-T-x c r i t i c a l surface of eight different binary systems were selected f o r study. The systems selected represent s i x of the nine different c r i t i c a l behavior types discussed by van Konynenburg and Scott (8) and summarized i n Table I. For each of these projected c r i t i c a l l o c i , the c r i t i c a l temperature, c r i t i c a l pressure, and c r i t i c a l volume were calculated from the five cubic equations of state i n Equations (5) - (9). As indicated i n Table I, the Class 1 Type I projections are characterized by a single continuous c r i t i c a l l i n e connecting the pure component c r i t i c a l temperatures. Binary mixtures which exhibit Class 1 Type I behavior have pure component c r i t i c a l temperatures

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Prediction of Binary Critical Loci

PALENCHAR ET AL.

6.

are i n the range studied are: (1) (2) (3)

T

c l

/T

c 2

<

2 (8).

The

Class

149 1 Type I systems

Propane and n-Hexane, Kay (16) Methane and Carbon Dioxide, Donnelly and Katz (17) η-Butane and Carbon Dioxide, Poettmann and Katz~~(18) and Redlich, K i s t e r , and Lacy (19)

More complicated Class 1 systems are those of Class 1 Type I I , which exhibit not only a continuous gas-liquid c r i t i c a l l i n e between the pure component c r i t i c a l s , but also a l i q u i d - l i q u i d c r i t i c a l l i n e from C to the UCEP termination of a three phase l i q u i d - l i q u i d - s o l i d l i n e , as described i n Table I. One system of Class 1 Type II was studied i n this work. This system i s : m

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

(4)

n-Decane and Carbon Dioxide, Reamer and Sage (20)

Class II behavior i s exhibited i n binary systems i n which the c r i t i c a l temperature of one of the pure components i s usually more than twice the c r i t i c a l temperature of the other. In Class II systems there i s no continuous c r i t i c a l l i n e connecting the two pure component c r i t i c a l points. Four different examples of Class 2 systems were studied i n this paper. These Include a Class 2 Type IIIa-HA system i n which the l i q u i d - l i q u i d c r i t i c a l l i n e between C2 and C , as described i n Table I, actually passes through a minimum temperature before diverging to the immeasurably high pressure at C. It i s therefore characterized as a Type IIIa-HAm system. The example chosen was: m

m

5)

Benzene and Water, Rebert & Kay (21), and Schneider (22)

Class 2 Type III systems, as indicated i n Table I, are the same as Type III-HA except that the three phase liquid-liquid-gas l i n e l i e s between the two pure component vapor pressure curves so that there i s no hetero-azetropic behavior, as discussed by Rowlinson ( 23). Two sample systems representing Type I l i a and Type I l l b were selected. These are: 6) 7)

Carbon Dioxide - Water, Takenouchi (24), Type I l i a Methane-η-Heptane, Kohn (27); Chang, Hurt & Kobayashi (28); and Reamer, Sage and Lacey (29), Type I l l b

The f i n a l c r i t i c a l behavior studied i n this work i s that of Class 2 Type V. In this case there are two c r i t i c a l l i n e s , one from the pure component c r i t i c a l CI to a UCEP and a second from the other pure component c r i t i c a l C2 to a LCEP, as described i n Table I. One example of this behavior was examined. This was: 8)

Methane-n-Hexane, Shim & Kohn (25) and Poston & McKetta (26) —

Results and C r i t i c a l Loci Predictions The average percent r e l a t i v e absolute error i n predicting the selected binary c r i t i c a l l o c i i s shown i n Table II and the c r i t i c a l

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D

150

APPLICATIONS

methane-carbon dioxide mixtures are plotted as a function of composition as predicted by the Teja equation. The c r i t i c a l values were f i t t e d exactly at a methane mole f r a c t i o n of 0.457. As shown i n Figures 10 and 11, the equation generally follows the trend of the data but predicts c r i t i c a l temperatures s l i g h t l y too high at mole fractions less than 0.457 and s l i g h t l y too low at larger mole f r a c t i o n s . The same trends are exhibited by the other equations and the shapes of the predicted l o c i are s i m i l a r . Accuracy of the c r i t i c a l l o c i predictions i s considerably enhanced by the use of two unlike pair parameters, instead of a single ζ parameter only, as i s a common p r a c t i c e . This i s p a r t i c u l a r l y true when the members of the binary mixture have large d i s s i m i l a r i t i e s i n s i z e and character. Usually, the assignment of ζ i2 1·0 * ΐ2 " predicts the c r i t i c a l temperature much more accurately than the c r i t i c a l pressure. The c r i t i c a l pressure i n the cases studied here, i s always extremely low, frequently by more than 30%. These large Pc errors can be remedied by changing ζΐ2> usually decreasing i t s value. Unfortunately, when the calculated and experimental Pc values are brought into good agreement by this process, the error i n the calculated Tc i s made larger than the i n i t i a l r e s u l t obtained with both X and ζ equal to 1.0 . 1 2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

=

a n <

λ

i2

Χ 2

Discussion As indicated i n Table I I , the c r i t i c a l temperatures and c r i t i c a l pressures i n systems of Class 1 Type I are predicted accurately by a l l equations of the van der Waals family. Best results for the predicted c r i t i c a l temperatures and pressures of these systems were obtained with the Teja equation. It i s important to note that this comparison applies only to c r i t i c a l l o c i tested by the procedures described here. It does not imply that this ranking i s v a l i d over the entire P-T-x diagram for these systems. The excellent results for the Class 1 Type I systems gives some support to the mixing rules adopted i n Equations (10)-(11). The results indicate that the two f i t t e d unlike pair c o e f f i c i e n t s i n these mixing rules resemble true molecular properties i n that the same values are v a l i d over the entire composition range. A remarkable feature of the van der Waals family of equations i s the a b i l i t y of a l l the equations tested to predict the l o c a l minimum i n the c r i t i c a l locus of the η-butane-carbon dioxide system at high concentrations of carbon dioxide, as shown i n Figure 3. This phenomenon i s well established experimentally for Class 1 Type I systems involving carbon dioxide. Morrison and Kincaid (30) have shown experimentally that this minimum exists both for the n-butanecarbon dioxide and also for the propane-carbon dioxide systems. It i s clear from Table II that the excellent accuracy of the c r i t i c a l temperature and c r i t i c a l pressure predictions for Class 1 Type I systems does not extend at a l l to the c r i t i c a l volumes. A common feature of the van der Waals family of equations i s that they predict pure component c r i t i c a l compressibility factors which are

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

6.

Prediction of Binary Critical Loci

PALENCHAR ET AL.

151

too large for components whose molecules contain more than one atom. This results i n c r i t i c a l volume predictions which are always too high. The best prediction of c r i t i c a l volumes i s given by the Peng-Robinson equation which predicts a c r i t i c a l compressibility factor of 0.307 (5). Although this value i s higher than the c r i t i c a l compressibility factor of the components i n the test systems, i t i s the lowest among the equations tested. For a system belonging to Class 1, Type I, a more accurate method of predicting i t s mixture c r i t i c a l volume V i s to use a simple linear average i n the form: c m

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

V

cm

X

V

- l cl

+

x

V

2 c2

( 1 6 )

Equation (16) gives a r e l a t i v e absolute error of 3.56% for the propane-hexane system and 6.5% for η-butane-carbon dioxide. These errors are much less than any of those i n Table II obtained with a van der Waals type equation of state for these systems. Results of Equation (15) for the propane-hexane system are shown graphically i n Figure 5. Unfortunately, Equation (16) i s not an e f f e c t i v e method for predicting c r i t i c a l volumes i n systems of more complicated types. It obviously f a i l s for Class 2 systems where there i s no continuous c r i t i c a l l i n e between the pure component c r i t i c a l s . For the Class 2 Type V system methane-n-heptane, for example, Equation (16) predicts a c r i t i c a l volume which i s generally worse than the equation of state predictions shown i n Table II for the c r i t i c a l volume of this system. A l l equations tested predict c r i t i c a l l o c i i n Class 2 systems and i n Class 1 Type II systems with much less accuracy than for the Class 1 Type I systems. In contrast to the results with Type I systems, the two-constant equations of state with the smallest deviation from the form of the o r i g i n a l van der Waals equation, such as the Redlich-Kwong and Soave equations, give better predictions of c r i t i c a l temperature and c r i t i c a l pressure i n Class 1 Type II and i n Class 2 systems than the more elaborate three-constant equations and equations which deviate farther from the o r i g i n a l van der Waals form. An example of a Class 1 Type II system i s shown i n Figure 4 for the n-decane-carbon dioxide system, where predictions of the Soave equation are i l l u s t r a t e d . The c r i t i c a l l i n e between the pure component c r i t i c a l s was f i t exactly at a mole f r a c t i o n of carbon dioxide • 0.5 by the unlike pair c o e f f i c i e n t s shown i n Table I I I . At lower carbon dioxide concentrations the predictions are excellent, as shown i n Figure 4. At carbon dioxide mole fractions greater than 0.5, approaching the pure carbon dioxide c r i t i c a l , the predicted c r i t i c a l pressures are much too low. No predictions at a l l are possible for the l i q u i d - l i q u i d c r i t i c a l l i n e from the UCEP to higher pressures. Among the Class 2 systems, the benzene-water system i s an example of Type III-HAm behavior, where the "m" indicates a minimum

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

152

i n the c r i t i c a l l i n e between the c r i t i c a l of water (C2) and the high pressure point at C . The results obtained with the Soave and Redlich-Kwong equations, which gave the best results for this system, are shown i n Figure 6. Although the f i t t i n g point at 0.4895 m.f. benzene l i e s on the vapor-liquid c r i t i c a l l i n e between the UCEP and the benzene c r i t i c a l (CI), the predictions are q u a l i t a t i v e l y correct along the other c r i t i c a l l i n e between C2 and C . It i s remarkable also that the Redlich-Kwong equation shows the required minimum i n this c r i t i c a l l i n e , although i t does not occur at the precisely correct pressure and temperature values. m

m

C r i t i c a l temperature and pressure predictions for the Class 2 Type I l i a system carbon dioxide-water are shown i n Figure 7 for the Soave equation. Because there i s no c r i t i c a l locus at X ^ q o = 0.5 the unlike pair c o e f f i c i e n t s for the Soave equation were obtained at C02 0*321, corresponding to the highest pressure data point on the c r i t i c a l l i n e between C and the c r i t i c a l point of water, C2. With c o e f f i c i e n t s evaluated at this concentration, the Soave equation can predict both parts of the c r i t i c a l l i n e , although the UCEP i s not located precisely so that results along the c r i t i c a l l i n e from CI to UCEP are only q u a l i t a t i v e l y predicted.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

X

=

m

Among the other equations, i t was possible to find unlike pair c o e f f i c i e n t s which could f i t both the c r i t i c a l pressure and c r i t i c a l temperature at this same C0 composition only for the Teja equation. The nearest composition where most of the others could be f i t to both the c r i t i c a l pressure and temperature was the data point i n Figure 7 at 0.270 m.f. C0 , corresponding to a c r i t i c a l pressure of 12,836 psia, although the predictions at other compositions are poor, p a r t i c u l a r l y for the steeply r i s i n g pressure. For a l l equations except the Soave and Teja equations the errors i n Table II are those obtained when using unlike pair c o e f f i c i e n t s f i t t e d at 0.270 m.f. C0 . Only for the Soave equation could the unlike pair c o e f f i c i e n t s , which were f i t t e d exactly to the single data point at 0.321 m.f. C0 , produce accurate pressure predictions along the c r i t i c a l l i n e from C to C2 at other compositions. For the other equations, reasonable predictions over the entire c r i t i c a l l i n e cannot be obtained from unlike pair c o e f f i c i e n t s f i t t e d at a single composition. 2

2

2

2

m

C r i t i c a l behavior of the Type I l l b methane-n-heptane system has been explained i n d e t a i l by Chang, Hurt and Kobayashi (28). Calculated results are shown i n Figure 9. Accurate results are obtained only at low methane concentrations along the c r i t i c a l l i n e from the heptane c r i t i c a l (C2) to the UCEP at the termination point of a l i q u i d - l i q u i d - s o l i d l i n e when the two l i q u i d phases become i d e n t i c a l . Reasonable results are obtained only along this c r i t i c a l l i n e between the heptane c r i t i c a l and a methane concentration of about 0.7 mole f r a c t i o n at a temperature of 775°R. At higher methane concentrations the calculated c r i t i c a l pressures are much too large. Reasonable results are also obtained along the very short second c r i t i c a l l i n e from the methane c r i t i c a l to the UCEP at the l i m i t of the liquid-liquid-gas l i n e . Results for the Class 2 Type V me thane-n-hexane system are shown i n Figure 8. This system i s predicted only q u a l i t a t i v e l y by

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

6.

P A L E N C H A R ET A L .

Prediction of Binary Critical Loci

153

the procedure used here for evaluating the unlike pair coefficients. On the c r i t i c a l l i n e between the LCEP and the hexane c r i t i c a l (C2), methane concentrations above the f i t t i n g point of CH4 0*^25 at a c r i t i c a l temperature of 762°R give predicted pressures which are much too low with a l l the equations. It i s possible to calculate some points on the gas-liquid c r i t i c a l l i n e between the UCEP and the methane c r i t i c a l (CI) but the accuracy i s extremely poor with the calculated c r i t i c a l pressure much too small. Reasonable results are obtained only at low methane concentrations near the hexane c r i t i c a l . Accuracy deteriorates rapidly as methane concentrations approach those where three phase conditions occur between the LCEP and the UCEP when the unlike pair c o e f f i c i e n t s are evaluated at conditions far from the three phase region.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

X

s

Predictions of Class 2 c r i t i c a l l o c i by equations of the van der Waals family are limited by their i n a b i l i t y to predict c r i t i c a l end points of three phase l i n e s . As an example, Figures 12 and 13 show the experimental and predicted UCEP conditions f o r the termination of the l i q u i d - l i q u i d - s o l i d l i n e i n the Type I l l b methane-n-heptane system. Results obtained with the Soave and Teja equations are shown. None of the equations tested here produced accurate UCEP values. Conclusions The results of this study may

be summarized as follows:

1. C r i t i c a l temperatures and c r i t i c a l pressures of Class 1, Type I, binary systems can be predicted successfully by a l l cubic equations of state of the van der Waals family tested i n this paper. The mixing rules used i n each equation studied here are those of the o r i g i n a l van der Waals equation with an added correction. This correction i s for the deviation of the contributions of unlike pair interactions from the form of these contributions assumed i n the o r i g i n a l van der Waals equation. It i s described by two adjustable unlike pair interaction c o e f f i c i e n t s . 2. In most, but not a l l , of the cubic equations tested the unlike pair interaction c o e f f i c i e n t s required for Class 1, Type I systems do not depend appreciably on composition, temperature, or pressure. This i s shown by the fact that when these c o e f f i c i e n t s are evaluated by f i t t i n g them to a single experimental data point for an approximately equi-molar mixture, the c r i t i c a l temperature and pressure at a l l other compositions on the c r i t i c a l locus are predicted with an average absolute r e l a t i v e error of less than 1%. Best accuracy i n predicting Class 1, Type I c r i t i c a l temperatures and pressures i n this way was obtained by using the Teja equation. 3. C r i t i c a l volumes predicted at a l l compositions, when the unlike pair c o e f f i c i e n t s are evaluated to predict the c r i t i c a l temperature and pressure at a single data point, are always too large for a l l the equations tested. For a l l Class 1, Type I systems

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

154

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

the Peng-Robinson equation gave the best prediction of the c r i t i c a l volume with the unlike pair c o e f f i c i e n t s f i t t e d to the c r i t i c a l temperature and pressure at a single data point. In Class 1, Type I systems a simple l i n e a r average of the pure component c r i t i c a l volumes gave a s i g n i f i c a n t l y better prediction of the mixture c r i t i c a l volume than any of the equations of state tested.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

4. In Class 1, Type I I systems and i n a l l Class 2 systems, a l l the equations predict c r i t i c a l temperatures and pressures with much less accuracy than for Type I systems when using the test procedures of this study. The r e s u l t s , showed that c r i t i c a l temperature predictions are much more accurate than those for c r i t i c a l pressure. 5. In Class 1, Type I I , and i n a l l Class 2 systems, the region of a c r i t i c a l locus near the formation conditions f o r a second l i q u i d phase or a s o l i d phase cannot be predicted r e l i a b l y when using unlike pair c o e f f i c i e n t s evaluated at a data point on a vaporl i q u i d or l i q u i d - l i q u i d c r i t i c a l locus far removed from the point of appearance of a third phase. 6. In Class 1, Type I I , and i n a l l Class 2 systems, either the Redlich - Kwong or the Soave equation shows the least v a r i a t i o n of the unlike pair c o e f f i c i e n t s with composition, temperature, and pressure and gives best predictions of c r i t i c a l properties at a l l compositions from pair c o e f f i c i e n t s evaluated at a single data point. For these more complicated systems, i t appears that the simpler two-constant equations, l i k e the Redlich - Kwong and Soave, give better c r i t i c a l l o c i predictions than the more elaborate equations whose form deviates farther from that of the o r i g i n a l van der Waals equation. Acknowledgments The authors appreciate the support of University by the Gas Research I n s t i t u t e .

this

research

at

Rice

The authors also appreciate the h e l p f u l suggestions made by Dr. J.M.H. Levelt Sengers.

Literature Cited 1. Chang, R. F.; Morrison, G.; Levelt Sengers, J.M.H. Chem. 1984, 88, 3389.

J. Phys.

2. Joffe, J . ; and Zudkevitch, D. Chem. Eng. Prog. Symp. S. 63, 43.

1967,

3. Spear, R. R.; Robinson, R.L.; Chao, K.C. Ind. Eng. Chem. Fundam. 1969, 8, 2. 4. Hissong, D. W.; Kay, W. B. AIChE J. 1970, 16, 580. 5. Peng, D.-Y.; Robinson, D. B. AIChE J. 1977, 23, 137. 6. Teja, A. S.; Rowlinson, J. S. Chem. Eng. Sci.

1973, 28, 529.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

6. PALENCHAR ET AL. Prediction of Binary Critical Loci 155

7. Teja, A. S. Smith, R. L.; Sandler, S. I. Equilibria 1983, 14, 265.

Fluid Phase

8. van Konynenburg, P. H.; Scott, R. L. Phil. Trans. Roy. Soc. Lond. 1980, 298, 495. 9. Gibbs, J. W. "Collected Works"; Longmans, Green, and Company; New York, 1928, Vol. 1, 129. 10. Redlich, O.; Kister H. Ind. Eng. Chem. 1948, 40, 345. 11. Redlich, O.; Kwong, J. N. S. Chem. Rev. 1949, 44, 233.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

12. Soave, G. Chem. Eng. Sci.

1972, 27, 1197.

13. Adachi, Y.; Lu Β. C.-Y.; Sugie, H. Fluid Phase Equilibria 1983, 13, 133. 14. Peng, D.-Y; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. 15. Teja, A. S.: Patel, N. C. Chem. Eng. Sci.

1982, 37, 463.

16. Kay, W. B., J. Chem. Eng. Data 1971, 16, 137. 17. Donnelly, H. G.; Katz, D. L. Ind. Eng. Chem. 1954, 46, 511. 18. Poettmann, F. H.; Katz, D. L. Ind. Eng. Chem. 1945, 37, 59. 19. Redlich, O.; Kister H.; Lacey, W. N. Ind. Eng. Chem. 1949, 841, 475. 20. Reamer, H. H.; Sage, B. H. J. Chem. Eng. Data 1963, 8, 508. 21. Rebert, C. J.; Kay, W. B. AIChE J. 1959, 5, 285. 22. Schneider, G. M. Ber. Bunsenges. Physik. Chem. 1972, 76, 325. 23. Rowlinson, J. S. "Liquids and Liquid Mixtures" 2nd Edition, Butterworth and Co., 1969, 203-216. 24. Takenouchi, S.; Kennedy, G. C. Am. J. Sci.

1964, 262, 1055.

25. Shim, J.; Kohn, J. P. J. Chem. Eng. Data 1962, 7, 3. 26. Poston, R. S.; McKetta, J. J. J. Chem. Eng. Data 1966, 11, 362. 27. Kohn, J. P. AIChE J. 1961, 7, 514. 28. Chang, H. L.; Hurt, L. J.; Kobayashi, R. AIChE J. 1966, 12, 1212. 29. Reamer, H. H.; Sage, Β. H.; and Lacey, W. N. Chem. Eng. Data S. 1956, 1, 29. 30. Morrison, G.; Kincaid, J. M. AIChE J. 1984, 30, 257. RECEIVED November 8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

7 Equation of State for Supercritical Extraction J. S. Haselow, S. J. Han, R. A. Greenkorn, and K. C. Chao

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

School of Chemical Engineering, Purdue University, West Lafayette, IN 47907

Nine equations of state are evaluated regarding their ability to describe supercritical extraction. Experimental data on 31 binary mixture systems are compared with calculations from the equations of Redlich-Kwong, Soave, Peng-Robinson, Schmidt-Wenzel, Harmens, Kubic, Heyen, Cubic Chain-of-Rotators, and Han-Cox-Bono-Κ wokStarling. Interaction constants of the equations determined from the experimental data in the course of the evaluation are reported.

Supercritical extraction has received much attention recently for its many applica tions and potential applications. Supercritical carbon dioxide is used to decaffeinate coffee, denicotinize cigarettes, and extract spices ( j_j2 ). Carbon dioxide flooding has assumed major importance in petroleum production ( 3 ). Extraction of coal with a supercritical solvent has been the subject of investigation by the British Coal Board ( 4J) ), Maddocks et al. ( 6 ), Blessing and Ross ( 7 ), Ross and Blessing ( 8 ), and Vasilakos et al. ( 9 ). Kerr McGee Oil Company developed supercritical extrac tion for the deashing of coal liquefaction reactor products in the ROSE process (10). The design and operation of a supercritical extraction process requires the estima tion of solute solubility in a supercritical solvent. Equation of state can be useful to satisfy this need. In this work, equations are screened regarding their applicability to the calculation of supercritical solubility, and the applicable equations are evaluated regarding their ability to quantitatively describe experimental solubility data. 0097-6156/ 86/ 0300-0156S07.00/ 0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

7.

H A S E L O W ET A L .

Supercritical Extraction

157

The Equations of State

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

From the large number of equations of state that have been proposed in the litera ture, we have found only a small number of them to show promise of being useful for the estimation of supercritical solubility. A severe limitation is the availability of the equation constants. Equations that require their constants to be determined by fitting extensive experimental data on the pure substances to the equation are not useful, for rarely are experimental data available for the complex solutes of interest. We are thus forced to discard complex equations such as the Benedict-WebbRubin ( U ), Jacobsen ( 12 ), Goodwin ( JL3 ), Perturbed-Hard-Chain ( 14 ), and Chain-of-Rotators ( 1_5 ). For detailed evaluation, we have selected nine equations from among those that have been in use in phase equilibrium calculations and newer equations that show promise. Eight of these are cubic equations, and one (Han-Cox-Bono-Kwok-Starling) is complex. But all have been generalized to express the equation constants in terms of critical properties, which are known for a large number of substances. Group con tribution ( 16,17 ) and other methods ( 18,19 ) are widely used for the estimation of the critical properties where experimental values have not been reported. Table I presents the nine equations of state that are studied in this work. Table II shows the critical properties of the substances that are used in the calculations. For many higher molecular weight compounds T , p , and ω were estimated, T and p by the method of Lydersen ( 18 ) and υ by the method of Edmister ( 19 ). c

c

c

c

Mixing rules described in the original papers are used in the calculations reported here for the Redlich-Kwong, HCBKS, Peng-Robinson, Kubic, Heyen, and CCOR equations. No interaction constant was employed by Soave in the mixing rules for his equation. We introduced an interaction constant k to the constant a, as we found it quite necessary. The Schmidt-Wenzel and Harmens-Knapp equations con tained no mixing rules at all. We employ the classical one-fluid mixing rules to extend these equations to mixtures as follows: 1 2

i J

Where θ may be either a or b, and the cross parameter θ

ι}

a

is given by

= (l-kyKauajj) ^ 1

y

<

2)

and h

= (H b

+

b

ij)/

2

< 3 )

These are, in fact, the same rules employed in the other equations studied in this work.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

158

Table I. Equations of State

1. Redlich-Kwong equation Source: Redlich, O., and Kwong, J.N.S., Chem. Rev. (1949) 44 , 223. RT Ρ

a

(_) v

b

T

o.5 ( v

v +

b

)

2. HCBKS equation Sources: Cox, K . W . , Bono, J.L., Kwok, Y . C . , and Starling, K . E . , Ind. Eng. Chem. Fundamen. (1971) 10 , 245.

Starling, K . E . , and Han, M.S.,

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

Hydrocarbon Processing (1972), 4 , 129.

p^RT + ÎBjtT-A.-^ + ^ - ^ p »

+ ( b R T - a - ^ ) / + a(a +

2

e

^V

+ -^"(1 + 7P )exp( - ΊΡ ) 2

3. Soave equation Source: Soave, G . , Chemical Engineering Science (1972) 27 , 1197.

P

RT _ a(T) (v-b) v(v+b)

4. Peng-Robinson equation Source: Peng, D., and Robinson, D.B., Ind. Eng. Chem. Fund. (1976) 15 , 59. RT (v-b)

P

a(T) [v(v-b)+b(v-b)]

5. Schmidt-Wenzel equation Source: Schmidt, G., and Wenzel, H., Chemical Engineering Science (1980) 35 , 1503. RT P

(v-b)

a 2 v

+ ubV+wb

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

7.

HASELOW ET AL.

Table I

159

Supercritical Extraction

Continued

6. Harmens-Knapp equation Source: Harmens, Α., and Knapp, H . , Ind. Eng. Chem. Fundam. (1980) 19 , 291.

P

RT

a

(v-b)

v + cbV-(c-l)b

2

2

7. Kubic equation Source: Kubic, W.L., Fluid Phase Equilibria (1982) 9 , 79. RT

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

P

_

(v-b)

a (v+c)

2

8. Heyen equation Source: Heyen, G., "A Cubic Equation of State with Extended Range of Application" in "Chemical Engineering Thermodynamics", Ann Arbor Science.

P

RT

a

(v-b)

v +(b + c)v-bc

(Ann Arbor), (1983).

2

9. C C O R equation Source: Kim, H., Lin, H.M., and Chao, K . C . , Ind. Eng. Chem. Fundam., (1985) in press. _ R T ( l + 0.77b/v) P

(v-0.42b)

R

0.055c RTb/v

a

bd

(v-0.42b)

v(v+c)

v(v+c)(v-0.42b)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

160

Table II. Critical Properties Used and Sources

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

Réf.

Pc(atm)

Carbon Dioxide

304.2

72.8

0..225

94.0

a

Ethylene

282.4

49.7

0,.085

129.0

a

Benzoic Acid

752.0

45.0

0..620

341.0

a

Phenanthrene

890.0

32.5

0 .429

644

2,3-Dimethylnaphthalene

785.0

31.748

0 .42403

522

2,6-Dimethylnaphthalene

777.0

31.796

0 .42013

516

c

Naphthalene

748.4

40.0

0 .302

410.0

a

Ethane

305.4

48.2

Fluorene

821

1,4-Naphthoquinone

792*

e

29.5

ω

257.0

31.5*

0,.498*

694.2

60.5

0 .440

Hexamethylbenzene

752

Triphenylmethane

229 785

f

0 494

751

0 .576

936

e

25.7

Trifluoromethane

299

48.2

0 .275

133

Chlorotrifluoromethane

302

38.7

0 .180

180

Hexachloroethane

698.4

e

e

32.97

a

f

e

22.1

e

399 659

0 498

e

d

d

f

e

863

a d

0 .575*

883*

23.5

c

d

590.0

Phenol

e

b

d

0 407

f

40.7*

d

148.0

0 .091 e

Acridine

Pyrene

a

3

Vc(cm /mol)

Tc(K)

Compound

h

0 .255

d

d

470

d

a a d

Reid, R . C . , Prausnitz, J.M., and Sherwood, T . K . , "The Properties of Liquids and Gases",

3rd éd.; McGraw-Hill: New York, 1977. b

G P A Research Report RR-30, "High Temperature V - L - E Measurements for Substitute Gas

Components"; Tulsa, Oklahoma, 1978. c

Driesbach, R.R., "Physical Properties of Chemical Componds", American Chemical Society,

Washington, D.C., 1955. d

Estimated by Zc = 0.291-0.008u;,Vc = ZcRTc/Pc .

e

Estimated by Lydersen's method.

f

Estimated by Edmister's method.

* Estimated by method of Joback (R.C. Reid, Personal Communication). h

Perry, R . H . , and Chilton, C H . , "Chemical Engineers Handbook", 5th ed.; McGraw-Hill:

New York, 1973, for vapor pressures, definition of acentric factor.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

7.

HASELOW ET AL.

161

Supercritical Extraction

Calculation of Supercritical Solubility

Solubility of a solid solute in a supercritical solvent is given by (4)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

y, =

p^fexp^p-pn/RT]

/(ρΦ,)

where the symbols are defined in Glossary of Symbols. In Equation 4 it is assumed that the solvent is not dissolved in the condensed phase, which is valid when the condensed phase is a solid. We exclude solubility data of liquids. Equation 4 addi tionally assumes that the solid phase is incompressible. The fugacity coefficient of the saturated pure vapor of the solute is set to be equal to 1 in view of the small vapor pressures. The calculation of solubility V j according to Equation 4 requires the \ apor pres sure pf and molal volume V of the solid solute to be known. Table III shows the sources from which vapor pressures of the solutes were obtained. Table IV shows the solid volume data that were used in the calculations. Since data were not available at the various temperatures of interest and since the effect of inaccuracy of solid volume is small, variation of the solid volume with temperature was ignored in the calculations. r

s

The variation of supercritical solubility with pressure is given by ρ and Φι in Equation 4. The characteristic supercritical effect of solute-solvent interaction is expressed by Φ^. The calculation of Φγ by an equation of state, and the use of it in Equation 4 to calculate V j for comparison with experimental data constitutes the test of the equation of state. Calculation of Φ γ by an equation of state follows the standard thermodynamic procedure described in textbooks. The calculation is sensitively dependent on the value of the interaction constant(s). A value of kjj of an equation of state is deter mined for each solute + solvent system from experimental solubility data for the best fitting of the data. The objective function

Ω = Σ [ln(yi,exp/yi,cal)]

2

(5)

is minimized in searching for the kjj value. Here η stands for number of solubility data points. The kjj thus determined is employed in a final round of calculations to give the calculated solubilities. The quality of the calculated supercritical solubility is expressed in terms of an average % deviation of the equation of state from the data given by

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

162

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table III. Vapor Pressure Data Sources Compound 2,3-Dimethylnaphthalene

a

2,6-Dimethylnaphthalene

a

Phenanthrene

b

Benzoic Acid

b

Hexachloroethane Naphthalene Fluorene

c d,e f

1,4-Naphthoquinone

g

Acridine

g h

Phenol

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

Reference

Hexamethylbenzene

i

Triphenylmethane

i h

Pyrene

* Osborn, A . G . , and Douslin, D.R., J. Chem. Eng. Data (1975) 20 , 229. b

deKrulf, C . G . , and van Grunkel, C.H.D., presented at the Quatrième Conference Intera-

tionale de Thermodynamique Chimique, 26 au 30, Montpellier, France, 1975. c

Sax, N.I., "Dangerous Properties of Industrial Materials", Van Nostrand Reinhold; Prince-

ton, 1979. d

Diepen, G.A., and Scheffer, F . E . C . , J. Am. Chem. Soc. (1948) 70 , 4085.

e

Fowler, L . , Trump, W., and Vogler, C , J. Chem. Eng. Data (1968) 13 , 209.

f

Johnston, K . P . , Ziger, D.H., and Eckert, C.A., Ind. Eng. Chem. Fundam. (1982) 21 , 191.

g

Schmitt, W.J., and Reid, R . C . , presented at Annual AIChE Meeting, San Francisco, C A

(1984). h

Weast, R.C., "Handbook of Chemistry and Physics", 55th ed.; Chemical Rubber Company:

U.S.A., 1975. Ambrose, D., Lawrenson, I.J., and Sprake, C.H.S., J. Chem. Thermodyn. (1975)8 , 503. Aihara, B., Chem. Soc, Japan, in Weast, op. cit.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

7.

Supercritical Extraction

HASELOW ET A L .

163

Table IV. Solid Volume Data and Sources s

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

Compound

Reference

8

Benzoic Acid

15

154.60

a

Hexachloroethane

20

113.22

a

Naphthalene

20

125.03

a

Phenanthrene

4

181.9

a

2,3-Dimethylnaphthalene

20

155.76

a

2,6-Dimethylnaphthalene

20

155.76

a

Phenol

20

89.0

a

Fluorene

0

138.18

a

1,4-Naphthoquinone

-

111.2

b

178.3

b

25

152.66

a

20

245

a

23

159.13

a

Acridine Hexamethylbenzene Triphenylmethane Pyrene a

V (cm /gmole)

c

Weast, R . C . , "Handbook of Chemitry and Physics", 55th éd.; Chemical Rubber Co.: U.S.A.

(1975). b

Schmitt, W.J., and Reid, R . C . , presented at AIChE Annual Meeting, San Francisco, C A

(1984). c

The density of diphenylmethane was used to calculate V

8

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

164

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

1

(6)

2

% deviation = 100" exp [(Ω/η) / ] -1

The use of logarithm in eq. 5 in preference to the sum of squares of relative devia tions avoids the bias that would be produced when the calculated values are either very large or very small when compared to data.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

Data and Results Table V presents the sources of experimental supercritical solubility data for the 31 systems studied. Many more systems have been measured. Reid (20) made an extensive compilation. But limitations on available vapor pressure or solid volume data prevented some of them from being analyzed here. Table VI presents the summary of the comparison of the equation of state in 31 parts. In each part, experimental y's of the solute in a supercritical solvent are com pared with calculations with the nine equations of state. The average % deviation is shown for each equation. Interaction constants are presented, one for each equation of state, except for the CCOR equation for which two interaction constants k j and k are given in this order. a

2

c l 2

The calculated results appear to depend on the source of the critical constants. The deviations are significantly larger for those solutes whose critical constants have been estimated by Lydersen's and Edmister's methods. The 31 systems studied are, therefore, arranged into two groups: the 14 systems with known T p and υ are shown in parts 1 through 14 in Table VI and the results are summarized in Table VII; the remaining 17 systems with estimated T p and ω are shown in parts 15 through 31 in Table VI. The overall summary comparison for all 31 systems is presented in Table VIII. c

c

c

c

For systems of known critical constants, the Redlich-Kwong appears to give the best overall description of supercritical solubility, showing an average deviation of 17% in Tble VII. The HCBKS, Peng-Robinson, and Heyen equations belong together showing average percent deviations in the middle twenties. Soave, Schmidt-Wenzel and CCOR equations form the next group with average percent deviations in the lower thirties. Harmen-Knapp equation follows with a somewhat higher percent deviation of 41. The Kubic equation, showing a very large deviation, does not even qualitatively describe supercritical solubility. The cross interaction constant of the Kubic equation is not given by Equation 2, but by a different for mula, which causes the failure.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

7.

Supercritical Extraction

HASELOW ET AL.

165

Table V . Supercritical Solubility Data Sources Mixture System (Solvent; Solute)

a

Ethylene, Benzoic Acid

a

C0

, Hexachloroethane

a

Ethylene, Phenanthrene

a

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

C0

2

2

, Phenanthrene

a

Ethylene, 2,6-Dimethylnaphthalene

a

C0

2

, 2,3-Dimethylnaphthalene

a

C0

2

, 2,6-Dimethylnaphthalene

a

Ethylene, Phenanthrene

a

Ethane, Naphthalene

b

Ethane, Phenanthrene

b

Ethane, Triphenylmethane

b

C0

, Triphenylmethane

b

Ethylene, Hexamethylbenzene

b

2

2

, Hexamethylbenzene

b

C0

2

, Fluorene

b

C0

2

, Pyrene

b

C0

2

, Phenol

c d

C C I F , Naphthalene

d

3

C0

2

, 1,4-Naphthoquinone

C H F , 1,4-Naphthoquinone 3

d d

C C I F 3 , 1,4-Naphthoquinone

d

Ethane, 1,4-Naphthoquinone

d

C0

d

2

, Acridine

Ethane, Acridine

d

C H F , Acridine

d

3

d

b

C H F , Naphthalene 3

c

b

Ethylene, Fluorene

Ethylene, Pyrene

b

a

Ethylene, 2,3-Dimethylnaphthalene

C0

a

Reference

Carbon Dioxide, Benzoic Acid

Kurnick, R,T., Holla, S.J., and Reid, R . C . , J. Chem. Eng. Data (1981) 26 , 47. Johnston, K.P., Ziger, D.H., and Eckert, C.A., Ind. Eng. Chem. Fundamen. (1982) 21 , 191. VanLeer, R.A., and Paulaitis, M . E . , J.Chem. Eng. Data (1980) 25 , 257. Schmitt, W.J., and Reid, R . C . , presented at AIChE Annual Meeting, San Francisco, C A ,

(1984).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table VI.

Deviations of Equation of State from S o l u b i l i t y Data 1. Benzoic acid

2. Benzoic acid

3. Naphthalene in

in Carbon Dioxide

in Ethylene

Trifluorochloromethane

Equation

av. % dev.

ki2

16.8

-.2686

11.3

.0145

10.5

HCBKS

.0118

21.5

-.0381

36.5

.0571

11.2

Soave

.0433

43.2

-.0080

19.6

.0959

17.7

Peng-Robinson

.0381

36.8

-.0118

14.8

.0983

16.4

Schmidt-Wenzel

.0256

279

-.0557

31.2

.0907

27.6

Harmens-Knapp

.0156

40.2

-.0633

37.9

.0883

35.2

Kubic

-.1257

409.8

-.1696

153.9

-.0582

139.7

Heyen

-.0509

10.3

-.0845

23.1

.0644

21.2

.3796

19.0

.3969

13.4

.3644

16.5

CCOR*

.5227

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

av. % dev

av. % dev. -.1797

Redlich-Kwong

Equation

4. Phenanthrene

5. Phenanthrane in

in Ethylene

Carbon Dioxide

ki2

Redlich-Kwong

.4351

.5178

av. % dev.

ki2

av. % dev.

6. 2,3-dimethyInaphthalene in Ethylene ki2

av. % dev.

-.0820

19.4

.0094

9.9

-.1108

30.2

HCBKS

.0256

26.5

.0571

18.5

.0207

25.7

Soave

.0782

284

.1371

37.5

-.0546

39.1

Peng-Robinson

.0734

20.5

.1284

30.0

.0509

29.6

Schmidt-Wenzel

.0658

36.1

.1384

31.9

.0457

30.0

Harmens-Knapp

.0519

44.3

.1246

45.7

.0357

37.2

Kubic

-.2374

1611.1

-.1935

13523.2

-.1585

439.8

Heyen

-.0207

23.4

.0283

14.8

-.0270

24.7

CCOR

.4422

34.8

.4724

21.5

.3572

110.0

.6242

7. 2,6-dimethylnaphthalene

8. 2,3-dimethylnaphthaIene

in Ethylene

in Carbon Dioxide

Equation

Redlich-Kwong

.5103

.6694

av. % dev.

k

1 2

a v. % dev.

9. 2,6-dimethylnaphthalene in Carbon Dioxide k^

av. % dev.

-.1045

18.7

-.0045

12.8

-.0031

9.1

HCBKS

.0181

16.0

.0506

26.1

.0495

19.4

Soave

.0544

40.0

-.1197

41.9

.1170

33.7

Peng-Robinson

.0509

31.1

.1132

36.5

.1108

28.5

Schmidt-Wenzel

.0443

25.6

.1284

22.9

.1257

20.6

Harmens-Knapp

.0332

33.0

.1184

32.4

.1159

29.9

Kubic

-.1561

467.4

-.1222

1638.6

-.1197

1733.9

Heyen

-.0218

17.6

.0308

23.3

.0319

12.9

CCOR

.3894

56.5

.4261

32.2

.4248

31.2

.5493

.6043

.6054

First constant is k ^ , second k 2cl

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Supercritical Extraction

HASELOW ET A L .

Table

VI

Continued

Equation

11. Phenanthrene in

Ethane

Ethane

ki2

Redlich-Kwong

av. % dev.

12. Naphthalene in Carbon Dioxide

av. % dev.

k^

av. % dev.

-.0519

29.9

-.0533

23.4

.0308

16.7

.0343

31.0

.0357

29.2

.0557

21.9

HCBKS Soave

.0405

32.8

.0831

23.4

.1208

38.3

Peng-Robinson

.0405

30.6

.0807

17.1

.1146

32.9

Schmidt-Wenzel

.0471

44.3

.1032

51.0

.1032

21.8

Harmens-Knapp

.0433

47.1

.0945

60.2

.0934

33.7

-.0796

84.3

-.2510

10600.3

-.0218

202.2

Heyen

.0083

20.7

-.0007

35.5

.0481

16.2

CCOR

.2932

31.9

.4285

30.2

.3518

60.6

Kubic

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

10. Naphthalene in

.3888

Equation

.5979

13. Naphthalene in

14. Phenol in

trifluoromethane

Carbon Dioxide

k^

av. % dev.

Redlich-Kwong

.0232

14.3

HCBKS

.0606

14.7

Soave

.1083

Peng-Robinson Schmidt-Wenzel Harmens-Knapp

.4815

kj2

15. Fluorene in Ethylene

av. % dev.

ki2

av. % dev.

-.0308

17.5

-.1121

.0495

23.4

.0183

32.6

18.8

.1007

16.2

.0433

57.8

.1070

17.8

.0983

17.4

.0419

43.0

.0896

39.5

.1059

32.9

.0405

47.2

33.0

.0820

53.3

.1032

39.2

.0294

58.5

Kubic

-.0332

156.9

-.0408

22.7

-.2245

10224.6

Heyen

.0644

36.2

.0533

37.2

-.0457

29.6

CCOR

.3543

19.7

.3010

3.0

.3922

118.4

.4552

Equation

.3462

.5843

16. Hexachloroethane

17. 1,4-Naphthoquinone in

18. 1,4-Naphthoquinone

in Carbon Dioxide

Carbon Dioxide

in Ethane

av. % dev.

kl2

av. % dev.

ki2

Redlich-Kwong

.0734

18.7

-.1135

19.0

-.0519

27.6

HCBKS

.0595

17.2

.0332

26.4

.0357

33.4

Soave

.1371

26.6

.0907

36.8

.1260

16.3

Peng-Robinson

.1322

21.4

.0834

30.6

.1233

15.8

Schmidt-Wenzel

.1322

9.5

.0758

68.7

.1308

55.5

1

ki2

av. % dev.

Harmens-Knapp

.1295

12.6

.0633

88.9

.1270

64.8

Kubic

0218

71.6

-.1371

1270.0

-.1797

2494.8

Heyen

.0820

11.2

-.0194

42.2

.0571

26.2

CCOR

.3795

37.4

.3849

351:

.4330

23.1

.5192

.5067

.5292

Continued

on next

page

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table VI Continued 19. 1,4-Naphthoquinone in

20. 1,4-Naphthoquinone in

21. Acridine in

Trifluoromethane

Trifluorochloromethane

Carbon Dioxide

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

Equation

av. % dev.

k^

av. % dev.

k^

av. % dev.

Redlich-Kwong

.1623

86.3

-.0433

31.1

-.0045

20.1

HCBKS

.0294

85.4

.0519

31.7

.0582

37.1

Soave

.0509

51.1

.1371

20.3

.1433

64.2

Peng-Robinson

.0481

56.3

.1357

22.0

.1357

51.7

Schmidt-Wenzel

.0180

147.9

.1260

63.8

.1485

36.7

Harmens-Knapp

.0045

173.3

.1208

77.3

.1346

54.1

Kubic

.1332

742.2

-.1384

749.2

-.2336

11211.1

Heyen

.0370

132.1

.0671

47.1

.0256

19.1

CCOR

.3323

69.5

.4248

15.1

.4836

26.7

.4414

Equation

Redlich-Kwong

.4579

.6632

22. Acridine in

23. Acridine in

24. Acridine in

Ethane

Trifluoromethane

Trifluorochloromethane

ki2

av. % dev.

ki2

av. % dev.

av. % dev.

ki2

-.0533

72.4

-.0332

34.4

-.0145

33.3

HCBKS

.0381

58.4

.0620

24.6

.0609

33.8

Soave

.1045

43.4

.1284

29.7

.1360

12.2

Peng-Robinson

.0997

43.9

.1233

21.7

.1333

19.2

Schmidt-Wenzel

.1146

106.8

.1083

96.0

.1360

74.0

Harmens-Knapp

.1059

117.3

.0921

124.9

.1260

91.6

-.2561

87120.0

-.1948

41167.0

-.1284

12516.7

Heyen

0183

54.9

.0256

70.0

.0544

48.4

CCOR

.4337

40.0

.4423

37.5

.4500

7.7

Kubic

.5805

Equation

Redlich-Kwong

.5581

.5553

25. Fluorene in

26. Hexamethylbenzene

Carbon Dioxide

in Ethylene

ki2

av. % dev.

ki2

av. % dev.

27. Hexamethylbenzene in Carbon Dioxide ki2

av. % dev.

-.0107

28.3

-.1246

28.5

-.0183

43.5

HCBKS

.0495

39.6

.0094

38.4

.0457

37.8

Soave

.1108

59.3

.0571

59.1

.1333

77.0

Peng-Robinson

.1059

49.1

.0533

41.6

.1257

63.4

Schmidt-Wenzel

.1184

18.0

.0658

55.6

.1409

28.0

Harmens-Knapp

.1083

28.6

.0557

66.7

.1295

33.4

Kubic

-.1609

27634.7

-.1911

6485.9

-.0772

1920.7

Heyen

.0142

35.6

-.0471

31.5

.0232

41.4

CCOR

.4387

44.8

.4009

73.3

.4422

33.8

.6431

.5930

.6393

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

7. H AS Ε LOW ET AL.

Table

VI

169

Continued

Equation

28. Triphenylmethane

29. Triphenylmethane

in Ethane

in Carbon Dioxide

ki2

av. % dev.

av. % dev.

30. Pyrene in Ethylene k^

av. % dev.

32.6

-.1159

15.5

-.0481

43.1

-.0921

HCBKS

.0169

15.6

.0470

46.8

.0495

19.3

Soave

.0807

33.6

.1284

85.1

.0959

90.4

Redlich-Kwong

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

Supercritical Extraction

Peng-Robinson

.0720

25.5

.1195

64.7

.0945

68.5

Schmidt-Wenzel

.0969

39.5

.1561

28.5

.0571

35.6

Harmens-Knapp

.0858

52.4

.1433

43.5

.0343

51.9

Kubic

-.2423

6066.4

-.2412

525956.7

-.2059

5907.1

Heyen

-.0370

28.1

-.0194

45.1

-.0495

39.0

CCOR

.4486

28.3

.4922

62.4

.4798

25.2

.6280

.6719

.6494

31. Pyrene in Carbon Dioxide Equation

k^

ave. % dev.

Redlich-Kwong

.0180

HCBKS

.0671

163.8

Soave

.1748

244.8

114.2

Peng-Robinson

.1658

202.1

Schmidt-Wenzel

.1696

45.5

Harmens-Knapp

.1534

43.5

Kubic

-.2250

1697621.0

Heyen

.0343

104.0

CCOR

.5498

34.0

.7319

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table VII. Summary of Equation of State Calculations for 14 Systems with Known Tc, Pc, and ω Equation of State Redlich-Kwong

17

HCBKS

24

Soave

31

Peng-Robinson

26

Schmidt-Wenzel

32

Harmens-Knapp

41

Kubic

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

Av. % Deviation

> 10,000

Heyen

25

CCOR

34

Table VIII. Summary of Equation of State Calculations for all 31 Systems Equation of State

Av. % Deviation

Redlich-Kwong

34

HCBKS

38

Soave

51

Peng-Robinson

43

Schmidt-Wenzel

49

Harmens-Knapp Kubic

60 > 10,000

Heyen

40

CCOR

38

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

7.

H A S E L O W ET A L .

Supercritical Extraction

171

The results shown in Table VIII include all mixtures. For seventeen of these mixtures the critical properties of the solutes have to be estimated. There is a general increase in the deviation of the calculated solubility from the experimental value for all equations when the relatively uncertain critical properties are employed in the calculations. The distinction among the equations becomes blurred at the same time. Thus, the Redlich-Kwong, HCBKS, and CCOR equation appear together with percent deviations in the thirties. The Peng-Robinson, Schmidt-Wenzel, and Heyen equations form the next group with percent deviations in the forties. The total summary for all of the 31 mixtures in Table VIII appears less meaningful than the partial summary of Table VII.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

Representation of the solubility data by an equation of state is examined in detail in Figures 1 through 10 for five equations - the Redlich-Kwong, HCBKS, PengRobinson, Heyen, and CCOR. Calculations by the R-K equation are compared with experimental data on benzoic acid solubility in C 0 in Figure 1. This mixture is chosen for illustration owing to the pure component properties being known, and the pressure range of the data being quite large. However, the temperature range is narrow, which is common for supercritical extraction experiments. Figure 1 shows that the equation does not describe the variation with temperature very well; the calculated temperature dependence is too weak. The calculated pressure dependence is qualitatively correct but misses the fine points. 2

Figure 2 shows the Redlich-Kwong equation in comparison with data on fluorene solubility in ethylene. This mixture is chosen for illustration for the large temperature range of the data. The critical properties of fluorene are estimated. The calculation describes the lower pressure data well, but departs from the data significantly at 200 bars and above. The Heyen equation is compared with the same two mixtures in Figures 3 and 4, and the Peng-Robinson equation in Figures 5 and 6. The HCBKS equation is likewise shown in figures 7 and 8. The CCOR equation is shown in Figure 9 for phenol in carbon dioxide, and in Figure 10 for acridine in C C I F 3 . The critical properties of phenol are known, but those of acridine are estimated. The CCOR equation calculations agree with data about equally well for both mixtures. The comparison shown in the two figures illustrates that the CCOR calculations are quite independent of the source of critical properties, estimated or known. This is also shown in Tables VII and VIII. It seems clear from this work that the quantitative representation of supercritical solubility by the currently available equations of state with their classical one fluid mixing rules is not adequate. In view of the many applications and potential applications of supercritical extraction, there is a need for new development of an equation and mixing rules for the improved representation of supercritical extraction.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

172

EQUATIONS O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

2

Δ

U

100

150

T=338 Κ

200

250

300

p, bars Figure 1. Solubility of Benzoic Acid in C 0 by the Redlich-Kwong Equation 2

1

10"

ρ

2

h

ίο"

5

1 0

1

1

1

=.

1 150

1 300

1 450

1 600

p, bars Figure 2

Solubility of Fluorene in Ethylene by the Redlich-Kwong Equation

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

H AS Ε LOW ET A L .

Supercritical Extraction

100

150

200

250

300

p,bars Figure 3.

Solubility of Benzoic Acid in C 0 by the Heyen Equation 2

ίο"

1 0

1

ρ

1

1

1

5

—

-5

I

1

1

1

0

150

300

450

Ξ -

1 600

p, bars Figure 4.

Solubility of Fluorene in Ethylene by the Heyen Equation

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

174

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

2

I

1

1

Δ

—

_

-

Ο

-

-

-

-

J/

Exp. Data

~-

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

—

•

Δ

ο

Τ=318 Κ

•

T=328 Κ

Δ

T=338 Κ

-

2

-

ι 150

100

ι 200

ι 250

ρ, bars Figure 5.

Solubility of Benzoic Acid in C 0 by the Peng-Robinson Equation 2

1

-

5

!

I

Ξ Δ

/

ζ 5

/

-

2

/

• \

Δ

~~~

>^J

Ο

_

o

Ο

-

3

jll 1 1 1 ι I

y, *°" 5

Ι Δ

2

-

Γ

ο

T=298 Κ

5

-

•

T=318 Κ

-

Δ

T=343 Κ

2

0

-

Δ

_

2

Exp

I 150

-

Data

I 300

Η50

— 600

ρ, bars Figure 6.

Solubility of Fluorene in Ethylene by the Peng-Robinson Equation

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

HASELOW ET AL.

Supercritical Extraction

Ι

10

C

Γ

ΙΟ"

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

Γ

Γ

100

150

200

250

300

ρ, bars Figure 7.

Solubility of Benzoic Acid in C 0 by the HCBKS Equation 2

10

10"° t r

io

150

300

450

600

p, bars Figure 8.

Solubility of Fluorene in Ethylene by the HCBKS Equation

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

10 · 75

150

225

300

p, bars Figure 9.

Solubility of Phenol in C 0 by the CCOR Equation 2

10

Vi Exp. Data

10

ο

T= 318 Κ

•

T= 328 Κ

10 ' 0

100

200

300

400

p, bars Figure 10. Solubility of Acridine in CC1F by the CCOR Equation 3

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

7.

H A S E L O W ET A L .

Supercritical Extraction

177

Glossary of Symbols a b c >

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

c

equation of state parameters

r

kjj

binary interaction parameter

η

number of data points in a binary system

ρ

pressure

p°

vapor pressure

R

gas constant

Τ

temperature

u

equation of state parameter

V

molar volume

V

s

w y

solid molar volume equation of state parameter fluid

phase mole fraction

Greek Letters ρ

molar density

φ

fugacity coefficient of fluid phase

φ°

fugacity coefficient of saturated vapor

ω

acentric factor

Ω

objective function

Subscripts 1 2 c

solid component fluid

component

critical property

Acknowledgment Funds for this research were provided by the Department of Energy through con tract DE-FG22-83PC60036.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

178

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Literature Cited 1. Zosel, K. Chem. Int. Ed. (1978) 17, 702. 2. Hubert, P.; Vitzthum, O.G. Agnew. Chem. Int. Ed. (1978) 17, 710. 3. Irani, C.A.; Funk, E.W. Recent Developments in Separation Science Volume 3, part A, p. 171; CRC Press, West Palm Beach, 1977. 4. Bartle,K.C.,Martin, T.G.; Williams, D.F. Fuel 54, 226. 5. Martin, T.G.; Williams, D.F. Phil. Trans. Roy. Soc. London (1981) A300, 183.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch007

6. Maddocks, R.R., Gibson, J.; Williams, F. Chem. Eng. Progress (1979) 75, 16, 49. 7. Blessing, J.E.; Ross, D.S. Organic Chemistry of Coal ,p. 171 et. seq., Am. Chem.Soc.,1978. 8. Ross, D.S.; Blessing, J.E. Fuel (1979) 58, 423. 9. Vasilakos, N.P., Dobbs, J.M.; Parisi, A.S. Ind. Eng. Chem. ProcessDes.Dev. (1985) 24, 121. 10. Adams, R.M., Knebel, A.M.; Rhodes, D.E. Hydrocarbon Processing (1980), May 150. 11. Benedict, M., Webb, G.B.; Rubin, L.C. J. Chem. Phys. (1940) 8, 334. 12. Jacobsen, R.B. Ph.D. Thesis, Washington State University, Pullman, WA (1972). 13. Goodwin, R.D. "The Thermodynamic Properties of Methane from 90 to 500K at Pressures to 700 Bar", NBS Tech. Note 653 (1974). 14. Donohue, M.C.; Prausnitz, J.M. AIChE J. (1978) 24, 849. 15. Chien, C.H., Greenkorn, R.A.; Chao, K.C. AIChE J. (1983) 29, 560. 16. Klincewicz, K.M.; Reid, R.C. AIChE J. (1984) 30, 1, 137. 17. Lydersen, A.L. "Estimation of Critical Properties of Organic Compounds", Univ. Wis. Coll. Eng. Exp. Stn. Rep. 3, Madison, WI, April, 1955. 18. Edmister, W.C. Pet. Refiner. (1958) 37, 4, 173. 19. Lin, H.M.; Chao, K.C. AIChE J. (1984) 30, 981. 20. Reid, R.C. "Creativity and Challenges in Chemical Engineering", University of Wisconsin, Madison, WI (1982). RECEIVED

November 5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

8 The Generalized van der Waals Partition Function as a Basis for Equations of State Mixing Rules and Activity Coefficient Models S. I. Sandler, K.-H. Lee, and H. Kim

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

Department of Chemical Engineering, University of Delaware, Newark, DE 19716

The generalized van der Waals (GVDW) partition function provides a basis for understanding the molecular level assumptions underlying presently used equations of state, their mixing rules including the new local composition (density dependent) models, and activity coefficient models. Further, our Monte Carlo computer simulation results for mixtures of squarewell fluids provide a method of testing these assumptions, many of which are found to be incorrect. Using the combination of the GVDW partition function and computer simulation results, we have formulated new equations of state and local composition models which are simple and accurate. Thermodynamic modelling of r e a l f l u i d s and mixtures, using both equations of state and a c t i v i t y c o e f f i c i e n t s , i s an area i n which there i s a strong temptation to introduce empiricism. The various modifications of the van der Waals equation, which now number i n the hundreds, i s one example of t h i s . S t a t i s t i c a l mechanics, on the other hand, leads to the detailed calculation of the properties of model f l u i d s with very simple intermolecular potential functions, but few generalizations to r e a l f l u i d s . For the l a s t several years we have been pursuing a d i f f e r e n t approach wherein we start from a firm s t a t i s t i c a l mechanical basis, but seek results of general v a l i d i t y , rather than to calculate numbers for an idealized i n t e r action potential model. In p a r t i c u l a r , we are seeking answers to such questions as: (1) Why are present equations of state and their mixing rules not applicable to mixtures of molecules of greatly d i f f e r i n g f u n c t i o n a l i t y or size?, and (2) What i s the molecular basis for l o c a l composition and density dependent mixing rules, and how can such rules be improved? This communication i s a progress report on our e f f o r t s . The basic idea i n our work i s the use of the rigorous generalized van der Waals p a r t i t i o n function, which we consider i n the following sections, to understand the molecular l e v e l assumptions imbedded i n presently used equations of state, their 0097-6156/ 86/ 0300-0180$06.50/ 0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

8.

Generalized van der Waals Partition Function

SANDLER ET AL.

181

mixing rules, and a c t i v i t y c o e f f i c i e n t models, and then to test these assumptions using computer simulation. In addition, we have been using the results of our simulations to formulate new and better molecular thermodynamic models. Our progress to date w i l l be reviewed here. Pure Fluids The s t a r t i n g point for our study of pure f l u i d s i s a new form of the generalized van der Waals p a r t i t i o n function we have derived [1]: 3 N

Q(N,V,T)

A- (q q q ) r

v

V

e x

f P<^f>

<

X)

where Q i s the canonical p a r t i t i o n function for Ν molecules i n a volume V at temperature T, q , q and q are the single p a r t i c l e r o t a t i o n a l , v i b r a t i o n a l and e l e c t r o n i c p a r t i t i o n functions, respec t i v e l y , Λ i s the de Broglie wave-length and k i s the Boltzmann constant. Also, Vf i s the free volume defined such that r

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

N

e

V f

N

v

e

z (N,V) HC

=

(2)

where i s the configurâtional i n t e g r a l for hard core molecules. The configuration i n t e g r a l for spherical p a r t i c l e s i s -u (£!,... r ^ / k T Z(N,V,T) -

J...Je

d£ ...dr 1

N

(3)

V where u(r_j, £2,...) i s the intermolecular potential function when a molecule i s located at position vector _ r j , a second molecule i s located at position vector _r_2 > etc. [The extension to nonspherical p a r t i c l e s i s considered b r i e f l y i n reference 1.] F i n a l l y , E^ONF^ the energy of interaction, for a pair-wise additive system i s given by CONF

M

E

J (r)g(r;N,V,T)clr u

(4)

and the mean potential

ψ

T

J ^

dT

(5)

i s the free energy of bringing the system from T=« (where only hard core repulsive forces are important) to the temperature of i n t e r e s t . In these l a s t equations, u(r) i s the two-body intermolecular poten t i a l and g(r;N,V,T) i s the two-body r a d i a l d i s t r i b u t i o n function, which i s a function of density, temperature and intermolecular separation distance. Once the p a r t i t i o n function i s known, a l l thermodynamic properties can be found. For example,

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

182

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

kT(4^) 3V

P(N,V,T) =

(6) N,T

and (7)

A(N,V,T) = -kTlnQ

The f i r s t of these equations provides a method of obtaining the volumetric equation of state from the p a r t i t i o n function, while from the second the fundamental equation of state i n the sense of Gibbs i s obtained. The importance of equations (1-5) i s that they focus attention on the two quantities which are needed f o r thermodynamic modelbuilding, the temperature and density dependence of the free volume Vf and the configurâtional energy, E^ONF Indeed, these are central to obtaining good equations of state, as w i l l be seen shortly. As an aside, we note that f o r the square-well potential Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

e

»

r
u(r)

(8)

oRo

we have that pCONF

Κα

2

-Ν ε ff-

J

g(r;N,V,T)dr

-Νε N (N,V,T) C

(9)

σ and -2kT Ν

/

Τ

gCONF

Τ N (N,V,T) / — τ, dT =i kT C

dT = ekT kT

z

(10)

2

T

T=oo where e

N (N,V,T) f C

J σ

g(r;N,V,T)dr

i s the number of molecules i n the potential well of central mole cule; we w i l l refer to this quantity as the coordination number. Thus, i n general, what i s needed f o r applied thermodynamic modelling i s the temperature and density dependence of the configuration energy; f o r the special case of the square-well molecule, tempera ture and density dependence of the coordination number s u f f i c e s . This comment has special relevance to mixture l o c a l composition models, which w i l l be considered l a t e r . To i l l u s t r a t e how the generalized van der Waals p a r t i t i o n function may be used, we r e s t r i c t our attention here to the squarewell f l u i d ; the extension to real f l u i d s i s simply accomplished as described i n reference 1. For the square-well f l u i d , the hardcore free volume i s best given by the Carnahan and Starling [8J expression

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

8.

Generalized van der Waals Partition Function

SANDLER ET AL.

v

- ν ·

f

e x

P ( η( - η) 4

183

3

(1-η)

}

2

with η • ïïpcrVo, though the less accurate van der Waals expression V

= V-b

f

(12)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

with b - 4η has been incorporated into many cubic equations of state used i n engineering. There has been l i t t l e consideration given to the coordination number i n the thermodynamic l i t e r a t u r e . Implicit i n the van der Waals model i s the assumption that the coordination number, that i s the number of molecules i n t e r a c t i n g with a central molecule, i s l i n e a r l y proportional to the density, N

= ! C = pC

c

(13)

Ra where C -

J

g(r;N,V,T)d_r i s assumed to be independent of temperσ ature and density. In this case, CONF . _ . ^ | Ç

E

where a = Ce/2. 2 7 kT

. _ ψ,

fφ

a

n

d p

.

p

H

S

_M|

(

1

4

)

Using the c r i t i c a l point conditions, we find that c

At low densities, we have that Lim

u

r

g(r;N,V,T) = e " ( ) /

k T

(15)

p+0 Assuming that this equation i s v a l i d at a l l densities f o r the square-well f l u i d yields N c

=|*l£

( R

3_

1 ) e

e/kT

(

1

6

)

and an equation of state whose volume dependence i s l i k e the van der Waals equation, but with the a parameter given by the temperature dependent function a = 2» 3 ( Β 3 - ι ) ( β σ

Μ

ε / Μ

-1)

(17)

The coordination number models b u i l t into other equations of state can also be obtained, and several are shown i n Table I. The f i r s t three, the van der Waals, Redlich-Kwong [3J and Peng-

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

184

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Robinson [41 equations, are representative of commonly used cubic equations of state, and the fourth i s a twenty-seven constant equation of the extended v i r i a l form developed by Alder et a l . [2] to f i t their computer simulation data for the square-well f l u i d . It i s interesting to note that i t i s this equation, with s l i g h t l y modified parameters, that i s used i n the perturbed hard chain theory of Beret and Prausnitz [5J and Donohue and Prausnitz [6], and largely gives r i s e to i t s complexity. The remaining two equations w i l l be discussed shortly. It i s now of interest to compare several of these coordination number models with data for the square-well f l u i d . We have obtained such data from our own Monte Carlo simulations described elsewhere [7J. Some of these data, together with the predictions of the various cubic equation of state models, are shown as a function of reduced temperature kT/ε and reduced density pcP i n Figure 1. This figure i s f o r an R • 1.5 square-well f l u i d , and we have used the estimates of Alder et a l [2J for the c r i t i c a l properties of the square-well f l u i d kT — -

V = 1.260, — 5 - = 3.006 Νσ

c

c

ε

P o3 — — =0.120 c

and

3

ε

to i n t e r r e l a t e the square-well and c r i t i c a l parameters. As we can see from the figure, none of the coordination number models b u i l t into existing cubic equations of state are very good. Based on a lattice-gas model, we have developed the simple expression [7J Z V m

N c

=

V + V

exp(e/2kT)

0

( 1 8 )

{exp( /2kT)-l}

0

e

where Z i s the close-packed coordination number ( Z - 18 for R • 1.5) and V = Νσ^//2. Figure 1 also shows the coordination number predictions for this model, which are better than any of the ones considered previously. Using this coordination number model i n the generalized van der Waals p a r t i t i o n function we obtain m

m

Q

Z

NkT-NkT

e / 2 k T

mVo(e -l) e/2kT.

v ( v + V o [ e

....

1 J )

Note that the density dependence of the a t t r a c t i v e term i n this new equation i s l i k e the Redlich-Kwong equation, but the denominator i s temperature dependent. Thus, at low density, N should be propor t i o n a l to exp (ε/kT) while i n equation (18) i t i s proportional to exp (e/2kT), which from our data appears appropriate at high density. Therefore, an empirical modification of Eq. (18) i s c

Z V exp( /akT) m

Ν

0

e

= V + ν {βχρ(ε/ο&Τ)-1} 0

y

where a • „ „ V-V

(20) 0

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

SANDLER ET AL.

8.

Generalized van der Waals Partition Function

TABLE I .

185

C o o r d i n a t i o n Number Models f o r t h e S q u a r e - W e l l F l u i d i n Various Equations of State

Equation of State 27 k T 2 -^r ( ) · ( ~) · po° Ροσ c

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

van der Waals

R

e

d

l

i

c

5

-

h

1

ε

3

3(0.4g8) ^ , 1 . 5 ^ , 0 . 5 0.0867 ε kT

Kwong

P

(

Robinson

£c,

(

1 + Κ ) Ι 1 + Κ

l n [ 1 + 0

0 8

67

<-JL^£ç, p „3 ε c

{1-/ I

ε

0.07789/2

.

i

}J 1 η [ ^ ] l - ( / 2 - 1)Δ 1

c

where κ - 0.37464 + 1.54226ω - 0.26992UJ

and

Δ - 0.07780

kT (—)(-£—)( Pco ε

ρ σ

+

(

+

1

)

Δ

2

3)

3

η=1 m=l (/2)

Z V exp( /2kT) m

E

q

n

#

(

1

8

)

0

£

V + V {exp( /2kT)-l} 0

£

Eqn. (20)

where α ε V + ν {βχρ<-^)-1} ^akT' 0

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ρ σ

3]

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

186

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Figure 1.

Predictions of coordination number f o r square-well fluid: (o) Monte Carlo simulation; (····) van der Waals; ( ) Redlich-Kwong; ( — ) Peng-Robinson; ( ) Eqn. (18); ( ) Eqn. (20)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

8.

Generalized van der Waals Partition Function

S A N D L E R ET A L .

187

The predictions of this model are also shown i n Figure 1, where they are found to be i n very good agreement with computer simulation. Using this expression we obtain v

Ρ

_

NkT

HS

Z

NkT

2

P

m

f

V ln[l

+ ^

0

t

o

^ε/ctkT ( e

e

' ° ^ - l ) ]

(V-V )2 0

V

+

e

/

a

k

T

o[(l-f^)e -l] — } (V-V )[V + V ( e -l)j

(21)

e / a k

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

0

0

To test the accuracy of these and other equations of state, we compare, i n Table II, the predicted compressibility factor for the square-well f l u i d with the data of Alder et a l . [2]. For this comparison we have used the Carnahan and Starling [8J expression for the hard-sphere pressure, and r e s t r i c t e d our attention to the vapor and l i q u i d one-phase regions of the Alder simulation data. We have also included i n Table II the predictions of the equations of Aim and Nezbeda [9] and Ponce and Renon [10J, both of which are meant to describe the square-well f l u i d . It i s interesting to note that equation (19) which i s one of the simplest equations considered, i s also the most accurate. Its accuracy i s further demonstrated i n Table I I . Absolute Average Deviation i n the Compressibility Factor of the Square-Well F l u i d as Predicted by Various Equations of State Equation of State

Average Absolute Deviation

van der Waals

8.380

Redlich-Kwong

1.332

Peng-Robinson

2.159

Alder, et a l .

0.380

Aim-Nezbeda w/o

3 body

Aim-Nezbeda with 3 body

0.418 0.323 0.378

Ponce-Renon This work, Eqn.

(19)

0.240

This work, Eqn.

(21)

0.269

Figure 2 where the compressibility factor along several isotherms f o r various equations of state are plotted as a function of density together with the results of our Monte Carlo simulations.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D

188

1

1

1

1

1

1

1

Γ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

τ

APPLICATIONS

Figure 2.

Predictions of compressibility f o r square-well f l u i d : (o) Monte Carlo simulation; (····) van der Waals; ( ) Redlich-Kwong; ( ) Peng-Robinson; ( ) Eqn. (19); ( ) Eqn. (21)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

8.

189

Generalized van der Waals Partition Function

S A N D L E R ET A L .

Based on our r e s u l t s , two conclusions can be drawn. F i r s t , i s that there i s an advantage to using the generalized van der Waals p a r t i t i o n function as a basis f o r developing equations of state. This i s demonstrated by the fact that by using this approach, we could develop simple equations with _no adjustable parameters, that are much more accurate than the empirical or semitheoretical equations f o r the square-well f l u i d that had been used heretofore. Second, from this analysis, we have developed an equation of state which, as a result of i t s accuracy, i s an obvious candidate f o r generalization to r e a l f l u i d s .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

Mixtures and Local Compositions Mixture behavior, and especially l o c a l composition effects and density dependent mixing rules, are of considerable p r a c t i c a l i n t e r e s t . Here the generalized van der Waals p a r t i t i o n function i s of great u t i l i t y i n understanding the assumptions contained i n presently used models f o r a c t i v i t y c o e f f i c i e n t s and equation of state mixing rules, and as a basis f o r improving upon them. The extension of the generalized van der Waals p a r t i t i o n function of the previous section to mixtures yields [ l j Q(N ,N ,...V,T) « 1

2

ι q q qp π [ ^ ^ i p - ) ι i* A r

l N

J

-N<&(T,V,NI,N..)

^ Ν

v

1

2

JV (T,V, f

Nl

, N ,...)exp(

9

!

2

l

2

*

)

1

(22)

where φ(Ν ,Ν ,·..ν,Τ) 1

2

= -^Γ _-2kT T r

Ν

l

T

ECONF k

œ

T

. , .* — - A i T

(23)

(N!,N ,...,V,T)

(24)

(

N

I

>

N

2

>

>

V

>

T

)

2

with CONF ECONF

( N I

,

N 2 >

..

# J V

,T)

= I

I

E

±i

2

i J and CONF Eij (N ,N ,...,V,T) X

Ν,· Ni j u (r)g (N ,N ,...,V,T;r)dr

2

ij

ij

1

2

(25) N

2

2v i j x

x

/ ij( )gij( l> 2>--«> > ; > £ u

r

N

N

v

T

r

d

The form of Eqn. ( 2 2 ) i s useful f o r the study of mixture equations of state and parameter mixing rules. For the study of excess free energy or a c t i v i t y c o e f f i c i e n t models, i t i s more convenient to use the difference between Equation ( 2 2 ) and the analogous equations f o r the pure components to obtain the expression below f o r mixing at constant temperature and t o t a l volume (so that V Vi)

=l

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

190

A

X

T

V

^ „ - %IX< > > T,V

N N ,...)-£ A i C T . V i . N i ^ k T j N i l n x i lf

2

±

Q(T,V,Ni,N ,...) 2

-

k

T

l

n

t

^ ( T TTQICT

^

)

Ni

0

.

N

^ Σ ^ "

J

1

i Z(T—,ν,Ν^,...)

-kTln[

, J wIZtd-.Vi.Ni)^/!]

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

E

C0NF

( T > v > N l

^

N 2 # e e )

_£

Τ kT / [

(26)

E i

CONF (

T > V i

,Ni)

i kT

dT

2

T=«

EX

EX ^T,V

ECONF( ^ N ,N ...)-jE T

V f

X

2

CONF iT,V ,N ) ±

±

±

dT T

T=oo

/ t-

(27)

T=oo

The f i r s t term on the righthand side of this equation i s the excess free energy of mixing when only repulsive forces are important. This term accounts f o r size and shape e f f e c t s , and may be modelled by the Flory-Huggins 11 l j , Guggenheim-Stave man [12] or other suitable expressions. The second term i s the excess free energy of mixing for bringing the system from T=<» to the temperature of i n t e r e s t at constant composition and t o t a l volume. This term i s the contribution of the soft portions of the intermolecular potential to the free energy of mixing. As shown by Hildebrand [13J, at low and moderate pressures, A| * G| p, where G| i s the excess Gibbs free energy change on x

X

X

v

p

x

mixing at constant temperature and pressure. It i s from G| that a c t i v i t y c o e f f i c i e n t expressions are most easily obtained. ' We want to stress that, i n Equations (22-27), i t i s the configurational energies p

CONF E

± j

2πΝιΝ4 V "

« Q

/

0 2

u (r)g (r)r dr i j

i j

(28)

that are important, rather than the (ambiguously defined) species coordination numbers

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

8.

Generalized van der Waals Partition Function

S A N D L E R ET AL.

4πΝ4

N

±j

Lij

J

'IT-

191

0

gij(r)r2 r

(29)

d

or l o c a l mole fractions N

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

x i Ji -

li (30)

plj

There are two problems with the species coordination numbers or mole f r a c t i o n s . F i r s t , since the cut-off distance L-y i s somewhat a r b i t r a r y , neither N-JJ nor X j _ j are uniquely defined. Second, i t i s the clearly defined configurâtional energies which appear i n the generalized van der Waals analysis, and, i n general, these cannot be gotten from the l o c a l compositions. It i s only f o r square-well molecules, with i t s well-defined range, for which CONF . ^ L

E

N

l

j

e

i

(31)

j

with N

N

R

, 0"ii

i

i j - V ~

/

g (N ,N ,...,V,T;r)dr ij

1

(32)

2

°±3 that the l o c a l species-species coordination numbers (or l o c a l compositions) are unambiguous and contain equivalent information to the configurâtional energies. This i s an important point since most l o c a l composition studies i n the past for the Lennard-Jones 6-12 f l u i d [14, 15, 16J have concentrated on the species coordination numbers or l o c a l mole f r a c t i o n s , rather than the configurational energies, which are really needed. A very useful c h a r a c t e r i s t i c of Equations (22-25) i s that they are i n a form which makes i t possible to determine the applied thermodynamic model which results from any molecular l e v e l assumption. We demonstrate this here by considering a mixture of square-well molecules. In this case, Equation (31) i s applicable, and the l o c a l coordination numbers, N-y, are useful i n thermodynamic modelling. Clearly, different models for the l o c a l coordination number w i l l give different equations of state, mixing rules, and a c t i v i t y c o e f f i c i e n t models. For example, i f we were, i n analogy with the pure f l u i d van der Waals model, to assume that My

-

±

C

where C i s a constant molecules), or even N

N

ij

"

(33a) (which i s reasonable

only for s i m i l a r size

i

7" C i j

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(33b)

192

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

where C J J can have a d i f f e r e n t numerical value for 1 - 1 , 2-2, 1-2, etc., interactions, we f i n d that the van der Waals equation of state i s again obtained with the a t t r a c t i o n parameter given by A

= I Σ i j ij x

x

a

34

< >

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

Equation (34) i s referred to as the van der Waals one-fluid mixing rule. Since i t i s known from s t a t i s t i c a l mechanics [ 1 7 ] that the second v i r i a l c o e f f i c i e n t depends quadratically on mole f r a c t i o n (and not on surface f r a c t i o n or volume f r a c t i o n ) , an important boundary condition on any cubic equation of state mixing rule i s that the a and b parameters for the mixture at low density reduce to the one-fluid form. Proceeding, we could instead assume that c

eEij/kT

(35

•υ -1 u 1

>

As indicated i n the previous section, Equation (35) i s the exact low density expression f o r the l o c a l coordination number i n a square-well f l u i d i f C

ij

a

*

R

ij"*( ij^-*) ·

Assuming this equation i s v a l i d for both the

pure f l u i d and the mixture, we f i n d that once again the van der Waals equation of state and one-fluid mixing rules apply, except that i n this case *ij

= s

I

| C

e

i j

(e iJ

/ k T

-l)

(36)

instead of being a constant. Continuing further, we f i n d that while d i f f e r e n t assumptions of the type N

±J

- y=- f ( N , V , T

>eij

,Oij)

(37)

where f i s any function of Ν, V, Τ and the potential parameters of a l l species, but independent of composition, lead to different forms of the a t t r a c t i v e term i n the equation of state, i n each case the van der Waals one-fluid mixing rule of Equation (34) applies at a l l densities. Defining N ^ to be the t o t a l coordination number for a species i molecule i n the mixture, from Equation (33a) we have (for similar s i z e molecules) that c

Nci

« »il

+

Nji -

f

C = N

cl

(38)

so that ο x

«il - i N

c i

ο

= xjN

c i

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(39)

8.

Generalized van der Waals Partition Function

S A N D L E R ET A L .

193

where N £ i s the coordination number for pure species i molecules at the same density. Using these results i n Equation (2 7) y i e l d s C

A

EX T-oo +

T,V

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

where 0 - j

1

N

° cl^ ll~ 12^ e

e

+

J

(40)

ΝΧΙΧ Θ 2

1 N

° c2^ 22" 12^ · ε

ε

F

o

r

R e c u l e s of s i m i l a r

size and shape, the f i r s t term on the right-hand side of Equation (40) vanishes, and we have the one constant Margules expansion. I f , on the other hand, Equation (33b) i s assumed, we then find that each species coordination number i s a l i n e a r function of mole fraction, N,

N

Nij

x

+

cj

(41)

Xiôj

that N

i cj

X j

N

(42)

c j

and that EX A

_ ^

EX

A

T,V

with Θ j

- Θ - j

1

(43)

!

+ Νχ^Χ2Θ +ΝχΐΧ2(χι~Χ2)Φ

T,V

1<5ι(ειι+ε2ΐ) + < $ 2 ( ε 2 2 1 2 ^ +ε

[ δ 2 ( ε 2 2 " " ε ΐ 2 ^ ~ ΐ ( 1 1 " " 2 Ρ J* δ

ε

T

ε

n

i

s

i s

t

n

e

a

n

d

Φ

=

two-constant

Margules

expansion i f the T - « term i s neglected, or an augmented two-constant Margules expansion i f the Flory-Huggins

A

EX T=oo

(44)

kjN ln ±

x

i

EX A

or

similar expressions are used for

T,V

Τ

In Equation (44),

Φι * i i / I i j > where v± i s some measure of the molecular volume of species i molecules X

v

x

v

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

194

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

If we assume that Equation (35) i s applicable to each species i n the mixture, then Equation (41) i s again obtained, and the two-constant Margules expansion results, though with s l i g h t l y d i f f e r e n t expressions for the Margules coeffcients. Indeed, whenever we s t a r t from l o c a l coordination number models of the type we have been considering, the excess Helmholtz (or Gibbs) free energy i s always of the Margules form, and the equation of state mixing rules are of the van der Waals one-fluid form at a l l densities. We now consider another class of l o c a l composition assumptions. Suppose, instead of Equations (33) we assume

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

N

N

lj

C

- i

ij

x

, i

C

ij

(45)

where C-y i s some constant. [A common choice f o r the constants C i s some measure of the molecular volume v, so that = C - M v-^, and C j j = C j i = V j . In this case the r a t i o of the l o c a l coordination numbers, that i s the l o c a l composition r a t i o , i s equal to the ratio of volume fractions.] By i t s e l f , Equation (45) i s i n s u f f i c i e n t , since each species coordination number ( i . e . , the separate Njj and Njj) i s needed i n the generalized van der Waals p a r t i t i o n function analysis. Implicit i n many a c t i v i t y c o e f f i c i e n t models i s the a d d i t i o n a l and quite separate assumption that the t o t a l coordination number for each species i n the mixture, N j = N - J J ^jj> * independent of mole f r a c t i o n , and therefore equal to tne pure component coordination number. That i s , the assumption i s β

+

s

c

N

cj -

N

(46) cj

Equation (46) has i t o r i g i n i n l a t t i c e theory where a molecule i s presumed to have a fixed coordination number; i t may be v a l i d for s i m i l a r size molecules at high density, but i s incorrect for molecules of d i f f e r e n t s i z e , or at low and moderate densities. Solving Equations (45 and 46) yields X±A

y*i and

x

=

c

N

j cj Xj+xiAij

3i

H

Xj+xiAij

(47)

c

where A i j ij/ jj« [Equations f o r Nj£ and N are gotten from the expressions above by an interchange of indices.] Assuming that A y " i / j yields iA

v

v

ΔΕΧ GEX

=

Τ,Ρ

AEX

Α

= τ

^T,V

τ,ν

Ι [Τ-»

x

t

x

Ν JVi jVj 2 XiVi+XjVj

(48)

where m

(eji-ejj)

n

^

+

(ejj-sjj)

If the Τ-» term i s set equal to zero, Equation (48) i s the excess free energy expression which gives r i s e to the van Laar and Hildebrand-Scatchard Regular Solution a c t i v i t y c o e f f i c i e n t models.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

8.

Generalized van der Waals Partition Function

SANDLER ET A L .

195

A l t e r n a t i v e l y , using Equation (44) the Flory-Huggins a c t i v i t y c o e f f i c i e n t model i s obtained. One could instead choose the molecular surface areas f o r the C^j's instead of molecular volumes, to obtain a variant of these models i n which surface area fractions rather than volume fractions appear. I f , instead, we use Equation (47) and the assumption that the t o t a l coordination number i s a l i n e a r function of density i n Equation (22) we obtain the following equation of state mixing rule for the a parameter

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

A

= <Σ i v i >

^f-

· Σ Î M

x

3

<«>

This mixing rule does not s a t i s f y the low density van der Waals one-fluid boundary condition (unless =* V j ) , which i s not surprising since Equation (47) on which i t i s based i s incorrect at low density. Next consider yet another l o c a l composition model

£ii = £i£ii < e

e i r e j j

)/kT

( 5 0 )

which i s the ratio form of Equation (35). If we choose C j ^ = Cjj v-£, we have the l o c a l composition assumption of Wilson. If one further assumes that the t o t a l coordination number i s constant (Equation (46)) then Equations (47) are again obtained, but with e

c

(e

i r e j j

)/kT

(

5

1

β

)

Using these results i n Equation (2 7) leads to the Wilson a c t i v i t y c o e f f i c i e n t model; when used i n Equation (22) with the assumption that the t o t a l coordination number i s a l i n e a r function of density we obtain an a parameter mixing rule i n the form of Equation (49) which, as already mentioned, does not s a t i s f y the low density one-fluid mixing rule boundary condition. While the various l o c a l composition models considered so f a r lead to different equation of state mixing rules and a c t i v i t y c o e f f i c i e n t models, none result i n density dependent mixing rules which have been of much interest l a t e l y . From the analysis using the generalized van der Waals p a r t i t i o n function, i t i s evident that density dependent mixing rules can only result from a density dependent l o c a l composition model. Two such density dependent mixing rules have been suggested recently. The f i r s t , due to Whiting and Prausnitz [18J i s of the form |u N

=a^i (e e

jj

N

i r e j j

)N

c j

/2kT

(

5

2

)

C

j JJ

where the t o t a l coordination number f o r eacn species N j = Njj + Njj i s assumed to be l i n e a r l y dependent on density, but independent of mole f r a c t i o n . This l o c a l composition model leads to the following mixing rule c

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

196

" Λ jV Xi

j

EXP

(

*^Γ$ *

E X P

( 5 3 )

where b i s a size parameter and » b^^N/4V. Also an a c t i v i t y c o e f f i c i e n t model i s obtained which i s similar to the three parameter Wilson equation. However, this model should not be expected to be very good since i t predicts that the effect of a t t r a c t i v e forces (the exponential term i n Equation (53)) vanishes at low density (where N j > 0), and i s largest at high density. This i s opposite to what should be expected since Equation (50) i s correct at low density, and as a t t r a c t i v e forces are of l i t t l e importance at high densities, the exponential term should approach unity (rather than increase i n value) i n this l i m i t . The recent " p r a c t i c a l " l o c a l composition model of Hu et a l . [19]

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

c

N

ij

s

3

f ^ R i j - ^ ^

3

ψ- expiaey/kT]

(54)

where 0.1865 a - 0.60-0.58 ( ρ ^ σ ϋ ) i 3

has a density dependence the observations above. density dependent mixing l o c a l composition models Table I I I .

which i s q u a l i t a t i v e l y i n agreement with This l o c a l composition model leads to the rule given i n reference 19. Many of the discussed above are summarized i n

Which Local Composition Model Is Best? In the previous section we considered numerous l o c a l composition models, and i t i s reasonable to ask which i s best? To answer this question we have been calculating l o c a l compositions i n mixtures of square-well f l u i d s using Monte Carlo simulation and i n t e g r a l equation methods. We report b r i e f l y some of our simulation results here. In Figure 3 we have plotted the quantity N

»21 l ll 2 N

N

a n

d

N N

1 2

22

N Νχ

2

as a function of density and composition for a mixture of squarewell molecules of equal diameter σ, but different well depths. The points represent our simulation results, and the arrows are the exact low density l i m i t s . For completely random mixing, both these ratios would be unity, which i s clearly not the case. Since the molecules are of equal diameter, = v , and the l o c a l composition ratios of Equation (45) also reduce to a unity at a l l densities. The Wilson l o c a l composition model of Equation (50) i s independent of density, and i s seen to be correct at zero density, but to 2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

8.

Generalized van der Waals Partition Function

S A N D L E R ET A L .

197

Table III Local Coordination Number Model

Equation of State Mixing Rule

Ni Nlj - V - C

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

N

N

ij

vdW

1-fluid

1-constant Margules

vdW

1-fluid

2-constant Margules

vdW

1-fluid

2-constant Margules

i

ν " °±3

=

Free Energy

Ν-; N

=

j

J

v

"a

i/kT

N

jj

- y -

ebrand Regular Solution and FIory=HuggIns

if N

I

+

J

Ν

NJJ

ν

N

i

j

" ^ ^

. < ' * ™

)

/

Κ

C

J

a = (Ix

i V i

)

Π

Φ ΐ Φ ^

Wilson

if N^

+ Njj -

N

cj N

cj - ν J C

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

198

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

Ζ CM

Ζ

Ζ

Z~

Ζ

ζ

Ζ

Ζ

(VI CVJ

0.7 0.0

0.1

0.2

0.3

0.4

REDUCED

Figure 3.

0.5

DENSITY,

0.6 ρσ

0.8

0.7

3

Extent of l o c a l composition i n square-well mixture f o r εχι/kT - 0.4, 622/ °· > o"ll <>22> l " 2 (·, 0 and +) Monte Carlo simulation; (····) Wilson (1964); ( ) Whiting and Prausnitz ( r e f . 18); ( ) Hu et a l . ( r e f . 19); ( ) Lee, Sandler and Patel (ref. 21); ( ) random mixing. (-• shows the theoretical low density l i m i t ) kT

β

8

=

R

R

=

U

5

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

:

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

8.

S A N D L E R ET A L .

Generalized van der Waals Partition Function

199

overpredict l o c a l composition effects at a l l other densities, random mixing at low density, and strong l o c a l composition effects at high density. As can be seen, this i s completely opposite from the computer simulation results which are i n agreement with both the exact low density result and the Weeks et a l . [20J conclusion that a t t r a c t i v e forces are of l i t t l e importance i n determining l o c a l composition at high density. Also, the Whiting-Prausnitz model shows no composition dependence. Since the underlying l o c a l composition mixing rule i s incorrect, i t i s not surprising that i n application, the density dependent mixing rule of Equation (53) i s found to be of l i t t l e u t i l i t y . Recently, Hu et a l . [19J have proposed the l o c a l composition model of Equation (54); the results of this model also appear i n Figure 3. There we see that for equalsize molecules, the Hu et a l . model predicts l o c a l composition e f f e c t s which q u a l i t a t i v e l y have the correct density dependence, but are too small at a l l densities. The model of Hu et a l . , l i k e the other models considered previously, shows no composition dependence i n the ratios we have plotted. Also shown i n Figure 3 are the predictions for the l o c a l composition ratios of Equation (2 0) from a model we have proposed recently [21J· This model has the correct low density and high density l i m i t s , a s l i g h t composition dependence, and i s i n better agreement with simulation results than the other models considered here. At present we are considering the extension of this model to molecules of unequal s i z e , and this work w i l l be presented separately [22]. Also, we consider elsewhere [23J the composition v a r i a t i o n of the t o t a l coordination number i n mixtures of unequal size molecules. Conclusions F i r s t , we have shown the u t i l i t y of the generalized van der Waals p a r t i t i o n function i n that i t allows us to i n t e r r e l a t e molecular l e v e l assumptions to applied thermodynamic models. This was used, i n Section II, to ascertain the coordination number models used i n a number of equations of state. These coordination number models were tested against our Monte Carlo simulation data for a square-well f l u i d , and none were found to be s a t i s f a c t o r y . A new coordination number model was proposed, and this was found to lead to an equation of state, with no adjustable parameters, which i s more accurate for the square-well f l u i d than multiconstant equations currently i n use. In Section I I I , the generalized van der Waals p a r t i t i o n function for mixtures was used to i d e n t i f y the molecular-level l o c a l composition and other assumptions imbedded i n commonly used equation of state mixing rules and a c t i v i t y c o e f f i c i e n t models. We then compared these assumptions for l o c a l composition effects with the results of our own Monte Carlo simulation studies for mixtures of square-well molecules i n Section IV. There we found that the models currently i n use do not properly account for nonrandom mixing due to a t t r a c t i v e energy e f f e c t s . This suggests that better l o c a l composition models are needed. Once they are obtained, the generalized van der Waals p a r t i t i o n function w i l l again be useful i n developing the macroscopic thermodynamic models which results from these molecular l e v e l assumptions.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

200

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Acknowledgments This work was supported, i n part, with funds provided by a National Science Foundation Grant (No. CPE-8316913) to the University of Delaware.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch008

Literature Cited 1. Sandler, S. I. Fluid Phase Eq. 1985, 19, 233. 2. Alder, B. J.; Young, D. Α.; Mark, M. A. J. Chem. Phys. 1972, 56, 3013. 3. Redlich, O.; Kwong, J. N. S. Chem. Rev. 1949, 44, 233. 4. Peng, D. Y.; Robinson, D. B. I&EC Fund. 1976, 15, 59. 5. Beret, S.; Prausnitz, J. M. AIChE J. 1975, 21, 1123. 6. Donohue, M. D.; Prausnitz, J. M. AIChE J. 1978, 24, 849. 7. Lee, K.-H.; Lombardo, M.; Sandler, S. I. Fluid Phase Eq. 1985, 21, 177. 8. Carnahan, N. F.; Starling, Κ. E. J. Chem. Phys. 1969, 51, 635. 9. Aim, K.; Nezbeda, I. Fluid Phase Eq. 1983, 12, 235. 10. Ponce, L.; Renon, J. J. Chem. Phys. 1976, 64, 638. 11. Flory, P. J. "Principles of Polymer Chemistry," Ithaca, N.Y., Cornell University Press 1953. 12. Staverman, A. J. Rec. Trav. Chim. Pays-bas 1950, 69, 163. 13. Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. "Regular and Related Solutions," Van Nostrand Reinhold Co. 1970. 14. Gierycz, P.; Nakanishi, K. Fluid Phase Eq. 1984, 16, 225. 15. Toukubo, K.; Nakanishi, K. J. Chem. Phys. 1976, 65, 1937. 16. Hoheisel, C.; Deiters, U. Molec. Phys. 1979, 37, 95. 17. See, for example, Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. "Molecular Theories of Gases and Liquids," J. Wiley & Sons, Inc., N.Y., 1953. 18. Whiting, W. B.; Prausnitz, J. M. Fluid Phase Equilibria 1982, 9, 119. 19. Hu, Y.; Ludecke, D.; Prausnitz, J. M. Fluid Phase Eq. 1984, 17, 217. 20. Weeks, J. D.; Chandler, D.; Anderson, H. C. J. Chem. Phys. 1971, 54, 5237. 21. Lee, K.-H.; Sandler, S. I.; Patel, N. C. Fluid Phase Equilibria, accepted for publication. 22. Lee, K.-H.; Sandler, S. I. Fluid Phase Eq., to be submitted. 23. Lee, K.-H.; Sandler, S. I.; Monson, P. A. Ninth Thermophysical Properties Meeting, Boulder, CO., June 1985. RECEIVED

November 5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

9 Local Structure of Fluids Containing Short-Chain Molecules via Monte Carlo Simulation 1

K. G. Honnell and C. K. Hall

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

Department of Chemical Engineering, Princeton University, Princeton, Ν J 08544

Monte Carlo simulation has been used to investigate local structure of model systems of butane and octane at volume fractions ranging from 0.001 to 0.3. At low and moderate densities the bead-bead intermolecular radial distribution function, g(r), near contact was found to be significantly less than one, indicating the presence of an excluded volume sur rounding each bead. The value of g(r) near contact increased with increased density, and decreased with increased chain length. In recent years there has been increased demand for general equa tions of state for f l u i d s and f l u i d mixtures of interest to the natural gas and petroleum industries. The techniques of s t a t i s t i c a l mechanics offer an a t t r a c t i v e route to the development of such equa tions of state since they are based upon fundamental physical con siderations. An h i s t o r i c a l survey of the s c i e n t i f i c l i t e r a t u r e shows however that up u n t i l the 1970 s most research i n s t a t i s t i c a l mechan i c s was limited to the development of theories for highly idealized monatomic f l u i d s containing spherically symmetric molecules, e.g., argon, which were r e l a t i v e l y uninteresting from an i n d u s t r i a l point of view. In the mid 1970 s however, research e f f o r t s began to focus on more i n d u s t r i a l l y relevant compounds including r i g i d asymmetric molecules, e.g., r i g i d polar molecules, e.g., H 2 O , and very long and very f l e x i b l e molecules, e.g., polymers. The perhaps more d i f f i cult problem of intermediate length molecules such as hydrocarbons, which are long and f l e x i b l e compared to diatomics but short and s t i f f compared to polymers, also began to receive some attention, largely because such f l u i d s are of great i n d u s t r i a l interest. The primary d i f f i c u l t y i n applying s t a t i s t i c a l mechanics to systems of short chains l i e s i n rigorously accounting for their asymmetric structure and their f l e x i b i l i t y . The two major a n a l y t i c a l 1

1

1

Current address: Department of Chemical Engineering, North Carolina State University, Raleigh, NC 27695-7905. 0097-6156/ 86/ 0300-0201 $06.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

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E Q U A T I O N S O F STATE: T H E O R I E S A N D

APPLICATIONS

approaches t a k e n to t h i s problem, the p e r t u r b e d h a r d c h a i n t h e o r i e s (1) and the l a t t i c e f l u i d t h e o r i e s ( 2 ) , of n e c e s s i t y must r e s o r t to r e l a t i v e l y crude a p p r o x i m a t i o n s i n o r d e r t o account f o r m o l e c u l a r asymmetry and f l e x i b i l i t y . In c o n t r a s t , computer s i m u l a t i o n i s cap a b l e of d e a l i n g e x p l i c i t l y w i t h t h e s e a s p e c t s of m o l e c u l a r shape. Thus, computer s i m u l a t i o n p r e s e n t s an a t t r a c t i v e way to d i r e c t l y examine the c a u s e - e f f e c t r e l a t i o n s h i p between the m o l e c u l a r s t r u c t u r e o f s h o r t f l e x i b l e c h a i n s and the m a c r o s c o p i c or thermodynamic b e h a v i o r of t h e s e systems. In a d d i t i o n , the s i m u l a t e d d a t a can be used as a r e f e r e n c e a g a i n s t which a n a l y t i c a l t h e o r i e s may be t e s t e d . Most of the e a r l y s i m u l a t i o n work on many-chain systems was done on a l a t t i c e ( 3 ) . The f i r s t o f f - l a t t i c e Monte C a r l o c a l c u l a t i o n s appear t o be t h o s e of C u r r o (4) who s i m u l a t e d c h a i n s o f f i f t e e n and twenty beads i n an e f f o r t t o compute the bead-bead r a d i a l d i s t r i b u t i o n f u n c t i o n and the r e s u l t i n g e q u a t i o n of s t a t e . H i s m o l e c u l a r model c o n s i s t e d of f i x e d bond l e n g t h s and a n g l e s , a r o t a t i o n a l i s o m e r i c s t a t e a p p r o x i m a t i o n and a h a r d - s p h e r e i n t e r m o l e c u l a r potential. C u r r o had d i f f i c u l t y s i m u l a t i n g a t r e d u c e d d e n s i t i e s ρ above 0.25 (where ρ = volume o c c u p i e d by chains/volume o f the system). B i s h o p , et a l . (_5) a l s o used a Monte C a r l o t e c h n i q u e to c a l c u l a t e the c e n t e r o f m a s s - c e n t e r of mass r a d i a l d i s t r i b u t i o n f u n c t i o n f o r f l e x i b l e m u l t i c h a i n systems but t h e i r r e s u l t i n g c u r v e s a r e h a r d to i n t e r p r e t . The m o l e c u l a r dynamics s i m u l a t i o n t e c h n i q u e has a l s o been used t o examine more s o p h i s t i c a t e d m o l e c u l a r models. R y c k a e r t and Bellemans (6) s t u d i e d s t a t i c and dynamic p r o p e r t i e s of butane and decane, u s i n g a Lennard-Jones i n t e r m o l e c u l a r p o t e n t i a l , a c o n t i n u o u s t o r s i o n a l r o t a t i o n p o t e n t i a l , and f i x e d bond a n g l e s and bond l e n g t h s . Weber (_7) a l s o i n v e s t i g a t e d butane and o c t a n e u s i n g a Lennard-Jones p o t e n t i a l c o u p l e d to a c o n t i n u o u s t o r s i o n a l r o t a t i o n p o t e n t i a l , but he r e l a x e d the c o n s t r a i n t s of c o n s t a n t bond a n g l e s and bond l e n g t h s . The r e s u l t s of R y c k a e r t and Bellemans and of Weber a r e r e a s o n a b l e f o r most q u a n t i t i e s measured w i t h the e x c e p t i o n of the p r e s s u r e , which i s quite poorly predicted. In t h i s paper we r e p o r t p r e l i m i n a r y Monte C a r l o r e s u l t s on the l o c a l s t r u c t u r e of model systems of butane, o c t a n e and dodecane. We have f o c u s e d our a t t e n t i o n on the i n t e r m o l e c u l a r r a d i a l d i s t r i b u t i o n f u n c t i o n , s i n c e t h i s may be used i n p r e d i c t i n g the p r e s s u r e . A f u t u r e paper w i l l r e p o r t more e x t e n s i v e s i m u l a t i o n r e s u l t s f o r the d i s t r i b u t i o n f u n c t i o n as w e l l as r e s u l t s f o r the p r e s s u r e . The

Model

A s i m p l i f i e d s k e l e t a l a l k a n e model was c o n s t r u c t e d c o n s i s t i n g of an η bead c h a i n (where η i s the number of c a r b o n s ) x^ith bond a n g l e s , Θ , and bond l e n g t h s , £, f i x e d a t 109°28' and 1.53A, r e s p e c t i v e l y . The d i a m e t e r of each bead (methyl g r o u p ) , σ, was a l s o f i x e d . Torsional r o t a t i o n s t h r o u g h the a n g l e <J>j_ were m o d e l l e d u s i n g the r o t a t i o n a l i s o m e r i c s t a t e p o t e n t i a l , which allows f o r j u s t three d i s c r e t e rotational states — t r a n s , gauche"*", and gauche". The p o t e n t i a l s employed were s i m i l a r to t h o s e of C u r r o ( 4 ) . The energy of the gauche s t a t e s was s e t a t 700 c a l / m o l above t h a t of the t r a n s . N e i g h b o r i n g g / g " c o m b i n a t i o n s a l o n g c h a i n backbones were g i v e n an a d d i t i o n a l energy of 2300 c a l / m o l , t o account f o r the s o - c a l l e d +

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

9.

HONNELL AND

HALL

Local Structure of Fluids

203

pentane effect a r i s i n g from s t e r i c interactions between beads sepa rated by four bonds (6). Intermolecular interactions and i n t r a molecular interactions between beads separated by more than four bonds were accounted for by a hard sphere p o t e n t i a l . The molecular model chosen for the Monte Carlo studies i s not substance s p e c i f i c but i s instead highly i d e a l i z e d . The goal has been to simplify the potential to the point where i t may be treated using some a n a l y t i c a l s t a t i s t i c a l mechanical technique but not beyond the point where those e s s e n t i a l features of the potential responsible for the f l u i d * s thermodynamic behavior are obscured. In this respect, these studies can serve as a bridge between developing a n a l y t i c a l theories and e x i s t i n g experimental and simulated data.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

Monte Carlo

Simulations

Simulations are performed within the general framework of the algorithm developed by Metropolis, et a l . (9). The Monte Carlo program randomly generates a large number of t r i a l configurations for a system of twenty-seven model molecules i n a box of size appropriate to give the desired density. Periodic boundary conditions were employed to minimize wall e f f e c t s . The program begins with an i n i t i a l configuration at a specified temperature and density. Sub sequent configurations are generated by moving one molecule at a time. After allowing the system to relax from i t s i n i t i a l state, the properties of each configuration are recorded and at the end of the simulation are averaged to y i e l d equilibrium properties. I n i t i a l configurations for the low density simulations were obtained by placing the molecules on a l a t t i c e with a l l t o r s i o n a l angles i n the trans p o s i t i o n . Higher density i n i t i a l configurations were obtained by compressing equilibrated low density configurations from previous runs. Starting with a low density system, t r i a l moves were performed u n t i l the molecule closest to the wall of the box was displaced toward the center. At t h i s point the wall of the box was moved i n toward the center r e s u l t i n g i n a smaller container. This shrinking process was repeated u n t i l the desired density was reached. The program uses four d i f f e r e n t methods to move the molecules: r i g i d t r a n s l a t i o n of the center of mass, r i g i d rotation about a central bead, t o r s i o n a l rotation of an end bead, and a reptation technique known as the s l i t h e r i n g snake (10). In the s l i t h e r i n g snake moves, a new torsional angle i s chosen for the chain, but rather than simply rotating into this state, the " t a i l " of the chain i s detached and reattached at the other end, creating a new t o r s i o n a l angle i n the process. The net effect i s a snake-like wiggling motion of the molecules along their axis throughout the box. After each attempted move, the new position of the chain i s checked against the positions of the other molecules to insure that there i s no overlap. In addition, i f a t o r s i o n a l rotation or s l i t h e r results i n an increase i n the intramolecular energy of the t r i a l chain, the move i s accepted with a p r o b a b i l i t y Ρ equal to the Boltzman factor, Ρ = exp(-AE/kT), where ΔΕ i s the change i n the intramolecular energy caused by the move. Moves which result i n a decrease i n the intramolecular energy of the chain but do not result in overlap are always accepted. For the r i g i d t r a n s l a t i o n a l and

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

204

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

r o t a t i o n a l moves, the maximum displacement parameters were adjusted so that, on average, half of the t r i a l moves were accepted. Butane, octane and dodecane were simulated at a temperature of 700K and at dimensionless densities ranging from 0.001 to 0.30. The r e l a t i v e l y high temperature was chosen to encourage t o r s i o n a l rotation to gauche states and does not influence intermolecular interactions. The dimensionless density i s defined to be the volume f r a c t i o n of the box occupied by molecules and i s given by

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

Ρ =

.3 Νηπσ' 6V

where Ν i s the number of molecules, η i s the number of beads per chain, σ i s the hard sphere diameter, and V i s the volume. The hard sphere diameter, σ, was set equal to the bond length for these studies. While a more r e a l i s t i c value of σ would be larger, t h i s "pearl necklace" model allows for a number of s i m p l i f i c a t i o n s i n theoretical treatments that larger diameter beads do not (11). For butane, two hundred thousand configurations were generated at each density, requiring approximately 45 minutes of CPU time each on the Princeton University IBM 3081. The computer time required to adequately generate and sample enough configurations to get r e l i a b l e s t a t i s t i c s increases with increasing chain length and/or density. For example, the generation of 75,000 configurations for dodecane required one hour and f i f t e e n minutes. E q u i l i b r a t i o n generally required between twenty and forty thousand configurations. In order to test the r e l i a b i l i t y of the program, several runs were repeated using d i f f e r e n t starting configurations and strings of random numbers. Approximate error estimates were obtained by p a r t i tioning each simulation into f i v e to ten blocks i n which subaverages of S i n t e r calculated. These subaverages were treated as independent samples contributing to the o v e r a l l ginter^ ")" Analysis of the standard deviation of the subaverages about the o v e r a l l average indicates that the results are accurate to ^ 4%. The structure of the simulated f l u i d was examined by calcu l a t i n g the bead-bead intermolecular r a d i a l d i s t r i b u t i o n function. The bead-bead intermolecular r a d i a l d i s t r i b u t i o n function, g i t e r ( ) i s proportional to the p r o b a b i l i t y of finding two beads (methyl groups) on d i f f e r e n t chains separated by a distance r. Thus, gi ( r ) i s a measure of the l o c a l density of the f l u i d . A value of g i n t e r ( ) 1 implies that the l o c a l density i s greater than the bulk density and conversely, when g i t e r ( ) 1» * l ° l density i s less than the bulk density. By monitoring the l o c a l density of the system of chains throughout the course of the simulation, g i n t e r ( ) can be found from w

e

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Ar

where <M> i s the average number of beads located at a distance between r and r + Ar from a neighboring bead on a d i f f e r e n t chain. Correlations were monitored out to a maximum distance of one half the

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

9.

HONNELL AND

HALL

Local Structure of Fluids

205

box length i n order to avoid the p e r i o d i c i t y introduced at larger separations by the boundary conditions. In addition to describing the l o c a l structure of the f l u i d , g i t ( ) ^ * calculate the pressure and i s the key to investigating other model potentials through perturbation expansions. r

n

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a

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Results The results for our Monte Carlo simulations are i l l u s t r a t e d i n Figs. 1 through 9. Figure 1 shows the intermolecular r a d i a l d i s t r i bution function for butane at reduced densities of 0.01, 0.10 and 0.30 as a function of r/σ where σ i s the bead diameter. Also shown for comparison i n the inset i s the r a d i a l d i s t r i b u t i o n function for a monatomic hard sphere f l u i d which i l l u s t r a t e s the familiar peak and v a l l e y structure, r e f l e c t i n g the s h e l l s of nearest neighbors, followed by a gradual loss of correlations as g i n t e r ( ) decays to 1. This behavior i s c h a r a c t e r i s t i c of a l l monatomic f l u i d s . What i s s t r i k i n g about the lower-density butane results i s that 8 i ( ) contact i s s i g n i f i c a n t l y less than 1, indicating a r e l a t i v e absence of neighboring beads i n the immediate v i c i n i t y of a central bead. The o r i g i n of this "correlation hole" can be seen by constructing a physical picture of what may be happening on a molecular l e v e l . A comparison of Figures 2a and 2b provides an explanation for the correlation hole. Figure 2a shows a cluster of monatomic mole cules surrounding a central (shaded) monatomic molecule. The mon atomic molecules are able to approach each other with r e l a t i v e ease resulting i n the familiar shells of neighboring beads, f i r s t nearest neighbors, second nearest neighbors, and so on, shown i n the inset on F i g . 1. In the case of chain molecules shown i n F i g . 2b, neighboring beads on the same chain as the central (shaded) bead are not counted i n computing intermolecular correlations. The presence of these adjoining beads, i n turn, leads to a shielding effect which makes i t more d i f f i c u l t for surrounding chains to approach the central bead. The net effect of this shielding i s the creation of an excluded volume surrounding each bead, the so called correlation hole. If this exclusion effect i s indeed the correct explanation for the correlation hole, then one might expect that the value of g ( r ) at contact ( i . e . at r = σ) would increase with density as more and more chains are packed together. This trend i s observed i n Figure 1; at a reduced density of 0.3, g i n t e r ( ) intercept greater than 1, and the curve begins to take on a shape reminiscent of a monatomic system. Another interesting feature of the l o c a l structure i s i l l u s trated i n Figure 3 which show cusps i n the g(r) curve for butane at σ = 0.25 occurring at distances r/σ = 2 and r/σ = 2.666. These cusps may be explained by r e c a l l i n g that d i s c o n t i n u i t i e s i n the i n t e r molecular p o t e n t i a l result i n corresponding d i s c o n t i n u i t i e s i n the i n the pair correlation function. For example, i n the monatomic hard-sphere model, the discontinuity i n the p o t e n t i a l at r = σ results i n a discontinuity i n g(r) at r = σ. Likewise, i n the mon atomic square well model, d i s c o n t i n u i t i e s i n g(r) occur at r = σ and at r = ko, ko being the width of the well (12). For chains however, the contact between a bead such as the one labeled 1 i n Figure 2b and the central bead can become an e f f e c t i v e contact between the central

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

r

r

n t e r

i n t e r

r

n

a

s a

n

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

a

t

206

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

F i g . 1. Comparison of g i t ( ) ^ butane Ρ 0.01, 0.15 and 0.3 obtained from Monte Carlo simulations at 700K. The inset shows the r a d i a l d i s t r i b u t i o n f o r a monatomic hard sphere system.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

r

n

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=

e r

Op J D - ® ÇrOm><£ (a)

(b)

F i g . 2. Local correlations f o r : (a) f l u i d of hard sphere monatomics, (b) f l u i d of hard bead chains.

1.0

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3.0

F i g . 3. Intermolecular r a d i a l d i s t r i b u t i o n function f o r butane at ρ = 0.25 predicted by Monte Carlo simulations.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

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Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

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were

diameter

In

o / l - 1 and

Consequently

The b u l k i e r molecules of

of

sphere

larger

separated

bead

of

and Weiss

to

the

the

a value

length

for

refer

conformation. length

r

i n which studies

length.

and a m o l e c u l a r

because,

bond

varying

a Lennard

vs.

298K

several

bond

hard

by V e r l e t

g/cc.

between

of

the

alkanes,

diameter of

o / l - 1 there

interactions.

effect

density

contact,

tend

the of

previously

length,

value

developed g i

described bond

3.7889Â.

isomeric

volume

the

models

sphere

chain to

to

corresponds

overlap

at At

The

3.

Pratt,

at

of

seen

torsional

r

the

temperature

appearance

chains

are

chains

realistic

packing

(r)

is

extent

local

determine

a

structure.

tighter

(as

1

dis

angle-averaged

bead

length

shapes

potential,

broad

the

to

hard

with

available

s

"*" p l °

700K

a method

diameter

fluid

more

shielding

the

set

= 0.631

of

bead

on

to

reduced

amount changes

of

6 compares

a

0.00333/Â the

with

greater

sphere

diameter

effective at

chain

overall

temperatures

the

This

sphere

than

and dodecane

therefore

was try

considered.

octane

contact

by H s u ,

of

(r)

it

maintaining

chosen

whenever

and bead

the

for

will

closer.

length.

addition

to

predicted

correlation hole

the

like

in

effect

that

making

ture

performed

r

appear

r / σ = 2.66

octane

a

at

One w o u l d

In

e

bead

effective

the

fairly

300K

At

with

gi

a

however,

formation, other.

t

effect

results

note,

been

temperature

and

structure at

to

a hard

simulated

an

the

of

which

2 and

lengths

any

occur

central

These

and

bead

r > σ will

the

butane,

degree

gi

0.2

of

to

cusps

angles

central

at

with

2b).

Although the

of

bond

(13).

for

0.15. size

effect

expected

hard

cusps

Longer chains,

density

relate

the

4 illustrates

5, w h e r e

while

in

Fig.

has

increased

very

2 in

rigid

potential

colinear

these

themselves

The Fig.

pair

of

of

an

bead

the

approach the

bead

density

to

the

4 to

r / σ = 2 for

are

length.

in

reflected

Results

around

3 or

at

using

the

4 since

2,

4 become

for

Figure

same,

3 or

2,

beads

207

Local Structure of Fluids

HONNELL AND HALL

9.

s

t

i

in

of c

the

box

steric °f

a

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

l °

w

EQUATIONS O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

208

r

F i g . 6. Effect of bead diameter on g i t e r ( ) ^ number density of 0.00333/Â at Τ = 298K.

o r

o

c

t

a

n

e

a

t

n

3

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

a

9.

HONNELL AND

Local Structure of Fluids

HALL

209

density system, with only a small change i n slope at a distance r = σ + £. These results indicate that thermodynamic properties of such model systems, p a r t i c u l a r l y the pressure, w i l l be very s e n s i t i v e to the size of the hard sphere diameter employed. In F i g . 7, g i ( ) for butane at a temperature of 298K, reduced density of 0.25 and o/l = 1 i s compared with the predictions of RISM, a computationally convenient i n t e g r a l theory for l o c a l structure (15). At values of r/σ > 1.3, good agreement between theory and simulation i s obtained, including the prediction of the cusps at r/σ = 2 and r/σ = 2.67. Near contact however, RISM predicts a tighter clustering of chains than i s observed i n our simulations. Since for hard chain systems the pressure i s related to the value of the d i s t r i b u t i o n function at contact, these two curves w i l l lead to d i f f e r e n t perdictions for the pressure. Similar discrepancies near contact between RISM predictions and simulation results have been observed i n systems of dimers and trimers (16). In Figure 8, RISM and Monte Carlo results are compared for octane at a temperature of 298K, reduced density of .05, and o/l r a t i o of one. Both simulation and theory predict a s i z a b l e correlation hole although the o v e r a l l agreement i s not as good as i t was for butane. Again the discrepancy between the two curves i s largest near contact. Further insights into the l o c a l structure of short chain f l u i d s can be gained by examining the i n d i v i d u a l correlations between beads at s p e c i f i c locations along the chain backbone. Figure 9 shows g ( l , l ) , g ( l , 2 ) , and g(4,4) for octane at N/V = 0.0033/Â3 and o/l - 2.48, where the pairs of numbers refer to the locations of the two beads along the chain backbone, r e l a t i v e to the ends of the chains. For example, g(1,1) represents the d i s t r i b u t i o n function for end beads on d i f f e r e n t chains, while g(4,4) refers to correlations between center beads on different chains. See Figure 10. Because the chains can be numbered from either end, beads 1 and 8, 2 and 7, 3 and 6, and 4 and 5 are equivalent, r e s u l t i n g i n only ten d i s t i n c t bead-bead pairs comprising g ^ n t e r ^ * r

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

n t e r

The d i s t r i b u t i o n function g(1,1) has i t s greatest value at contact, as i t i s r e l a t i v e l y easy for two end beads to approach each other. However, at larger distances g(1,1) drops below one, since i f two "heads" of chains are adjacent, their " t a i l s " must be r e l a t i v e l y far apart. The (1,2) correlation has i t s maximum at a distance of a diameter plus a bond length, which corresponds to the distance that would be between beads 1 and 2 i f the two heads of the chain were i n contact. Near contact, g(1,2) i s less than one, since i t i s more d i f f i c u l t for a bead i n the second p o s i t i o n to approach another chain end without s t e r i c interference from i t s bonded neighbors. At larger distances g(1,2) begins to resemble g(1,1) since, i f beads 1 and 2 are i n r e l a t i v e proximity, beads 7 and 8 are l i k e l y to be far apart. Steric hindrance plays the biggest role i n the (4,4) c o r r e l a t i o n . Near contact g(4,4) i s much less than one, implying that the centers of chains rarely come into contact. At larger values of r however, g(4,4) has a r e l a t i v e l y broad peak. This i s because the distances between equivalent pairs (4,4), (4,5), (5,4) and (5,5) are l i k e l y to be about the same, unlike the pairs contributing to g ( l , l ) and g ( l , 2 ) . The behaviors of the seven remaining correlation functions l i e i n between that of g(1,2) and

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

l.5r-

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

σ>

0.5

Fig. 7. Comparison of RISM and Monte Carlo predictions of g ( r ) f o r butane at ρ = 0.25, o/l = 1, Τ = 298K. Solid curve i s RISM, boxes are simulation results, The two curves coincide for r/o>2. i n t e r

1.5

1.0 - -

0.5 - -

xlO

F i g . 8. Comparison of RISM and Monte Carlo predictions f o r octane at ρ = .05, a/I = 1, Τ = 298K. Solid curve i s RISM, boxes are simulation r e s u l t s .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

HONNELL AND

HALL

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

0

I Ο

Local Structure of Fluids

1

!

1

1

1

0.5

1.0

1.5

2.0

2.5

r/σ

F i g . 9. Comparison of g ( l , l ) , g ( l , 2 ) , and g(4,4) bead correla tions f o r octane at a number density of 0.00333(1/Â ), all = 2.48 Τ = 298K. 3

Fig. 10. Location of i n d i v i d u a l beads along chain backbone.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

212

g(4,4). Each has a c o n t a c t v a l u e l e s s than one, which, when imposed, l e a d s t o t h e c o r r e l a t i o n h o l e seen i n ginter( )«

super

r

Conclusions

and D i s c u s s i o n

Monte C a r l o s i m u l a t i o n has been used t o o b t a i n t h e i n t e r m o l e c u l a r r a d i a l d i s t r i b u t i o n f u n c t i o n f o r systems o f i n t e r a c t i n g s h o r t c h a i n molecules. A t low and moderate d e n s i t i e s , t h e v a l u e o f g i n t e r ( ) near c o n t a c t i s s i g n i f i c a n t l y l e s s than one, i n d i c a t i n g t h e p r e s e n c e o f an e x c l u d e d volume s u r r o u n d i n g each bead. The e x i s t e n c e o f t h i s " c o r r e l a t i o n h o l e " appears t o stem from a s h i e l d i n g e f f e c t a s s o c i a t e d w i t h n e i g h b o r i n g beads a l o n g t h e same c h a i n backbone. An e x a m i n a t i o n o f t h e i n d i v i d u a l bead-bead c o r r e l a t i o n s c o m p r i s i n g g i n t e r n ) i n d i c a t e s t h a t i n t e r i o r beads a l o n g t h e c h a i n s a r e much more h e a v i l y s h i e l d e d t h a n beads n e a r t h e c h a i n ends.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

r

The s i z e o f t h e c o r r e l a t i o n h o l e i s seen t o i n c r e a s e w i t h i n c r e a s i n g c h a i n l e n g t h and d e c r e a s e w i t h i n c r e a s i n g d e n s i t y . One would expect t h e c o r r e l a t i o n h o l e t o be q u i t e s m a l l f o r dimers and trimers. L o c a l s t r u c t u r e i s s t r o n g l y dependent on t h e h a r d s p h e r e diameter. P r e l i m i n a r y comparisons w i t h RISM c a l c u l a t i o n s show o v e r a l l good agreement e x c e p t n e a r c o n t a c t . C a l c u l a t i o n s a r e c u r r e n t l y underway t o more t h o r o u g h l y i n v e s t i g a t e t h e e f f e c t o f bead d i a m e t e r on l o c a l s t r u c t u r e as w e l l as t o more c l o s e l y examine t h e i n d i v i d u a l end-end, end-middle, and m i d d l e m i d d l e c o r r e l a t i o n s w h i c h comprise g i n t e r ( ) * Simulations are being performed a t h i g h e r d e n s i t i e s and a t l o n g e r c h a i n l e n g t h s i n o r d e r t o more f u l l y i n v e s t i g a t e t h e e f f e c t o f d e n s i t y and c h a i n l e n g t h on l o c a l s t r u c t u r e . We a r e a l s o t r y i n g t o develop s i m p l e c o r r e l a t i o n s w h i c h summarize t h e e f f e c t o f v a r y i n g m o l e c u l a r parameters on t h e radial distribution function. F i n a l l y , we s h o u l d p o i n t o u t t h a t due t o o u r r e l a t i v e l y s m a l l sample s i z e and r e l a t i v e l y s h o r t runs t h e r e s u l t s p r e s e n t e d i n t h i s paper s h o u l d be c o n s i d e r e d s u g g e s t i v e r a t h e r t h a n c o n c l u s i v e . More e x t e n s i v e s i m u l a t i o n s w i l l be r e p o r t e d i n a f u t u r e p a p e r . r

Acknowledgments The a u t h o r s would l i k e t o thank t h e Gas R e s e a r c h I n s t i t u t e (Grant No. 5082-260-0724) and t h e P e t r o l e u m R e s e a r c h Fund (Grant No. 14851-AC5) f o r s u p p o r t i n g t h i s r e s e a r c h . Ms. D. C h i r i c h i l l a i s g r a t e f u l l y acknowledged f o r h e r h e l p i n p e r f o r m i n g t h e RISM c a l c u l a t i o n s . The a u t h o r s a l s o w i s h t o thank D r . L . P r a t t , E . H e l f a n d , and J . C u r r o f o r h e l p f u l d i s c u s s i o n s and s u g g e s t i o n s .

Literature Cited 1. Donohue, M. O.; Prausnitz, J. M. AIChE J. 1978, 24, 849. 2. Sanchez, I.; Lacombe, R.; J. Chem. Phys. 1976, 80, 2352, Lacombe, R.; Sanchez, I., J. Chem. Phys. 1976, 80, 2568; Simha, R., Macromolecules, 1977, 10, 1025. 3. DeVos, E.; Bellemans, Α., Macromolecules 1975, 8, 651; ibid, 1974, 7, 812. 4. Curro, J. G., J. Chem. Phys. 1976, 64, 2496.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch009

9. HONNELL AND HALL Local Structure of Fluids

213

5. Bishop, M.; Ceperley, D.; Frisch, H. L.; Kalos, M. H., J. Chem. Phys. 1980, 72, 3228. 6. Ryckaert, J.-P.; Bellemans, Α., Chem. Phys. Lett. 1975, 30, 123; Faraday Soc. Disc 1978, 66, 95. 7. Weber, Τ. Α., J. Chem. Phys. 1979, 69, 2347; 1979, 70, 4277. 8. Flory, P. J. "Statistical Mechanics of Chain Molecules", John Wiley & Sons, 1969, pp. 55-56. 9. Metropolis, N.; Metropolis, Α.; Rosenblush, M.; Teller, Α.; Teller, E.; J. Chem. Phys. 1953, 21, 1087. 10. Wall, F. T.; Mandel, P., J. Chem. Phys. 1975, 63, 4593. 11. Curro, J. G.; Blatz, P. J . ; Pings, C. J., J. Chem. Phys. 1969, 50, 2199. 12. Nelson, P. A. Ph.D. Thesis, Princeton University, New Jersey, 1967. 13. Hsu, C. S.; Pratt, L. R.; Chandler, D., J. Chem. Phys. 1978, 68, 4213. 14. Verlet, M.; Weis, J., Phys. Rev. A 1972, 5, 939. 15. Ladanyi, Β. M.; Chandler, D., J. Chem. Phys. 1975, 62, 4308. 16. Gray, C. G.; Gubbins, Κ. E. "Theory of Molecular Fluids"; Oxford: Clarendon Press, 1984, Vol. I, pp. 400-403. RECEIVED November 18, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

10 Local Composition of Square-Well Molecules by Monte Carlo Simulation R. J. Lee and K. C. Chao

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

School of Chemical Engineering, Purdue University, West Lafayette, IN 47907

Local composition in mixtures is determined for square-well molecules by means of Monte Carlo simulation at 29 states from dilute gas to dense liquid at a wide range of temperature for divergent energies of interactions and varying compositions. The results are compared with models in the literature, and a new local composition model is proposed.

Local composition about a molecule in a mixture can differ from the total bulk composition as a result of divergent energies of interaction and molecular size differences. The difference between local composition and bulk composition reflects molecular segregation and preferential orientation. Debye and Huckel ( ! ) developed their ionic solution theory by recognizing the local composition of ions. Wilson ( 2 ) postulated the dependence of partial entropy on local composition. Whiting and Prausnitz ( 3 ) based their mixture thermodynamics on local composition. The occurrence of local composition is fundamental to thermodynamics and transport properties of mixtures. Local composition cannot be determined by experiments in the laboratory, but can be simulated by computer calculations. Nakanishi and co-workers ( A£J_ ) and Hoheisel and Kohler ( 8 ) obtained local composition in mixtures of Lennard-Jones molecules by means of molecular dynamics simulation. In this work we determine local composition of square-well molecules by means of Monte Carlo simulation. Square-well molecules offer the special advantage of an unambiguously defined neighborhood in which local composition occurs. We carry out calculations to states previously unexplored by extending to wide ranges of density, temperature, bulk composition, and energies of interaction. 0097-6156/ 86/ 0300-0214$06.00/ 0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

10.

LEE AND CHAO

Local Composition of Square- Well Molecules

215

Model and Method The system investigated in this work is a binary mixture in which molecules interact via the square-well potential,

oo

ÎTjj < Γ < K<7jj

(1)

K^j < r where φ is the potential energy between molecules i and j at center-to-center dis tance r; e is the well depth; σ is the diameter and Κ is the energy well width factor. In this work Κ = 1.5, and all σ are equal; the investigation is directed at energy effects. χ ]

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

Χ]

The number of molecules j at a distance r to r + dr from a molecule i is given by dz^ = 4πρβ {τ)τ άτ

^

2

}]

where p^ is the number density of molecules j and g is the radial distribution func tion. The local composition of j at r from i is therefore

Σ PkSki(r)

(3)

k

where χ denotes a mole fraction. We have obtained the total number of molecules j in the energy well of i , Κσ Zji

= 4π

j gji(r)r dr 2

Ρ}

(4)

from which we have evaluated the average local composition Κσ

Pi / S j i ( ) ο r

r 2 d r

(5)

Κσ

Σ Pv / gki(r)rdr 2

In this work we report the average local composition as the local composition for brevity and in agreement with the usage of previous investigators ( 4^8 ). We have also obtained gjj (r) and Xjj (r) which we will report elsewhere.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

216

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

The upper limit of integration of Equations 4 and 5 is denned without ambiguity for the square-well molecules, while there is an element of arbitrariness in setting the upper limit for continuous potentials such as the Lennard-Jones. We simulated Equations 2 through 5 with the canonical ensemble method. The procedure of Metropolis et al. ( 9 ) was applied to 108 molecules in a central cell with periodic boundary conditions and following the minimum distance convention. The procedure was initiated by randomly displacing molecules from a regular face centered cubic lattice. About 1.0 χ 10 initial configurations were discarded for dilute gases. Up to 1.7 χ 10 initial configurations were discarded for high density liquids. The total Markov chain varied from 1.2 χ 10 configurations for a dilute gas to 4.5 χ 10 for a dense liquid. About 10 minutes of computing on a CYBER 205 was required to generate 1 million configurations. 5

6

6

6

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

From the Monte Carlo simulation we obtain the radial distribution function,

g ( r )

3

=

N

( *>)

(6)

with Γ

Γ

λ =

λ-ι +

i

r

r = y ( r \ + Γχ-ι) where ^Njj ^ is the ensemble average number of molecular pairs j i within the Xth distance interval from Γχ-j to Γχ. Equation 5 becomes (Ν^Κσ,))

=

"

Σ

N ,(K
ki

1

'

k

where ^ Ν ^ ( Κ σ ^ is the ensemble average number of molecular pairs j i at distances of separation up to Κσ^ and is obtained by summing the ^Njj^ . Results of Monte Carlo Simulation Pure square-well fluid pressure, internal energy, and radial distribution function were generated by our computer programs at a large number of states and compared with previous results by Alder et al. ( 10 ), Henderson et al. ( JT ) and Smith et al. ( 12 ). By setting Κ = 1 for some molecules in our computer programs we simulated mix tures of hard sphere and square-well molecules and compared the results with those of Alder et al. ( 13 ). Comparison of our results ( 14 ) with these previous studies enabled us to check out our programs.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

10.

Local Composition of Square- Well Molecules

LEE AND CHAO

217

Square-well mixtures have not been simulated before. We compared our results with Lennard-Jones mixtures obtained by Nakanishi et al. ( 6 ) at two comparable states. Table I shows that comparable local compositions were obtained. Our simulated local compositions x , x and number of nearest neighbors ζ z in square-well fluids are presented in Table II, where i is given by ji- Four varij ables — density, temperature, interaction factors, and composition — are varied suc cessively to generate the results of Table II. Density in reduced form ρ* ( = ρσ ) varies from 0.01 to 0.8; reduced temperature T* ( = k T / e ) from 1.0 to 4.0; compo sition from 0.25 to 0.75 mole fraction; unlike interaction factor E ( = e 12/^11) from 1.0 to 10.0 and alike interaction factor E ( = ^Λη) from 2.0 to 10.0. The interac tion factor ranges are sufficiently wide to include non-polar (except the quantum gases), associating, and hydrogen bonding energies. n

22

2

υ

z

x

ζ

n

x

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

2

The 29 states in Table II are made up of four sections in each of which one vari able is systematically varied while all others are held constant. In section 1 consist ing of states 1 to 9, density is the variable. At the lowest density of state 1 which is a dilute gas, there is the greatest departure of local composition from the bulk. As density increases, local composition departs less and less from the bulk and tends to approach it as a limit. In section 2, consisting of states 10 to 15, temperature is the variable. Two sub sections of three states each show the effect of temperature at a different density. The effect of increasing temperature is to homogenize local composition toward the bulk. This is the most expected part of our results. In section 3, consisting of states 16 to 22, the interaction factors are varied. In all these states E > 2.0, and x is always greater than x . The tendency to form alike pairs of greater energy is favored. For Lorentz-Berthelot mixtures with e i2 (ene22)1^2 both x and x increase with increasing alike interaction factor E . The states with a constant E ( = 2.0) show that x and x decrease as the unlike interaction factor Ej increases. 2

22

n

=

n

22

2

2

n

22

In section 4, consisting of states 23 to 29, bulk composition is changed. Local composition is shown to be strongly dependent on bulk composition. Deviation of local composition from bulk composition is predominantly on the part of the dilute component. In addition to the variation of one single variable within a section of Table II, the effect of simultaneous variation of two variables can be ascertained upon suitably scanning the table entries in different sections. Higher effects can be similarly dis cerned. The entire simulated results should be considered in the development of a new model.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

218

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Comparison with Previous Models Wilson ( 2 ), Gierycz and Nakanishi (GN) ( 7 ), and Hu, Ludecke, and Prausnitz (HLP) ( 15 ) have proposed models of local composition. The models are examined with our simulated results. Wilson expressed local composition by

J1

Σ Xk exp(-X /kT) ki

k

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

Gierycz and Nakanishi's local composition is given by

_ XjAjBjjexpt-aGji/kT) xji

(9)

xjexp(-aGii/kT)

(10) Aj -

(σ σ η

22

+

2 σ

^ϋ 22)/ ι 2 σ

Bji = exp[a VG„G (1

2

- G G /G?)/kT]

22

n

22

(12)

a = 0.4 Hu, Ludecke, and Prausnitz's model is 2

x,z + x z - [(x^! + x z ) -4xix ziz r il t

2

2

2

2x z r 1

j

1

2

2

2

1/2

2

21

ά

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(13)

10.

Local Composition of Square- Well Molecules

LEE AND CHAO

1 - exp(n/kT)

Ω

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

ll,hS 22,hS

x

12,hS 21,hS

(15)

x

x

(16) «2l(r«) + « 12(Γΐ2) ~ Cll(rii) "

x

a

x

219

J

c r

(17)

χ

|/Ek ^

(18)

0.60 - 0.58p',•0.1856

Figure 1 shows the substantial local composition variation with density of states 1 to 9 of Table II. Both the Wilson and the G Ν models are independent of density and completely miss this variation. The Wilson model exactly represents the low density limiting behavior of L C according to Chao and Leet ( 16 ). The limiting behavior is confirmed by the simulated data.

Figure 2 shows the variation with temperature of our simulated data and the models at two liquid-like densities. For the relatively small energy difference shown in Figure 2A the HLP model agrees better with the simulated data than the other two models. With larger E ( = f / i ) shown in Figure 2B the HLP model tends to be low for both x and x d the deviation becomes larger at lower temperature. The G Ν model gives good results for x t l ° ι ι · Wilson's model departs widely from data for these high density conditions. t

1 2

n

a n

n

22

DU

w χ

22

Figure 3 shows the effect of unlike energy factor E while ρ*, T*, and E are kept constant. For low Ej the HLP equation stays closer to the data than both the Wil son and the G Ν equations. All equations become worse with increasing E ^ apparently due to the Boltzmann exponential factor lending too much weight. The best results of the GN equation are attributable to the use of a factor of a = 0.4 to reduce the weight of the Boltzmann term. 1

2

Figure 4 shows the effect of bulk composition while /?*, T*, E and E are kept constant. All three models are in reasonable agreement with the data in part A of Figure 4 where the interaction factor values are not large. The HLP equation appears particularly good. However, the equations fail at larger values of the interaction factors as shown in Figure 4B. 1 ?

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2

220

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table I Local Compositions in Square-Well and Lennard-Jones Fluids

T* *

Ρ Ει E 2

*i ll X

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

X

22

sw

LJ

SW

LJ

1.5 0.7 1.414 2.0 0.5 0.491 0.518

1.44 0.716 1.414 2.0 0.5 0.492 0.518

1.5 0.8 2.0 2.0 0.5 0.438 0.458

1.43 0.75 2.0 2.0 0.5 0.449 0.465

.6000

.5500

.5000

.4500

EQUATIONS) HILS0N MODEL 6N MODEL HLP MODEL

MC DATA .4000

-

.3500 ι .000

I1

.200

11 .400

χ

I1

1

1

.600

.800

1.000

M

Figure 1.

Local Composition as a Function of Density and Comparison of MC Data with Four Models at T = 2.0, E = 1.414, E = 2.0, and x = 0.5. t

2

t

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Local Composition of Square- Well Molecules

LEEANDCHAO

Table II Simulated Number of Nearest Neighbors and Local Composition in Square Well Mixtures

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

State

Ζχ

z

0.5

0.18 0.89 1.74 3.27 5.24 5.93 7.28 10.40 11.78

0.23 1.14 2.28 4.13 6.26 6.92 8.18 10.86 12.17

0.447 0.449 0.447 0.456 0.466 0.476 0.482 0.499 0.500

0.571 0.572 0.576 0.568 0.553 0.552 0.539 0.520 0.516

2.0

0.5

2.0

2.0

0.5

10.62 10.20 10.09 12.60 12.19 11.56

11.23 10.52 10.31 13.28 12.64 11.77

0.491 0.500 0.497 0.420 0.438 0.470

0.518 0.516 0.508 0.450 0.458 0.480

0.8

1.0 2.0 2.0 3.0 5.0 10.0 10.0

2.0 2.0 4.0 10.0 2.0 2.0 10.0

0.5

11.65 11.97 11.75 11.48 12.71 13.39 12.72

12.10 12.30 12.95 14.58 13.02 13.83 15.48

0.579 0.448 0.563 0.607 0.363 0.315 0.364

0.595 0.462 0.603 0.691 0.378 0.337 0.477

2.0

0.7

10.0

2.0

2.0

0.8

2.0

2.0

0.25 0.333 0.50 0.67 0.75 0.25 0.75

14.05 13.30 12.38 11.48 11.11 12.10 11.84

11.59 11.88 12.73 13.83 14.25 12.25 12.38

0.074 0.145 0.298 0.498 0.620 0.189 0.726

0.626 0.521 0.318 0.167 0.111 0.733 0.216

Τ*

ρ*

E

1 2 3 4 5 6 7 8 9

2.0

0.01 0.05 0.1 0.2 0.35 0.4 0.5 0.7 0.8

1.414

2.0

10 11 12 13 14 15

1.5 3.0 4.0 1.0 1.5 4.0

0.7

1.414

0.8

16 17 18 19 20 21 22

2.0

23 24 25 26 27 28 29

2

X\

2

x

x

tl

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

22

222

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

«~ .000

ι 1.500

*

I 3.000

1 4.500

« 1 .000

1 1.500

*

1 3.000

1 4.500

Figure 2 . Local Composition as a Function of Temperature and Comparison of MC Data with Four Models. (A): p* = 0.7, E = 1.414, E = 2.0, and x = 0.5. (B): p* = 08, E = 2.0, E = 2.0, and x = 0.5. T

T

x

2

2

' .00

2.00

4.00

Figure 3. Local Composition Changing with E E = 2.0 and x = 0.5.

l

2

1

x

1

6.00

6.00

10.00

for Four Models at p* — 0.8, T* = 2.0,

x

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Local Composition of Square- Well Molecules

LEE A N D CHAO

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

10.

.000

.250

.500

.750

223

1.000

*1

.000

.250

.500 X

.750

1.000

1

Figure 4. Local Composition Changing with Bulk Composition for Four Models. (A): p* — 0.8, T* = 2.0, E = 2.0, and E = 2.0 (B): p* - 0.7, T* = 2.0, E = 10., and E = 2.0. Equation(19); Wilson Model; GN Model; HLP Model. 1

2

x

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

224

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

The comparisons show that the Wilson model is correct at the low density limit. Being density independent, it diverges from data at all other densities. The GN model gives good results at a liquid-like density. Though density independent it is much improved over the Wilson model. The HLP model is density dependent and displays the qualitative trend of the data at varying density, temperature, and ener gies. Quantitatively the HLP model diverges from the data in density effect, and over estimates the effect of high energy factors. A New Model

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

The comparisons of the last section suggest the need for a new model for the representation of the MC data. We propose a new local composition model as fol lows,

χ

=

Xjexpfrjifrji

" cjî)/kT]

S*kexp[7ki(*ki "

Jl

k

€ °i)/kT] k

where =

+ (1 - e)

<

fjJ

20)

and α is density dependent and is given by 2

3

a = 1 + 0.1057/ - 2.1694/ + 1.7164/>

^

For Lorentz-Berthelot mixtures 7μ = 1, Equation 19 reduces to the Wilson model as p—*0 and a—*1. Comparison of the new model with data is reported in Figures 1 to 4. The den sity dependence of local composition is shown in Figure 1. Quantitative representa tion of the data in the entire composition range studied is achieved with the new model and the low density limit is correctly approached. The temperature dependence of local composition is shown in Figure 2 where the new model agrees with data better than the other models. Figure 3 shows that the new model brings about a substantial improvement at high values of E . In Figure 4A all models show good agreement with data at changing bulk composition when the energy differences are not excessive. x

Figure 4B shows a case of large energy difference where the new model holds out better than the others. The weight of the Boltzmann factor is reduced in the new model to achieve the improved result but the reduction tends to be on the high side.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

10.

Local Composition of Square- Well Molecules

LEE AND CHAO

225

Conclusion 1. Local composition in square-well fluid mixtures has been simulated with the canonical ensemble Monte Carlo method. 2. Local composition is found to depend on the state of the system and local com position values are determined for 29 states at various density, temperature, interaction energy, and bulk composition. 3. The predictions of three models are tested with the simulated results. The Wil son and G Ν models do not show any density dependence. The HLP model does, but underpredicts the dependence.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

4. A new model is proposed for improved representation of local composition. Acknowledgment Funds for this research were provided by the National Science Foundation through grant CBT-8516449. Glossary of Symbols Ej

= € /f ii, unlike interaction factor

E

= ^22/ n> alike interaction factor

12

c

2

Gjj

= energy parameter in G Ν model

g

= radial distribution function

Κ

= energy well width factor

k

= Boltzmann's constant ensemble average number of molecular pairs j i within the Xth distance interval

r

intermolecular separation

r , r

diameter of the first and second shells respectively

Τ

temperature

T*

k T / f , reduced temperature n

bulk composition of molecule i local composition of molecule j around molecule i coordination number of molecule i number of molecules j around a central molecule i a

parameter in GN and HLP model

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE. T H E O R I E S A N D A P P L I C A T I O N S

226

7

= energy parameter in the new model

e

= well depth of square-well potential

Xjj

= energy parameter in Wilson's equation

p-

= number concentration of molecule j

ρ*

— ρσ , reduced density

σ

= hard-core diameter

^ij

= potential energy between molecules i and j

3

Literature Cited Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch010

1. Debye, P.; Hückel E. PhysikZ.1923, 24, 185, 334. 2. Wilson, G.M. J. Am. Chem. Soc. 1964, 86, 127. 3. Whiting, W.B.; Prausnitz, J.M. Fluid Phase Equil. 1982, 9, 119. 4. Nakanishi, K.; Toukubo, K. J. Chem. Phys. 1979, 70, 5648. 5. Nakanishi, K.; Toukubo, K. Polish J. Chem. 1980, 54, 2019. 6. Nakanishi, K.; Tanaka, H. Fluid Phase Equil. 1983, 13, 371. 7. Gierycz, P.; Nakanishi, K. Fluid Phase Equil. 1984, 16, 255. 8. Hoheisel,C.;Kohler, F. Fluid Phase Equil. 1984, 16, 13. 9. Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H. J. Chem. Phys. 1953, 21, 1087. 10. Alder, B.J.; Young, D.A.; Mark, M.A. J. Chem. Phys. 1972, 56, 3013. 11. Henderson, D.; Madden, W.G.; Fitts, D.D. J. Chem. Phys. 1976, 64, 5026. 12. Smith, W.R.; Henderson, D.; Tago, Y. J. Chem. Phys. 1977, 67, 5308. 13. Alder, B.J.; Alley, W.E.; Rigby, M. Physica 1974, 73, 143. 14. Lee, R.J. M.S. Thesis, Purdue University, 1985. 15. Hu, Y.; Lüdecke, D.; Prausnitz, J.M. Fluid Phase Equil. 1984, 17, 217. 16. Chao, K.C.; Leet, W.A. Fluid Phase Equil. 1983, 11, 201. RECEIVED November 5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

11 Equations of State for Nonspherical Molecules Based on the Distribution Function Theories 1

1

2

S. B. Kanchanakpan , L. L. Lee , and Chorng H. Twu 1

School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, OK 73019 Simulation Science, Inc., Fullerton, CA 92633

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

2

Based on the distribution function theories of the liquid state, we are able to derive an expression for the contribution from the attractive energy to the pressure of anisotropic fluids. We have adopted the nonspherical square-well potential with Gaussian overlap to model moderately anisotropic molecules. The repulsive pressure is shown to be given essentially by the underlying hard core repulsion and is represented, for example, by the Nezbeda equation of state for hard convex bodies. It is found that the background correlation function, y(12), plays a major role in the determination of attractive pressures. We exhibit the cluster series for this correlation function and derive therefrom a resummat ion formula for the pressure. To obtain adequate description of real fluid properties, a two-step form of the square-well potential as proposed by Kreglewski is adopted. The final equation consists of three terms, clearly separated into the hard core, repulsive and attractive parts. It is used to correlate the P-v-T and thermal behavior of some 69 substances, including hydrocarbons, ketones, alcohols, amines and polar solvents. For pressures up to 69 MPa, the errors in density and vapor pressure predictions are, with few exceptions, within 1%. Comparison with similar equations of state, such as the Peng-Robinson and Mohanty-Davis equations, shows that the present equation is uniformly superior. The use of hard convex molecules as a reference f l u i d f o r r e a l f l u i d s has received much attention l a t e l y , e s p e c i a l l y i n perturbation theory formulations (1-4) dealing with nonspherical molecules. This i s due to the recognition that i n l i q u i d s the structure i s e s s e n t i a l l y determined by the repulsive part of the molecular i n t e r a c t i o n potent i a l . On the other hand, the a t t r a c t i v e i n t e r a c t i o n makes a major 0097~6156/86/0300-0227$06.75/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

228

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

contribution to the energy of the l i q u i d (Note: the energy derived from the hard convex bodies, HCB, i s zero, and i s therefore inade quate for thermal properties.) In order to achieve a complete de s c r i p t i o n of the thermodynamics of the l i q u i d state, both c o n t r i butions, a t t r a c t i v e as well as repulsive, should be considered. Beret and Prausnitz (5) proposed the perturbed hard chain (PHC) equation of state for long chain hydrocarbons. It contained a modified hard sphere term of repulsion and a square well term for a t t r a c t i v e pressure. I t was applied to hydrocarbons such as methane, ethane, n-decane and eicosane as well as molecules involving polar forces, such as carbon monoxide and water. Later the approach has been extended (Donohue and Prausnitz (6) 1978, Gmehling and Prausnitz (7) 1979) to mixtures and highly polar substances such as methanol, ethanol, acetone and acetic acid by using a chemical theory. The BACK (Boublik-Alder-Chen-Kreglewski) equation proposed by Chen and Kreglewski (8) and Simnick, L i n and Chao (9) was based on the Boublik (10) equation of hard convex bodies. It has been success f u l l y applied to f l u i d s such as methane, neopentane and hydrogen s u l f i d e . In these formulations the a t t r a c t i v e contribution to the pressure was expressed i n a 24-parameter temperature-density double power series o r i g i n a l l y given by Alder, Young and Mark (11). This a t t r a c t i v e term was l a t e r modified to a 21-parameter or a 10parameter series. These series, however, are cumbersome to use and do not reveal the underlying physical basis. We propose here a t h e o r e t i c a l formulation of the a t t r a c t i v e contributions to the equations of state based on the d i s t r i b u t i o n function theories of l i q u i d s and to derive equations that are a p p l i cable to moderately anisotropic f l u i d s . F i r s t we use the Gaussian overlap potential of Berne and Pechukas (12) to represent aniso tropic forces. We note that a continuous potential (see Figure 1) can be approximated by an η-step p o t e n t i a l . In the l i m i t η—> oo , the o r i g i n a l p o t e n t i a l i s recovered. The v a l i d i t y of this repre sentation i s closely associated with the Riemann integration theory i n r e a l analysis. Secondly, we apply the cluster theories of c o r r e l a t i o n functions f o r the derivation of the a t t r a c t i v e pressure. The approach i s therefore d i f f e r e n t from the perturbation approach. These developments w i l l be presented i n Sections 2 and 3. E a r l i e r studies on r i g i d nonspherical molecules were based on the so-called scaled p a r t i c l e theory (13) (SPT). This theory made use of the fact that the chemical potential of an N-body system i s related to the energy required to create a cavity i n the f l u i d i n order to accommodate an additional p a r t i c l e . As developed by Reiss et a l . , (13) SPT contains certain approximations. Thus the results of the theory are not exact. Gibbons (14,15) f i r s t applied this theory to hard convex bodies and obtained an equation of state of the form P/pkT = l / ( l - y ) + 3 a y / ( l - y )

2

2

2

+ 3a y /U-y)

3

(1)

where y i s the packing f r a c t i o n , y=pb, b i s the volume of a single hard convex molecule, α i s a r a t i o of geometries of the HCB, α = rs/3b

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(2)

11.

KANCHANAKPAN ET AL.

229

Nonspherical Molecules

where r i s the mean radius of curvature, and s the surface area of the HCB molecule. For d i f f e r e n t shapes, the mean radius of curva ture i s given by d i f f e r e n t geometric formulas. As an example, for spherocylinders, the geometric factors are 2

3

r = R + L/4, s = 4πϊ1 +2πί&, and b = (4/3)πϊ1

2

-HrR L

(3)

where R i s the radius of the hemispheres and L the bond length be tween the centers of the hemispheres. For spheres, r = R, the radius of the hard sphere (HS). Equation 1 i s accurate at low densities but deteriorates at high densities. Thus i t must be improved. Boublik (10,16) proposed a modified form of (1) f o r prolate spherocylinders that reduces to the Camahan-Starling (17) equation at a=l:

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

P/pkT = l / ( l - y ) + 3 a y / ( l - y )

2

2

2

+ 3a y /(l-y)

Nezbeda (18) upon considering the v i r i a l an alternative resummation formula 2

2

3

2

3

-a y /(l-y)

3

(4)

c o e f f i c i e n t s , proposed

P/pkT = [ l + ( 3 a - 2 ) y + ( a + a - l ) y - a ( 5 a - 4 ) y ] / ( l - y ) 3

3

(5)

which, i n comparison with simulation r e s u l t s for spherocylinders, i s more accurate than Equation 4, especially at high densities. These equations have been developed for the specialized geom etry of spherocylinders. In l a t e r studies (Nezbeda and Boublik (19, 20) on fused diatomic hard spheres, i t was found that Equation 4 r e mains applicable i f one substitutes for the dumb-bell an equivalent spherocylinder that has the "neck" f i l l e d i n . Thus the fused spheres can also be described by the equations developed for hard convex bodies. A recent study [Wojcik and Gubbins (21)] showed that Equation 4 i s also applicable to mixtures of hard dumb-bells. Nez beda and Boublik (22) (see also Nezbeda, Smith and Boublik (23), Nezbeda, Pavlicek and Labik (24), Boublik (25)) c l a s s i f i e d hard con vex bodies into three major types (i) l i n e a r molecules (e.g. prolate spherocylinders, l i n e a r fused hard spheres and diamonds), ( i i ) disk l i k e molecules (e.g. oblate spherocylinders) and ( i i i ) cubes. They found that Equation 4, derived for prolate spherocylinders, i s accurate for l i n e a r molecules, and reasonable for disk molecules, but i s less satisfactory f o r cube-like molecule. We s h a l l adopt the Nezbeda Equation 5 here to describe the harsh repulsive forces i n r e a l molecules not of the cubic shape. Theoretical Developments We formulate our approach i n terms of a square-well (SW) p o t e n t i a l . The SW potential has been extensively studied (for a review, see Luks and Kozak (26) and references contained therein). This i s a simple p o t e n t i a l embodying the essential features of the i n t e r a c t i o n forces i n r e a l molecules, i . e . , the excluded volume and a t t r a c t i v e forces. For nonspherical molecules, the o r i e n t a t i o n a l variations of the pair interaction should also be accounted f o r . One class of angle-dependent potentials that can be generated from simple spheri cal ones i s the so-called Gaussian overlap model of Corner (27) and

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

230

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Berne and Pechukas (12). They b u i l t the molecular anisotropy into the potentials by incorporating angle dependence into the potential parameters. Therefore a Gaussian overlap model f o r the square-well (GOSW) potential i s , (See Figure 2) r <ά(ω ω )

u(12) =oo,

±

ά(ω ω )<

-εζογ^),

1

0,

(6)

2

r ^(ω^)

2

r > λίω^)

where the potential parameters, d, λ and ε are functions of r e l a t i v e orientation, i . e . the Euler angles and ω · We note that although Equation 6 i s a simple model, i t can be generalized to more elabo rate potentials by using multistep square wells. For example, the spherical Lennard-Jones potential depicted i n Figure 1 can be approximated by an η-step spherical square well p o t e n t i a l , p a r t i c u l a r l y as n~* oo . Thus by increasing the number of steps, one can approximate a variety of smooth potentials. The strengths of i n t e r action can also be approximated by the heights of the f l i g h t s (which y i e l d impulsive forces at the distances of d i s c o n t i n u i t i e s ) . The same procedure can be applied to angle dependent smooth potentials when the same angle dependence i s reproduced by the Gaussian overlap parameters. These multistep SW have been applied to simulate hydro gen bonding i n water (Dahl and Andersen (28)) and v i b r a t i n g dumb-bells (Chapela and Martinez-Casas (29)> Kincaid, S t e l l and Goldmark (30) have shown that with a p o s i t i v e shoulder at λ^, while setting a l l a t t r a c t i v e energies, ε-^=0, the f l u i d can support a c r i t i c a l point (see also Kreglewski (31)). For the purpose of t h i s study, we con sider a two-step SW p o t e n t i a l consisting of a p o s i t i v e shoulder and a negative w e l l (Figure 3).

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

2

u(12) = oo,

r < dtoy^)

ε (ω ω ), 1

1

(7)

dCcy^ r ^((γ^)

2

-ε (ω ω ), λ ^ ω ^ ω ^ r ^ ( ω ^ ) 2

χ

2

0,

r >

λ

ω

2

ω

^ 1 2^

The v i r i a l equation f o r anisotropic molecules i n s t a t i s t i c a l mechanics i s given by [Hansen and MacDonald (32)].

p/6J*dr

P/pkT = 1 -

4iTr

2

2

12

(8)

where the angular brackets <·> indicate indicate the angle average <...>

12

2

(4π)" / d 9

1

= 1/(4π) s i n 6 άφ 1

2 2

χ

Ιάω Jdo^

1

d6

2

άω d«

(...)

2

s i n 0 àj> 2

2

(9) (...)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

11.

Nonspherical Molecules

K A N C H A N A K P A N ET A L .

231

Figure 1. Approximation of the Lennard-Jones potential by a s i x step square-well p o t e n t i a l . Note that there are three repulsive shoulders and three a t t r a c t i v e wells following thereafter. The repulsive step acts e f f e c t i v e l y as a temperature dependent hard core dimension commonly used i n perturbation theories f o r soft repulsion.

d' d" d"

Or

!

Χ λ" X"

ι -" -€ 1 e

-e'

L _

...!

;û _ ο

r

-

Figure 2 . Schematic drawing of the Gaussin overlap anisotropic square-well p o t e n t i a l , u(r,θχ,Θ2,Φΐ2)» at three values of Θ2 ( 0 , π/6 and π / 2 ) . θ]_ i s kept at 0 . The dependence on the azimuthal angle, <j>i2» i s not shown. The parameters, d, ε, and λ are the Kihara hard core dimension, the a t t r a c t i v e energy and the range of a t t r a c t i v e i n t e r a c t i o n , respectively. The primed quantities are values at d i f f e r e n t angles.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

232

E Q U A T I O N S O F STATE. T H E O R I E S A N D A P P L I C A T I O N S

For nonlinear molecules, one should also integrate over the r o t a t i o n a l angles and normalize by (8π^)2. Note that the gradient of the pair p o t e n t i a l can be written i n terms of the Boltzmann factor, e(12) = exp[-$u(12)], as 33u(12)/9r

= -

(3e(12)/8r)exp[$u(12)]

(10)

However, due to the form of the two-step SW p o t e n t i a l , the derivative 8 e ( 1 2 ) / 3 r i s a sum of three Dirac deltas, one a r i s i n g from the hard core repulsion at dia^a^), the others from the steps at λχ(ωιω2) and λ2(ω^ω2) 9e(12)/8r

= 6 ( r , d ) e x p i - ^ ) + 6(r,X ) [ e x p ( 3 e ) exp(-3e )]

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

1

(11)

2

1

-6(r,X )[exp(3e )-l] 2

2

Substitution of Equation 10 and Equation 11 i n Equation 8 and change of r-integration with angle averaging gives P/pkT = 1 + (4π/6)ρ
3

y (d,o)l,u)2)> sw

3

+ (4π/6)ρ < l ( e λ

3

- (4π/6)ρ < 2 (

3 e 2

-e"

3 e l

)y

s w

inter

(12)

12

(Xl, l, 2)> W

W

1 2

3ε2

-1) (λ2,ω1,ω2)> sw where y i s the background d i s t r i b u t i o n function for the squarewell p o t e n t i a l defined as 6

Υ

± z

y (12) = ( 1 2 ) gw

8

β υ ( 1 2 )

(13)

β

The second term i s the repulsive contribution at the contact of hard cores. We note that for SW, the contact value, y ( d ) , i s not the same as that for hard spheres. In the following we s h a l l ana lyze the simulation data for spherical SW to determine the r e l a t i o n between y g and y^g. The t h i r d term i s due to the repulsive shoulder at λ^ω^α^)· This term actually plays the role of the temperaturedependent hard core dimension often used i n conventional perturba t i o n theories for soft p o t e n t i a l s . The fourth term i s the a t t r a c t i v e contribution. It can be treated as a mean f i e l d value i n case of continuous p o t e n t i a l s . To gain a better understanding, we analyze simulation data on spherical square wells next. g w

W

Contact Values of Correlation Functions for the SW Potential Henderson et a l . (33,34) have carried out computer simulations for the spherical one-step square well p o t e n t i a l with d i f f e r e n t widths, λ* = λ/d = 1.125 - 2.0, over wide ranges of state conditions. The p o t e n t i a l well i s a negative step, - ε, beginning at d and vanish ing at λ. Table 1 shows the contact values of the background corre l a t i o n functions for SW and HS, respectively. The hard spheres are taken to have the same c o l l i s i o n diameter, d, as that of the SW ( i . e . , at 3 ε = 0, the SW reduces to the HS p o t e n t i a l ) . We make the following observations :

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

11.

KANCHANAKPAN

233

Nonspherical Molecules

ET A L .

(i) The data c l e a r l y show that the Weeks-Chandler-Andersen zeroth order approximation

i s inadequate. In f a c t , i t i s true only i n the l i m i t of high tem peratures (or low d e n s i t i e s ) . ( i i ) For a given density, the simple empirical formula y ( d ) = y (d)exp(C3e) s w

(15)

HS

i s found to apply with surprising accuracy for wide ranges of well widths, (λ* from 1.125 to 2.0, see Table 1). The index ζ has the values of -1.18 at p* = pd =0.8, -1.16 at p* =0.6 and -1.1 at p* =0.4 (for the well width λ* = 1.5). Since at low densities, both y and y approach 1, ζ must approach zero at zero density. Figure 4 shows the variations of the index, ζ, with density. The contact value of yg^W (Note that the y function i s continuous at d) i s also dependent on the well width, λ. We have displayed the ζ values for cases λ* = 1.375, 1.5, 1.625 i n Figure 4. The three curves are f a i r l y consistent i n their density dependence, ζ i s around -1.0 at medium densities ( i . e . , near p* = 0.5). ( i i i ) The values of ygy(X) at the a t t r a c t i v e wall can be represented by SW ' Υ <λ)εχρ(ξ$ε) (16) 3

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

s w

H S

y

(X)

Ηδ

The index ξ has the values -0.6 at p* = 0.8, -0.525 at p* = 0.6 and -0.535 at p* = 0.5 (for λ/d = 1.5). Similar results are observed for other well widths (see Table 1). The value of ξ i s around -0.5 at high densities. Again, as ρ* -Κ), ξ must be zero. Thus ξ i s a function of p*. Equation 12 reduces, f o r an angle-independent one-step squarewell p o t e n t i a l , to 3

P/pkT = 1 + (47r/6)pd y (d)e

3e

sw

3

(17)

y^(X)

3e

-(4TT/6)pA (e -l)

The above observations can be u t i l i z e d to transcribe the pressure equation (for one-step SW) into 3

P/pkT = 1 + ( 4 ^ 6 ) p d y 3

-(4π/6)ρλ (

3ε β

-1) Υ

H S

Η 8

(d)e

(λ)β

( 1

^

) 3 e

(18)

ξ 3 ε

We i d e n t i f y the second term as related to the hard sphere pressure, written as 3

(W6)pd y (d) = Z H S

R S

-1 = Z'

HS

f

(19)

where Z i s the configurational part of the compressibility factor. To carry out the c a l c u l a t i o n , i t i s necessary to f i n d an expression for the value of yns(X) at the a t t r a c t i v e w a l l . From d i s t r i b u t i o n H S

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

234

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

€,(ω,,ω ) 2

λ (ω,,ω )

λ |(ω,,ω )

2

2

2

ω )

r->

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

2

-€ (ω,,ω ) 2

Figure

3.

parameters.

The two-step This

square-well

potential

is

used

1

potential in

the

1

2

with

present

angle

dependent

correlation.

I

2.5

- - - λ'= 1.375 — λ"= 1.5 λ*= 1.625

2.0

-

1.5 /

1.0

/ / / / //

//

I

0 ' )

I 0.5

I 1.0

I 1.5

Ρ* Figure

4.

square-well

The v a r i a t i o n o f potential.

index

Three

well

ζ with widths

the are

density

for

one-step

studied.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

11.

K A N C H A N A K P A N ET AL.

Nonspherical Molecules

235

function theories, we know that the background c o r r e l a t i o n function has the following cluster representation (35) yHS(X) = e ^

+

<Π

+

<^

The c l u s t e r i n t e g r a l

+

^ +

Ι

^

+

Ι

^ + °(P ) 3

(20)

for hard spheres i s known (36) 3

c/\> = (ir/12)ρ* (λ* -12λ* +16)

(21)

where p* = pd . For λ* = 1.5, οΛ> = 0.36 p*. Higher density terms of the cluster integrals have also been published (37). However, for s i m p l i c i t y we propose to approximate Equation 20 by 3

y (h) = e ^ [ l H S

+

f ] = e ^

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

where f i s a function of density. e

+ e<\

(22)

We next approximate f by

cA>. -a"p* f = a p* e ο

2

, . (23) 9 r

Obviously, other forms of density dependence could be introduced. We used simulation data (33) to determine the constants, a and a" and found Q

a

ο

- -0.34378 and a" = 1.2284

for the case λ* = 1.5. written as

(24)

Thus the complete pressure equation can be

,

P/pkT = 1+Ζ ε(1+ζ)βε

(25)

Η8

3

2

tt /^\^ r 0.36p* ^ . a"p* ] - (4π/6)ρλ (e -1) [e + a p* e ο 1 Λ

r

ξ£ε e

This equation i s used to calculate the pressure for SW f l u i d at p* = 0.8 and for temperatures $ε = 0.25 to 1.75 at w e l l width λ = 1.5d. The r e s u l t s are compared with simulation data i n Table I I . The agreement i s quite s a t i s f a c t o r y over the range of conditions com pared. For other densities, similar r e s u l t s are obtained. We there fore adopt the form Equation 25 as one of our working equations f o r the equation of state studies. Generalization of Equation 25 to angle-dependent GOSW p o t e n t i a l i s straightforward. For example, the one-step form (with negative ε) of Equation 12 i s P/pkT = 1 + (4π/6)ρ
3

3

e

- (4π/6)ρ < X ( e

3 e

y (d,o) u) )> sw

3 e

r

2

-1)7 (λ,ω ,ω )> δν

1

(26)

12

2

12

By mean value theorem, we may extract out the angle averages 3

P/pkT = 1 + (4π/6)ρ3 e

3 e

(27)

s w

3

- (4π/6)ρλ (

y (d)

3ε β

-l)Y

s w

â)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Table I. The Contact Values of Correlation Functions f o r the One-Step Spherical Square-Well Potential λ* = 1.5 3ε

+

y

(d ) SW

n,s«>«

tBe

(MC) 3

pd = 0.5 0.0 2.12 0.25 1.62 0.50 1.225 0.75 0.95 1.00 0.72

y

HS

( A ) e

ζ = -1.1 2.12 1.61 1.223 0.93 0.71

0.98 0.864 0.77 0.675 0.574

ξ = -0.535 0.98 0.857 0.75 0.66 0.574

pd = 0.6 0.0 2.56 0.25 1.92 0.50 1.41 0.75 1.02 1.00 0.80

ζ = -1.16 2.56 1.91 1.43 1.07 0.80

0.92 0.802 0.716 0.638 0.544

ξ = -.525 0.92 0.807 0.708 0.621 0.544

Pd = 0.8 0.0 3.97 0.25 3.01 0.50 2.21 0.75 1.67 1.00 1.20 1.25 0.90 1.50 0.72 1.75 0.58

ζ = -1.18 3.97 2.96 2.20 1.64 1.22 0.908 0.68 0.50

0.73 0.63 0.55 0.487 0.423 0.35 0.315 0.27

ξ = -0.6 0.73 0.63 0.54 0.47 0.40 0.35 0.3 0.26

3

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

(X)

*sw (MC)

3

λ* = 1.625 3

pd = 0.4 0.0 1.812// 0.333 1.30 0.5 1.10 0.80 0.85

ζ = -1.0

Pd = 0.6 0.0 2.56 0.333 1.78 0.50 1.44 1.02 0.80

15 ζ = -ι. 2.56 1.74 1.44 1.02

3

3

pd = 0.8 0.0 3.97 0.333 2.77 0.50 2.58 0.80 1.86 1.25 1.178

ζ = -0. 96 3.97 2.88 2.46 1.84 1.195

—

0.72 0.67 0.56

—

0.62 0.58 0.51 0.40

ξ = -0.538 (0.861)* 0.72 0.66 0.56 ξ = -0.478 (0.727) 0.62 0.57 0.50 0.40

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

11.

KANCHANAKPAN ET AL.

Nonspherical Molecules

237

λ* = 1.375 /Λ\

+

3ε

y (d )

ζ$ε

s w

ί\\ y

y

ο (MC) pd = 0.8 0.00 3.97 0.667 1.72 1.00 1.21 1.25 0.82

HS

ζ3ε

x

sw< > (MC)

( d ) e

J

ζ 3.97 = -1.19 1.795 1.21 0.897

0.65 0.52 0.46

ξ = -0.58 (0.942) 0.64 0.527 0.46

1.14 0.79 0.58

ξ = -0.684 (2.26) 1.138 0.81 0.575

1.11 1.02 0.92

ξ = -0.495 (1.2) 1.105 1.017 0.937

λ* = 1.125

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

3

pd = 0.8 0.0 3.97 1.0 1.58 1.5 1.02 2.0 0.66

ζ = -0.90 3.97 1.61 1.029 0.656

λ* = 2.00 3

pd = 0.8 0.0 0.1667 0.333 0.5

ζ = -0.67 3.97 3.55 3.175 2.84

3.97 3.65 3.17 2.95

*Values i n parentheses are estimated.

Table I I . The Pressure Prediction for One-Step Square-Well Potential λ* pd

= 1.5 3

=0.8

3ε

Ζ (MC)

0.25 0.50 0.75 1.00 1.25 1.50 1.75

Ζ From Equation

6.47 5.08 3.84 2.34 1.35 0.18 -0.65

6.34 5.09 3.86 2.65 1.44 0.22 -1.0

L

l

25

(From second v i r i a l ) /attractive -1.61 -3.67 -6.32 -9.71 -14.08 -19.69 -26.88

*We have used ζ = -1.18 and ξ = -0.6 for the calculations. #Ζ^ i s the part of second v i r i a l c o e f f i c i e n t due to a t t r a c t i v e contribution. i.e. Z

3

2

= -(4π/6)ρλ (ε

3ε

-1)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

238

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

This results i n temperature (and possibly density) dependent mean value parameters d=d(T), ε=ε(Τ) and λ=λ(Τ). We approximate the r e pulsive pressure by an equivalent hard convex body pressure accord ing to the relations Equation 15 and Equation 19 and the a t t r a c t i v e pressure according to Equation 25 p/pkT = ι ζ · +

Η

ε

Β

β< 3

1

+

^

ε

28

< >

βε

b

- (Απ/6) λ (β -1) [e 'P*

e e

Ρ

+ b

h

e "^

p*

Ο

Note the replacement of the hard sphere Z^ by the hard convex body Ζ . Ζ i s set to the Nezbeda form, Equation 5. The term i n volving D , b and b" i s a representation of the temperature and density dependence of y ^ ( A ) . Generalization to the two-step SW potential mentioned above i s less obvious. One should use simulation data (these being unavailable at the present time) to v e r i f y the correct parametrization. I f we i n s i s t on the same functional form as Equation 28 but treat the parameters as empirical constants, we obtain s

β

f

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

q

Ζ = 1 + Ζ·

Η ε Β

β-

( 1 + ζ ) β ε

1

ρν β 2 -β- ΐ)[β-ν * βε

+

(29) /Τ

βε

ι(

-ρν (^ 2-1)[β-ν * ε

/Τ

2

+

d y e"V

2 / T

o

ν

Λ

?

2

/

Ρ

*]

'

where and are, respectively, the repulsive volume of the shoulder at λ]_ and a t t r a c t i v e volume of the well at λ2· The con stants CJL and d^, i=0,l,2, are introduced to account for the state dependence of the c o r r e l a t i o n functions y „ at the locations of d i s c o n t i n u i t i e s . This form of functional dependence i s a r b i t r a r y . The form i s suggested by Equation 25. However, i t mimics the cluster form Equation 20. T* i s a reduce temperature, to be specified l a t e r . We note the s i m i l a r i t y of Equation 29 with those derived by Reijnhart (38,39) and Henderson and Chen (40) from d i f f e r e n t methods. Due to the temperature and density dependence of the mean value parameters ε^(Τ), λ^(Τ) and ygw^i^» combined the density and tempera ture terms i n the exponentials. Equation 29 gives the pressure expression for the two-step Gaussian overlap square well p o t e n t i a l . In the following we s h a l l discuss i t s applications. s

w

e

n

a

v

e

The P-v-T Behavior of Real Fluids We have applied Equation 29 and i t s modifications to the c a l c u l a t i o n of the pressures and enthalpies of r e a l f l u i d s , such as hydrocarbons (e.g. methane, ethane, n-decane, and eicosane), nonhydrοcarbons (e.g. carbon dioxide, and hydrogen s u l f i d e ) , alcohols (1-butanol and phenol), ketones (acetone and 2-butanone), amines (methylamine and diethylamine), ethers (dimethylether and diphenylether) and polar solvents (e.g. tetrahydrofuran). Some 69 substances have been i n vestigated. The average conformation of a polyatomic molecule i n the f l u i d state i s characterized by a geometric r a t i o , a. We do not assign a

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

11.

KANCHANAKPAN

Nonspherical Molecules

ET A L .

239

s p e c i f i c "shape" to the molecules since conformational changes (e.g. bond rotation and solvent forces) constantly "deform" the molecule, α should r e f l e c t the averaged effects of such changes. Other molecu lar parameters for use i n Equation 29 are b, the volume of a single molecule, ε-^ and ε , energies of interaction, and λ , the widths of potential walls, and the constants c^ and d^ (i=0,l,2) specifying the temperature and density dependence. However, i n practice, i t soon becomes apparent that the repulsive second v i r i a l , (l+3a)b, given by Z » i s inaccurate, as pointed out by Kohler and Haar (41). To obtain an adequate description of the second v i r i a l c o e f f i c i e n t of gases, we modify Equation 29 to give t n e

2

2

H c B

Ζ = 1 + Ζ·

HLD

( 1 + ζ ) β ε

β"

1 -(1 3α) β+

( 1 + ζ ) ί ? ε

Υ

e e

1 + yB*e- l r

Be

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

+yV*(e 2 - e - ^ l M e - V ^ * ^ y £

-yV*(/ 2 - 1 ) [ β - ^

/ Τ

* c y +

o

e ^ '

e ^ '

1

1

(30)

* ]

* ]

i.e. we have taken out the HCB second v i r i a l , (l+3a)b exp[-(l+ζ)fte^], and substituted for i t the repulsive term Β εχρ(-3ε ) Γ

(31)

1

Β being an e f f e c t i v e repulsive second v i r i a l c o e f f i c i e n t . (Note, B£* = Br/b, V]_* = Vi/b, and V * = V /b) Equation 30 automatically reduces to a second v i r i a l expression, i . e . P/kT = ρ + B p with 2

2

2

B

2

[εχρ($ε ) - e x p t - f ^ ) ] - ν [ β χ ρ ( β ε -1)]

= Β εχρ[-3ε ] + V Γ

1

±

2

2

2

(32)

At the moment, we have thirteen parameters to use i n the equa t i o n of state. This number of parameters i s considered excessive i n p r a c t i c a l calculations, especially when some of the parameters (e.g. the C j / s and d^'s) are empirical and have no physical meaning. They could have been evaluated i f simulation data for the 2-step SW f l u i d were available. In the absence of information we embark on a re duction of parameters by setting e

d

±

(33)

i-°

= c ,

i «

±

T* - kT/e We have also found from f i t t i n g c r i t i c a l temperature by e /k z n

(34)

0,1,2

(35)

2

the data that e /k i s related to the 2

« Τ 12.18601 c

(36)

and V

= 1.4 b (Kreglewski 1984)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(37)

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

240

We define an a t t r a c t i v e volume (between V or

= V a

V

0

2

9

2

and λ )»

as

2

- V1

= 1.4b

T

(38)

+ V a

(39)

The above prescription has reduced the number of parameters from 13 to 7, i . e . , B , V , Cj[ (i=0,l,2), α and b. Therefore the f i n a l working equation i s given by r

a

3

2

2

3

Ζ = (l-y)" [l+(3a-2)y+(a +a-l)y -a(5a-4)y ] -(l+3a)y +pB - V Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

P

a

(e

1 / T A

r

(40)

2

C

-l)[e- l

y / T

*

+

e

V "

2

7

/T

*

]

where y=bp, V i s given by Equation 38 and T* by Equation 35 and Equation 36. There i s no compelling reason for the choices made i n Equation 33 to Equation 39 except that we have achieved economy of parameters. Future research s h a l l address such questions i n more d e t a i l , prefer ably based on detailed simulation data. We f i r s t used the P-v-T data of p a r a f f i n hydrocarbons from C^ to C Q (methane to eicosane) to determine the parameters. (N.B. We have excluded the c r i t i c a l region i n the f i t . ) As the c r i t i c a l behavior i s nonanalytical, we do not expect Equation 40 to apply. Normally, we are +2°C away from the c r i t i c a l temperature. The results are l i s t e d i n Table I I I . These values are empirically determined. How ever, they are not e n t i r e l y a r b i t r a r y . The molecular volumes, b, thus determined are d i r e c t l y proportional to the c r i t i c a l and/or van der Waals volumes (Bondi (42)) for hydrocarbons. (See Figure 5). The energy parameter, ε/k, i s proportional to the c r i t i c a l tempera ture as mentioned above. (ε = ε ) . The agreement i n density predictions for twenty hydrocarbons i s around 1%, and i n vapor pressures, for the most part, less than 1%. We are also able to predict the configurational enthalpy to within 1 cal/g. (See Table V). This equation i s p a r t i c u l a r l y e f f e c t i v e i n the l i q u i d state, e.g., for l i q u i d ethane and n-pentane up to 69 MPa. In a number of perturbation formulations, the contribution from the a t t r a c t i v e second v i r i a l c o e f f i c i e n t overwhelmed other contributions at low temperatures and l i q u i d densities, thus requiring a high order density correction (up to the sixth v i r i a l i n , e.g., Bienkowski and Chao 1975). As an example, Table II exhibits the values of the a t t r a c t i v e second v i r i a l i n SW f l u i d . It reaches -26.9 at 3ε = 1.75. Use of the exponential form Equation 25 has moderated such influences and obviated the necessity of retaining e x p l i c i t l y high order density terms. The same behavior i s obtained by Equation 40. We have made calculations with other equations of state, such as Peng-Robinson (43) (PR), Mohanty-Davis (44) (MD), De Santis, Gironi and M a r r e l l i (45) (DGM) and the BACK (Simnick et a l . (1*46,47,48)) equation. The results based on the same data sets for density and vapor pressure are compared i n Table IV from methane to n-decane. Equation 40 i s uniformly better than a l l the equations compared. PR equation i s reasonable for vapor pressure calculations (e.g., 1.5% a

2

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Nitrogen o-Xylene Bicyclohexyl Fluorine Diphenylmethane Dichlorod i f1uo rorae t hane Carbon d i o x i d e Ammonia Methyl f l u o r i d e Butanol Phenol p-Cresol

H S

Methane Ethane Propane n-Butane n-Pentane n-llexane n-lleptane n-Octane n-Nonane n-l)ecane n-Llndecane n-dodecane n-Trideeane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane Eicosane Ethylene Propylene Lsobutane isopentane Benzene Toluene

16..325 23..629 31..795 39.,831 49.,330 56.,797 66.,381 74.,956 84.,341 94..133 101., 72 113.,43 123.,38 132.,02 141.,11 150.,65 160.,94 169.,83 180.,02 193.,74 21.,170 27.,870 40. 237 48.,689 36.,117 44.,494 14. 722 14.,762 57.,6142 97. 3920 83.,0885 83.,7528 30.,9599 11.,0023 10.,7183 17.,9824 47.,9597 41..6151 57..2452

2,.20215 1,.49024 1,.93017 1,.13685 1,.51368 1.1934

b,cc/gmol

4 .8590 61.0547 7.8860 6..0482 81.0264 6..2618

4.4186 5.4571 6.3561 6.4804 6.8402 7.3511 7.3663 8.3639 8.3276 8.6238 8.7832 9.2483 9.3391 10. 044 10. 651 11. 270 11. 201 11. 844 11. 983 12. 457 5.3742 5.9988 6.1462 6.9562 6.3696 8.0161 5.9335 5.2785 7.8234 8.7008 12 .4446 8 .6256 7.0824

r

B /b

20..8229 28..7928 31..3100 15..6550 18,.4726 32,.1420

20,.036 24,.213 24,.277 25,.995 27,.438 28,.842 29,.549 32,.183 33,.405 34,.592 35..242 37,.098 37..462 40..291 42.,723 45.,207 44.,931 47..512 48.,067 49..967 22.,722 26.,160 27.,186 27.,182 26.,729 32..300 25.,050 21.,174 28.,4512 33.,7047 39.,2337 33.,5330 28..4097

Vjb Q

4.8934 3.8722 3.9492 3.2727 3.1471 2.7713

3.,4908 3.5874 3.6030 3.6669 3.6114 3.7117 3.6766 3.7150 3.6863 3.6627 3.7154 3.5195 3.6206 3.6004 3.5717 3.5873 3.5585 3.5761 3.5600 3.4807 3.3543 3.7196 3.6441 3.5997 4.6586 4.0585 3.6959 3.4955 3.6678 3.3930 4.0983 3.6182 4.0143

C

9.5355 4.8739 6.0694 2.4495 3.1105 2.8097

4.,0623 4.,7173 4.,3511 4.,3736 4.,1813 3.,6129 3.9829 4.3176 4.1272 4.0298 3.9781 3.6788 3.7317 3.9976 4.1281 4.3009 4.0536 4.3169 4.1916 4.1385 4.0591 4.8769 4.6742 4.3044 5.7104 5.7809 4.9114 3.7681 4.2050 3.8834 6.1894 4.0255 5.6238

2

10784.0 -85.944 -48.39 1429.4 4996.2 5869.6

0 .0 0 .0 0 .0 0 .0 0 .0 0,.0 0,.0 0,.0 0,.0 0,.0 0,.0 0,.0 0,.0 0..0 0,.0 0..0 0..0 0..0 0..0 0..0 0..0 0..0 0.,0 0..0 0..0 0..0 0.,0 0.,0 0.,0 0..0 0..0 0..0 0.,0

2

D/b , Κ

Continued on next page

14..153 17..299 15..114 1..0914 4..5436 17..998

11 .652 14,.840 11,.715 12,.950 13,.377 8,.2301 13..768 14..547 15,.303 15..471 15,. 539 15..069 15..123 16..189 16.,733 17..477 16..590 17..528 16..961 16.,808 11..930 14.,069 15.,352 13.,145 13.,991 15.,230 12.,920 13..016 12.,6131 12.,396 16,,972 13..566 13..830

C

The M o l e c u l a r P a r a m e t e r s * t o Be Used i n t h e E q u a t i o n o f S t a t e , E q u a t i o n AO

I,.02008 .1,.10299 1,.19523 1..33117 1,.37924 1..46968 1..56411 1..62814 1,.73361 1,.79693 1..88289 1,.96885 2..05458 2,.14076 2..21226 2..29972 2..38626 2..46985 2.,57054 2.,65651 1.,05120 1.,40083 1.,33986 1..33484 2..05852 1..95468 1..36413 1..10870 1..38407 1.,89254 1..66415 1.,82590 1..69801

a

T a b i c 111.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

^

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

α

where ε/k = ε/k +

Z

-. b T

r

7.5929 7.5816 7.5123 8.6534 5.7769 8.4338 7.0549 5.4309 6.5979 7.7616 4.9072 10.8006 5.8656 6.9561 6.9604 5.1196 8.6346 7.1343 9.0103 8.2691 8.9344 12.2423 0.8686 8.1989 6.5736 5.9428 7.8006 7.3413 20.0423 9.3165

B /b V /b a 30,.4572 30,.4120 30.1339 35,.7104 23..1723 26,.3432 28..2993 23,.0285 30..7533 32..3502 30..2940 44.,8423 23..5286 27..7942 27.,8841 29.,1512 26.,4511 31.,8251 30.,3674 27.,9398 35.,8262 35..3462 22..2851 30..7596 29..3502 27..3692 28..3144 26..9661 49..7294 32..3696

Continued c ο 2,.8199 3,.3792 4,.0656 4,.3692 4,.7482 3,.4796 4,.2381 3,.4387 3..9140 4..1240 3..4141 4.,3707 3..4105 3.,6289 3.,6215 3.,7969 3.,1887 4.,1165 3.,1100 4.,0630 3..6023 4..0133 4.,2058 4..5221 4..1849 3..4524 4.0240 3..7304 4,.3846 4..3589

c r i t i c a l temperature by ε/k == Τ /2..18601 c

57.2771 52.8778 50.5423 27.3953 32.2047 49.5422 24.0541 36.8069 40.9721 47.4490 86.2730 22.4067 23.5104 29.5128 28.1500 45.4039 49.2220 35.3380 57.4625 56.8107 99.2549 77.4694 20.4789 26.2166 27.5070 43.4649 54.8143 34.3903 68.9898 66.1668

b,cc/gmol

i s c a l c u l a t e d as T* = kT/ε

ε, i s r e l a t e d to the

1..01932 1,.45342 1,.51523 2,.41148 1..30404 1,.34478 1..87662 1,.20617 1..86761 1..93416 1.,54833 2.,77183 1..08011 1.,64228 1.,44901 1.,76156 1.,00785 2.,21624 1..35605 1..65480 2.,03042 1..85980 1..08191 2..16017 1.,85038 1..28571 1..38088 1..04294 1..56977 2..40006

Note that the reduced temperature

*The parameter,

o-Cresol m-Cresol 2,3-Xylenol Acetone 2-Butanone 2-Pentanone Dimethyl ether Methylethyl ether D i e t h y l ether E t h y l p r o p y l ether Diphenyl ether Acetic acid Methylamine Dimethylamine Ethylamine Diethylamine Aniline Pyridine 4-Methyl p y r i d i n e i-Quinoline Carbazol Acridine Formamide E t h y l mercaptan Dimethyl s u l f i d e Tetrahydro thiophane Thianaphthene Dibenzo-thiophene Tetrahydrofuran Dibenzofuran

Table I I I .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

C

l 14 .249 14 .595 19 .978 16,.110 16,.2711 9 .8353 13,.7299 12,.7078 16,.7066 16,.6954 20..4675 18,.4607 13..0372 12..4523 13..4346 19..1798 9..2086 14..8389 10..8422 11..4332 14..6698 10,.1169 16..0239 13,.4265 16..0913 15..0371 13,.9649 10..8115 19,.4136 9,.3444 C

2 2,.7697 3,.7238 5,.9300 7..2340 5..3465 3..7188 5..7141 3,.2635 5..2008 6..5101 3.,1343 8.,1122 2..8898 3.,5060 3.,8865 4.,1423 2.,8025 4.,7775 3.,7898 4.,7670 4.,1678 4.,6394 3..2574 7..6142 6.,8876 3.,4758 4..9626 4..5357 8..6362 5..3389

2

2181.6 8209. 19978.2 1008.9 5249.2 107.4 268.2 3045.3 1138.7 2680.5 255.94 -798.44 917.3 150.6 177.4 490.4 -2274.8 -5068.1 633.56 -3145.9 144.4 -15394.1 -13245.1 410.9 2469.5 -4452.9 1430.3 -28998.1 -8.72: -17853.6

D/b , Κ

11.

KANCHANAKPAN

I

! I

I I I I I

1

I I I

Q20

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

243

Nonspherical Molecules

ETAL.

o~

0J5 Β ^

010

-û

0.05

1

0

I 1 1

Q2

04

1

1 I

0.6

I

I I 1

Q8

|.o

Figure 5. Variations of the hard core volume, b, with the c r i t i c a l volume V , for n-paraffins.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

244

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

for methane and 3% for ethane). However, Equation 40 gives about half the deviations for a l l hydrocarbons studied (1% for methane and 0.7% for ethane). The BACK equation performs well for lower hydro carbons (e.g. from methane to n-hexane), but deteriorates for higher hydrocarbons (e.g., to 14% i n deviations for density and 6.7% for vapor pressure of n-decane). The vapor pressures predicted by BACK i s of the order of 4% as compared to 1% by Equation 40. We note that the PR equation used i s i n i t s generalized form, which has some ad vantages i n application. In contrast to previous equations, such as PHC and BACK, the present equation i s based on d i s t r i b u t i o n function theory. Thus further improvements can be made by resummation of cluster integrals and/or inclusion of i n t e g r a l equation r e l a t i o n s . The contributions from i n d i v i d u a l terms can also be checked against simulation data. The same equation i s applied to other nonpolar substances i n cluding branched chain hydrocarbons, unsaturated hydercarbons, and ring compounds. The parameters are presented i n Table III and the results are summarized i n Table V. The o v e r a l l deviations for 607 data points of density, 435 points of vapor pressure and 147 points of enthalpy are 0.7%, 0.6% and 3 J/g, respectively. We have shown some sample compounds here, the detailed comparison can be found i n a thesis (49). In order to apply the equation to polar f l u i d s , i t i s necessary to use a mean potential form of the energy parameter. It i s given by

2

ε/k = ε/k + D/(Tb )

(41)

where one additional parameter, D, i s introduced. This formulation i s designed to introduce the temperature dependence i n the mean value parameter ε(Τ). It accounts i n part for the dipolar moment c o n t r i butions i n polar f l u i d s . The temperature i s now reduced according to T* = kT/ε. For example, for ammonia D/b = -85.94 K. Some t h i r t y - s i x amines, ketones, alcohols and ethers have been chosen for correlation. The results are given i n Tables III and V. For over a thousand density data and a thousand vapor pressure data, the o v e r a l l deviation i s 1.2% and 0.4%, respectively. Further comparison i s made with the Peng-Robinson and BACK equation i n Table VI. The results again show that Equation 40 i s more accurate based on the same set of experimental data. We have shown here that, by empirically f i t t i n g the two-step square-well equation to experimental data, we are able to reproduce quantitatively the P-v-T information. The given formula can be generalized to highly polar molecules according to known s t a t i s t i c a l mechanics. To treat strong molecular anisotropy, i t w i l l be neces sary to l e t b and λ depend on temperature. We also anticipate further developments of the present formulation to mixtures i n the future. 2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Ρ VP

Ρ VP

Ρ VP

Ρ VP

Ρ VP

Ρ VP

Ρ VP

Ρ VP

Ρ VP

Ethane

Propane

n-Butane

n-Pentane

n-Hexane

n-Heptane

n-Octane

n-Nonane

n-Decane

Methane

Type Ρ VP

3..7 4.,11 3..3 1..35 1.,6 4..92 1..5 1..1 4..1 3,.73 3,.39 4,.05 4,.7 2,.09

0.,56 0.,80 1.,18 0.,67 0.,34 0.,42 0..77 0.,77 0..72 0..8 0..67 0..05 0,.47 0..04

0.1-48.2 0.001-3 0.1-68.9 0.001-3.4 0.1-27.4 0.0001-3 0.1-21.2 0.00006-2.4 0.1-1.6 0.00001-2.4 0-50 0.00001-0.2 1.4-41.3 0.003-0.2

144-494 194-411 166.5-511 223-470 177.6-433 220-508 205-511 231-531.6 216.5-538.7 230-566.5 253-573 245.7-452.6 311-511 350-477

40 52

39 45

41 53

41 40

48 56

43 18

32 18

— — — —

-

8.7 8.75

-. 2.15

3.1

—

4.6

14.3 6.7

— —

6.0 3.6

2.6 3.4

1.0 3.6

3.6

—

1.9 3.9

1.2 5.1

1.1 4.4

2.0 4.6

BACK 0.4 1.07

9.1

—

5.9

—

7.2

—

32.0

—

DGM 8.7

5.9

3.5 3.85

7.0 5.28

2.7 6.11

2.8 6.12

9.9 7.7

4..3 3..1

0..82 0..79

4.5 5.8

5,.6 3..0

0..95 0..62

0.1-73 0.00001-4.26

46 46 90-398 144-494

MD 3.3 2.3

PR 5..4 1..5

Error AAD% Equation 40 0..93 0..99

134 55

Press. Range, MPa 0.9-14.3 0.1-4.6 0.9-68.9 0.003-4.9

Temp. Range, Κ 115-623 112-190.6 133-427.6 139-305

No. Data 39 32

Table IV. Comparison of Density (p) and Vapor Pressure (VP) Calculations Obtained from the Peng-Robinson (PR), Mohanty-Davis (MD), De Santis-Gironi-Marrelli (DGM), BACK Equations of State and Equation 40

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

246

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

T a b l e V. P-v-T and Thermal P r o p e r t i e s C a l c u l a t e d by E q u a t i o n 40 f o r Some S e l e c t e d Compounds Type n-Hexadecane

Ρ

VP Eicosane

Ρ

VP Ethylene

Ρ

VP Η

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

Benzene

Ρ

VP Carbon dioxide

Ammonia

Ρ

VP Η Ρ

VP Phenol

Ρ

VP Acetone

Ρ

VP Acetic

acid

Ρ

VP Diethyl amine Aniline

Ρ

VP Ρ

VP Pyridine

Ρ

VP TetrahydroΡ f uran VP *AAD i s d e f i n e d as

No. Data 9 10

Temp. Range, Κ 463-543 463-5553

P r e s s . Range, MPa 0.007-0.07 0.007-0.09

E r r o r , AAD* 0..17% 0. 17%

25 15

373-573 473-623

5.0-50.0 0.002-0.11

0. 98% 0.,42%

40 35 34

116.5-400 133-412 188.7-400

0.1-13.8 0.69-5.1 0.7-1.4

0.,91% 0.,94% 4. 7 J / g

102 67

280-545 280-562.6

0.005-4.0 0.005-4.9

0.,74% 0.,35%

243-413.2 216.5-302.6 243-413.2

1.5-38 0.5-7.1 3-50.6

0.,39% 0.,81% 4..1 J / g

238-598 221-403.6

0.09-81.4 0.04-11

2.,37% 0.,08%

14 15

323-673 323-694

0.0003-5.1 0.0003-6.1

0.,78% 1.,33%

28 20

273.2-503 329-508

0.1-4.5 0.1-4.8

0.,62% 0.,24%

21 23

392-583 392-595

0.1-5 0.1-5.8

0..52% 0..36%

17 18

329-483 329-493

0.1-3 0.1-3.5

0..50% 0,.80%

29 34

273.2-673 273.2-648.6

0-4 0-3

0..44% 0..76%

26 25

253-613 253-613

0.0001-5.3 0.0001-5.2

0,.76% 0,.88%

10 25

253-333 253-540

0.002-0.08 0.002-5.2

0,.27% 0..32%

39 33 39 300 172

η (1/n)

V

[V . .-V _ .]/V „ . calc.i expt,i expt,i

i=l For d e n s i t y (p) and vapor p r e s s u r e (VP) p r e d i c t i o n s , the p e r c e n t ( x l 0 0 % ) AAD i s used. For e n t h a l p y (H) p r e d i c t i o n s , t h e a b s o l u t e v a l u e i n J o u l e / g i s used. The s o u r c e s o f e x p e r i m e n t a l d a t a used here a r e l i s t e d i n r e f e r e n c e 49.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.

KANCHANAKPAN ET AL.

Nonspherical Molecules

Table VI. Comparison of Equation 40 with the BACK and Peng-Robinson Equations of State Component

Property

Number of Points

AAD

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

Equation 40

BACK

PR

Ethylene

RHO VP ENTH

40 35 34

0.91 0.94 2.00

3.19 5.22 2.77

Propylene

RHO VP

57 28

1.28 0.80

2.31 3.72

i-Butane

RHO VP ENTH

116 82 73

0.79 0.75 1.34

3.77 2.39 1.67

i-Pentane

RHO VP

38 22

0.84 0.29

3.01 0.90

Benzene

RHO VP

102 67

0.74 0.35

1.79 3.16

3.35 1.21

Toluene

RHO VP

13 33

0.31 0.89

1.82 5.24

0.54 2.31

o-Xylene

RHO VP

59 43

0.37 0.25

1.17 5.27

Dichloro difluoro methane

RHO VP ENTH

182 40 40

0.47 0.53 0.26

1.69 2.50 1.36

Carbon dioxide

RHO VP ENTH

39 33 39

0.39 0.81 1.76

1.89 0.61 3.09

Nitrogen

RHO VP ENTH

38 19 77

0.79 0.69 1.33

0.30 0.33 2.36

4.32 3.26 1.21

Hydrogen sulfide

RHO VP

40 24

0.53 0.39

0.74 2.12

3.69 4.23

Methyl fluoride

RHO VP

131 28

0.74 0.54

4.14 18.1

Phenol

RHO VP

14 15

0.78 1.33

10.5 6.11

Acetone

RHO VP

46 20

0.62 0.24

13.1 0.99

Dimethyl ether

RHO VP

34 19

2.90 0.26

3.77 3.46

Methyl ethyl ether

RHO VP

28 18

3.25 0.23

6.51 2.27

Diethyl ether

RHO VP

43 23

1.85 0.35

5.11 0.95

Aniline

RHO VP

29 34

0.44 0*76.

2.98 8.09

Library

1155 16th St., N.W. Washington, D.C. 20036

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

248

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

Kohler, F.; Quirke, N. J. Chem. Phys. 1979, 71, 4128. Boublik, T. Fluid Phase Equil. 1979, 3, 85. Tildesley, D. Mol. Phys. 1980, 41, 341. Chung, T. H.; Khan, M. M.; Lee, L. L.; Starling, Κ. E. Fluid Phase Equil. 1984, 17, 351. Beret, S.; Prausnitz, J. M. AIChE J. 1975, 21, 1123. Donohue, M. D.; Prausnitz, J. M. AIChE J. 1978, 24, 849. Gmehling, J . ; Liu, D. D.; Prausnitz, J. M. Chem. Eng. Sci. 1979, 34, 951. Chen, S. S.; Kreglewski, A. Ber. Bunsengesellschaft Phys. Chem. 1977, 81, 1048. Simnick, J. J . ; Lin, H. M.; Chao, K. C. Adv. Chem. Series 1979, 182, 209. Boublik, T. J. Chem. Phys. 1975, 63, 4084. Alder, B. J . ; Young, D. Α.; Mark, M. A. J. Chem. Phys. 1972, 56, 3013. Berne, B. J . ; Pechukas, P. J. Chem. Phys. 1972, 56, 4213. Reiss, H.; Frisch, J. L.; Lebowitz, J. L. J. Chem. Phys. 1959, 31, 369. Gibbons, R. M. Mol. Phys. 1969, 17, 81. Gibbons, R. M. Mol. Phys. 1970, 18, 809. Boublik, T. Mol. Phys. 1974, 27, 1415. Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. Nezbeda, I. Chem. Phys. Lett. 1976, 41, 55. Nezbeda, I.; Boublik, T. Chem. Phys. Lett. 1977a, 46, 315. Neabeda, I; Boublik, T. Czech. J. Phys. 1977b, B27, 1071. Wojcik, M.; Gubbins, K. E. Mol. Phys. 1983, 49, 1401. Nezbeda, I.; Boublik, T. Mol. Phys. 1984, 51, 1443. Nezbeda, I.; Smith, W. B.; Boublik, T. Mol. Phys. 1979, 37, 985. Nezbeda, I.; Pavlicek, J . ; Labik, S. Coll. Czech. Chem. Commun. 1979, 44, 3555. Boublik, T. Equation of State of Nonspherical Hard Body Systems in "Molecular-Based Study of Fluids"; Haile, J. M.; Mansoori, G. Α., Eds.; ADVANCES IN CHEMISTRY SERIES No. 204, American Chemical Society, Washington, D.C., 1983. Luks, K. D.; Kozak, J. J. Adv. Chem. Phys. 1978, 37, 139. Corner, J. Proc. Roy. Soc. (London) 1948, A192, 275. Dahl, L. W.; Andersen, H. C. J. Chem. Phys. 1983, 78, 1980. Chapela, G. Α.; Matinez-Casas, S. E. Mol. Phys. 1983, 50, 129. Kincaid, J. M.; Stell, G; Goldmark, E. J. Chem. Phys. 1976, 65, 2176. Kreglewski, A. "Equilibrium Properties of Fluids and Fluid Mixtures", Texas A&M University Press, College Station, Texas, 1984. Hansen, J.-P.; McDonald, I. R. "Theory of Simple Liquids", Academic, New York, 1976. Henderson, D.; Madden, W. G.; Fitts, D. D. J. Chem. Phys. 1976, 64, 5026. Henderson, D.; Scalise, O. H.; Smith, W. R. J. Chem. Phys. 1980, 72, 2431. Watts, R. O.; McGee, I. J. "Liquid State Chemical Physics", Wiley, New York, 1976.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

11. KANCHANAKPAN ET AL.

36. 37. 38. 39. 40. 41. 42. 43.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch011

44. 45. 46. 47. 48. 49.

Nonspherical Molecules

249

Kirkwood, J. G. J. Chem. Phys. 1935, 3, 300. Nijboer, B. R. Α.; van Hove, L. Phys. Rev. 1952, 85, 777. Reijnhart, R. Physica 1976, 83A, 533-547. Peters, C. J.; van der Kooi, J. J . ; Reijnhart, R.; Diepen, G. A. M. Physica 1979, 98A, 245. Henderson, D; Chen, M. J. Chem. Phys. 1975, 16, 2042. Kohler, F; Haar, L. J. Chem. Phys. 1981, 75, 388. Bondi, A. Physical Properties of Molecular Crystals, Liquids and Glasses (Wiley, New York) 1968. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59. Mohanty, K. D.; Davis, H. T. AIChE J. 1979, 25, 701. DeSantis, R,; Gironi, F.; Marrelli, L. Ind. Eng. Chem. Fundamen. 1976, 15, 183. Kreglewski, A; Chen, S. S. J. De Chim. Phys. 1978, 75, 347. Bienkowski, D. R.; Chao, K. C. J. Chem. Phys. 1975a, 62, 615. Bienkowski, D. R.; Chao, K. C. Fourth International Conference of Chemical Thermodynamics, Montpelier, 1975b, p. 28. Kanchanakpan, S. B. Ph.D. Thesis, "Molecular Theoretical Equation of State for Applications to Selected Pure Liquids and Dense Fluids of Natural Gas, Petroleum Hydrocarbons and Coal Chemicals", University of Oklahoma, Norman, Oklahoma, 1984.

RECEIVED November 8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

12

Application of a New Local Composition Model in the Solution Thermodynamics of Polar and Nonpolar Fluids 1

2

M. H. Li, F. T. H. Chung , C.-K. So , L. L. Lee, and Κ. E. Starling

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, OK 73019

A local composition model in solution thermodynamics is developed for the calculation of vapor-liquid equilibria of mixtures of molecules vastly different in size, polarity and strength of interaction. The concepts of nearest neighbor numbers, coordination shell, and pair interaction energies are interpreted in terms of modern liquid theory. For broad ranged applications, an accurate equation of state is introduced in the spirit of the mean density approximation by differentiation of the Helmholtz free energy. Calculations of the vapor-liquid equilibria of 83 binary and ternary systems, including nonpolar hydrocarbons, hydrogen-bonding alcohols, water, ammonia, and carbon dioxide show good agreement with experimental data. S i n c e i t s i n t r o d u c t i o n , the l o c a l c o m p o s i t i o n model (LCM) f o r the e x c e s s f r e e energy has been u s e d i n s o l u t i o n thermodynamics f o r the c a l c u l a t i o n o f v a p o r l i q u i d e q u i l i b r i a of h i g h l y n o n i d e a l m i x t u r e s . The systems i n c l u d e d p o l a r f l u i d s ( s u c h as a l c o h o l s , water and a c e t o n e ) (1-3) . The f o r m u l a t i o n was i n a form s u i t a b l e f o r c a l c u l a t i o n of a c t i v i t y c o e f f i c i e n t s . However, the o r i g i n a l method was n o t u s e f u l f o r the c a l c u l a t i o n of d e n s i t i e s . I t a l s o d i d n o t a p p l y to t h e near c r i t i c a l r e g i o n . On t h e o t h e r hand, m i x t u r e 'Current address: NIPER, PO Box 2128, Bartlesville, OK 74005 Current address: Aspen Tech, 251 Vassar St., Cambridge, MA 02139

2

0097-6156/86/0300-0250S08.75/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

12.

LI ET A L .

New Local Composition Model of Fluids

251

p r o p e r t i e s have been p r e d i c t e d by u s i n g e q u a t i o n s o f s t a t e w i t h m i x i n g r u l e s f o r the e q u a t i o n p a r a m e t e r s ( 4 ) . The p r e d i c t e d p r o p e r t i e s i n c l u d e d d e n s i t y , e n t h a l p y and v a p o r - l i q u i d e q u i l i b r i u m (VLE). The o v e r l a p p i n g a p p l i c a t i o n s o f t h e s e methods have c o e x i s t e d f o r many y e a r s w h i l e t h e two a p p r o a c h e s remained s e p a r a t e . Only r e c e n t l y have t h e r e been e f f o r t s t o a s s i m i l a t e the u s e f u l f e a t u r e s o f the l o c a l c o m p o s i t i o n a c t i v i t y c o e f f i c i e n t model i n t o e q u a t i o n s o f state(5.). The t h e o r e t i c a l groundwork c o n n e c t i n g b o t h a p p r o a c h e s was l a i d by L e e , et a l . (6) where a m o l e c u l a r t h e o r y was e s t a b l i s h e d f o r the LCM. In t h i s work, we d e m o n s t r a t e the f e a s i b i l i t y of the a p p r o a c h by c a l c u l a t i n g f o r the v a p o r - l i q u i d e q u i l i b r i a as w e l l as the d e n s i t i e s o f h i g h l y n o n i d e a l m i x t u r e s . We f i n d t h a t l o c a l c o m p o s i t i o n m i x i n g r u l e s p e r f o r m b e t t e r than c o n f o r m a i s o l u t i o n m i x i n g r u l e s f o r most of the systems studied.

Thfvry New F r e e E n e r g y M o d e l . The c o n v e n t i o n a l p r a c t i c e i n s o l u t i o n thermodynamics i s t o employ e q u a t i o n s o f s t a t e to d e s c r i b e the gaseous s t a t e of m i x t u r e s w h i l e u s i n g the s o - c a l l e d a c t i v i t y c o e f f i c i e n t m o d e l s ( e . g . , van L a a r , M a r g u l e s , R e d l i c h - K i s t e r ) t o d e s c r i b e the l i q u i d mixtures. We a t t e m p t a s y n t h e s i s h e r e by c o m b i n i n g t h e a c t i v i t y c o e f f i c i e n t model w i t h the e q u a t i o n o f s t a t e . The b a s i c a p p r o a c h i s o u t l i n e d below, and d i s c u s s i o n s t h a t f o l l o w w i l l g i v e the d e t a i l s . Outline

of T h e o r e t i c a l

Steps

1.

L o c a l c o m p o s i t i o n e x p r e s s i o n o f energy, (by i n t e g r a t i o n : A/NkT = / dp (U/N))

U *

2.

H e l m h o l t z f r e e energy (H.F.E.) (by the r e l a t i o n : Ρ = - 3 A / d V l )

A *

The pre s s u r e

Ρ

T

3.

We f i r s t g i v e a s t a t i s t i c a l m e c h a n i c a l definition of t h e l o c a l c o m p o s i t i o n s . F o r s i m p l i c i t y , we c o n s i d e r a b i n a r y m i x t u r e of s p h e r i c a l m o l e c u l e s o f t y p e s A and B. The number of n e i g h b o r i n g Β m o l e c u l e s s u r r o u n d i n g a c e n t r a l A m o l e c u l e i s g i v e n i n terms o f the r a d i a l d i s t r i b u t i o n function, 8βΑ^ ^' L Γ

r

BA - PB V BA where L i s t h e range ( r a d i u s ) o f t h e c o o r d i n a t i o n . S i m i l a r l y , t h e number of n e i g h b o r s A s u r r o u n d i n g t h e center A i s n

( L )

4

n

r

g

( r )

(

1

)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

252

n

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

r

« PA V

L

AA< >

4

2

"

g

(

AA<*>

2

)

We c a l l d n e a r e s t n e i g h b o r numbers. The c o o r d i n a t i o n number of the c e n t e r A i s t h e n t h e sum of i t s A n e i g h b o r s and Β n e i g h b o r s , a

Z

n

A " AA

+

n

n

Π

β

Α

t

n

e

(

BA

3

)

As a consequence of d e f i n i t i o n s E q u a t i o n s 1-3, ' l o c a l c o m p o s i t i o n s ' of Β m o l e c u l e s and A m o l e c u l e s s u r r o u n d i n g a c e n t e r of type A a r e X

B A

=

B A

N

/

Z

A

-

B A

N

/

(

N

A A

+

N

BA>

(

4

the

)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

and X

n

AA

" AA

/ z

n

A

n

- AA/< AA

+

n

S u b s t i t u t i n g the i n t e g r a l and 5, we have

X

B A

=

'S

BAdr

P B

Jo

[PA

P

+

X

A A

A

[PA

P

+

If X

A

=

χ

Λ

r

r

"

AAdr

Β ΒΑ

^0

/ ( χ

N

R

A A d r

^0

B

4

2

* A A <

4

B A d r

R

2

2

G

+

X

Equations 4

R

R

> ]

<

6

)

> /

A A < ' )

" «BA<«>] 2

through Α

* B A <

N

"

)

* Α Α < * >

2

4

4

expressions into

5

*BA<«>/

2

4

/î *

we d i v i d e

BA

d

n

BA r

B

P4

-

A

4

(

BA>

(

2

by / d r 4 n r g

A A

7

)

(r)

A

(

B BA>

8

)

and X

A A

=

*A'<*A

* B A

+

B

A >

(

9

)

wher e ^ A

-

ft***

4

"

2

8

B

A < ' ) / ^

A

A

D

R

4

N

R

2

G

A A (

R

>

(

1

0

)

S i m i l a r d e f i n i t i o n s c a n be made f o r x^g £ X^JJ . It i s i n t e r e s t i n g t o note t h a t E q u a t i o n s 8 and 9 a r e of the same form as g i v e n by W i l s o n (1). The a n a l o g y c a n be c a r r i e d f u r t h e r by n o t i n g t h e mean v a l u e theorem i n a

n

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

12. LI ET A L . integral

B A

calculus,

2

/|j dr

253

New Local Composition Model of Fluids

47rr g

B A

B A

(r) = /p

2

dr

4irr e x p [ - W ( r ) ] P

B A

ν

- ΒΑ**ρ[-β* ] where W = W ( r ) i s t h e p o t e n t i a l of mean f o r c e (7) e v a l u a t e d a t some mean v a l u e , t , a,nd V i s the s p h e r i c a l volume, ^ /3. Similarly, Β Α

B A

Q

Q

B A

4 π

Β Α

AA

/J dr

2

4nr g

A A

(r) = V

A A

exp[- W P

A A

]

(12)

Thus

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

A

BA =

(

W AA)"P[-P< BA V

V

f

(

- AA>]

1

3

)

i n a n a l o g y to W i l s o n ' s A . = ( j / ν ± ) e x p [ ( g ± g^/kT]. The energy e q u a t i o n i n s t a t i s t i c a l mechanics i s g i v e n by (6) i

ϋ· = [ N ( N A

- D / 2 V ] / dr g

A

A A

(r) u

A B

(r)

+ ( N N / 2 V ) / dr g

B A

(r) n

B A

(r)

+

B

[N (N b

-

B

By mean v a l u e =

(r)u

A B

B

A

'

A A

i

+ (N N /2V) / dr g A

ϋ

V

( 5

AA

/ 2 )

PA

1)/2V] / dr g

(r) u

B B

(r)

(14)

theorem,

V

1

d

r

*AA<*>

+

( i

BA

/ 2 )

PBPA

+

( 5

AB

/ 2 )

PAPB

+

( 5

BB

B B

(r)

/ 2 )

PB

V

V

V

f

/

d

r

*

d

r

d

r

«BA<

r )

r

*AB< > *BB

( r )

(

1

5

)

where U' i s t h e c o n f i g u r a t i o n a l f i r s t n e i g h b o r i n t e r n a l energy and u . . ( r ) i s t h e p a i r p o t e n t i a l between s p e c i e s i and j . F o r a n g l e - d e p e n d e n t and m u l t i - b o d y p o t e n t i a l s , an e f f e c t i v e p a i r p o t e n t i a l can be used. Using E q u a t i o n s 1-5, J

D'/N = ( z / 2 ) x x A

A

Û

A A

+ (z /2) x x B

B

A

B B

A

+ (z /2) x x A

u

B

B

A

e

B A

+ (z /2) x x B

B

B

A

A B

U

A

B

(16)

The key element o f W i l s o n ' s o r i g i n a l f o r m u l a t i o n was h i s f r e e energy e x p r e s s i o n . Here we d i s c u s s t h e form due t o W h i t i n g and P r a u s n i t z ( 5 ) : " N k T =

X

η

A 1 <*Αβ*Ι>[-β*ΑΑ]

+

*Β ΒΑ· »[-*'ΒΑ]> Ρ

Χ

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

254

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

+

X

1 η

B

χ

< Β

β χ

+

ρ[-β*ΒΒ]

χ

Ρ

Α ΑΒ

β χ

ν

ρ[-β Αβ]

)

(

1

7

)

At low d e n s i t i e s i t can be shown t h a t α i s r e l a t e d the c o o r d i n a t i o n number, z. <* = 2/z

to

(18)

S i n c e i n a l i q u i d , the c o o r d i n a t i o n number, z^, of m o l e c u l e s A i s not n e c e s s a r i l y the same as the c o o r d i n a t i o n number, ζ , f m o l e c u l e s B, we can g e n e r a l i z e the W h i t i n g - P r a u s n i t z (WP) f o r m u l a by i n t r o d u c i n g two α p a r a m e t e r s , β

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

" ÏÏIT

=

( χ

+

X

+

+

1 η

F

Β

/ α

Β

) 1 η

χ

< Α

β χ

ρ[-β*ΑΑ]

W

[^ BA]>

e

B BA *P ( χ

X

α

Α/ Α>

0

<*Β

F

A AB^P

β χ

ρ[-β*Ββ] (

[-β^Αβ]>

1

9

)

Under weak c o n d i t i o n s ( 6 ) , one can show t h a t E q u a t i o n i s c o n s i s t e n t w i t h E q u a t i o n 16 t h r o u g h the G i b b s Helmholtz r e l a t i o n . T h i s e q u a t i o n s a t i s f i e s the pure f l u i d l i m i t s of A°/NkT =

(l/a )

f°A

A

/ k T

(

2

0

19

)

E q u a t i o n 19 s e r v e s as a f r e e energy model f o r future calculations. Next we s h a l l l o o k a t the m i x i n g r ul e s · N - F l n i d T h e o r i e s and P r e u j a r e s . When an e q u a t i o n of s t a t e d e v e l o p e d f o r a pure s u b s t a n c e i s e x t e n d e d t o m i x t u r e s , one of the q u e s t i o n s i s the c o m p o s i t i o n dependence of the new e q u a t i o n . T h i s dependence i s u s u a l l y i n c o r p o r a t e d t h r o u g h m i x i n g r u l e s a p p l i e d t o the s t a t e v a r i a b l e s ( d e p e n d e n t , i n d e p e n d e n t or b o t h ) and/or the p a r a m e t e r s of the e q u a t i o n . A fundamental u n d e r s t a n d i n g o f c o m p o s i t i o n dependence can be g a i n e d t h r o u g h s t u d y i n g some model m i x t u r e s i n s t a t i s t i c a l mechani c s . The v i r i a l p r e s s u r e e q u a t i o n f o r a b i n a r y m i x t u r e of m o l e c u l e s o f type A and Ng m o l e c u l e s o f type Β i s g i v e n i n terms o f the p a i r p o t e n t i a l s and p a i r c o r r e l a t i o n f u n c t i o n s ( p c f ) as P/(pkT) = 1 - βρ/6

χ

2$»i îi ~ x

1

This

equation

d

r ( a u

3 3

serves

ii 3

/ d r

i i > 8 i i ( r ; Τ , ρ,χ) i , j = A,B

as our

3

3

starting

point

(21)

of

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

12. LI ET A L .

255

New Local Composition Model of Fluids

e x a m i n a t i o n of the m i x i n g r u l e s . We note t h a t the c o m p o s i t i o n dependence has an ' e x p l i c i t ' p a r t , under the d o u b l e summation ( u n d e r l i n e d by ^ and an ' i m p l i c i t ' p a r t , c o n t a i n e d i n the f u n c t i o n a l dependence of the p a i r c o r r e l a t i o n f u n c t i o n s , Sjj» A t t e m p t s have been made to a p p r o x i m a t e E q u a t i o n 21 oy the s o - c a l l e d van der Waals η-fluid t h e o r i e s ( i . e . o n e - f l u i d , twof l u i d and t h r e e - f l u i d t h e o r i e s ) where the c o m p o s i t i o n dependence i s s i m p l i f i e d and the p c f ' s a r e e v a l u a t e d a t r e d u c e d s t a t e s c h a r a c t e r i z e d by the c o r r e s p o n d i n g energy and s i z e p a r a m e t e r s . In the f o l l o w i n g we s h a l l examine f i r s t the e f f e c t s of m o l e c u l a r s i z e s on m i x i n g r u l e s i n the absence of a t t r a c t i v e e n e r g i e s ( e . g . i n h a r d sphere m i x t u r e s ) , and t h e n of a t t r a c t i v e e n e r g i e s i n a d d i t i o n to the s i z e d i f f e r e n c e s ( e . g . i n m i x t u r e s of L e n n a r d Jones m o l e c u l e s ) . The f o r m u l a t i o n due to H e n d e r s o n and L e o n a r d (9) f o r m u l t i f l u i d t h e o r i e s i s summarized below. Hard Sphere M i x t u r e s . For h a r d s p h e r e m i x t u r e s , the difference i n species i s manifested i n size. We s h a l l denote the b i g sph e r e s as t h o s e w i t h d i a m e t e r d ^ and s m a l l s p h e r e s w i t h d i a m e t e r ά" . The c r o s s i n t e r a c t i o n i s c h a r a c t e r i z e d by the d i a m e t e r d ^ ( (<*ΑΑ + d ) / 2 ) . The v i r i a l e q u a t i o n r e d u c e s i n the h a r d s p h e r e case to A

ΒΒ

=

B

P/(pkT) = 1 + < 4 π / 6 ) Ή

χ

x

p d

3

J

(

g 1 J

J

B B

+

,j) i , j = A,B ;p

d J

(22)

where S j ^ i d t . ) i s the c o n t a c t v a l u e of the i j p c f . E q u a t i o n 2 2 èan be c o n t r a s t e d w i t h the pure h a r d s p h e r e pressure equation J

3

+

P/(pkT) = 1 + ( 4 n / 6 ) p d g ( ; p ) o

(23)

d

where g denotes the pure h a r d s p h e r e p c f . The q u e s t i o n posed i n the η-fluid t h e o r i e s i s the r e l a t i o n between the m i x t u r e p c f , g ( ) , and the pure g ( r ) . As f a r as the p r e s s u r e i s c o n c e r n e d , i t i s t h e i r c o n t a c t v a l u e s t h a t are important. F o r example i n the t h r e e - f l u i d t h e o r y (vdw3) f o r b i n a r y m i x t u r e s , one assumes Q

i j

8

8

d

r

0

Ξ

*o< AA-P AA>

x )

Ξ

8o< BBiP BB>

AB< AB'P'*>

Z

«o< AB'P AB>

AA< AA-P'X> BB

( d

;

BB P'

d

d

(

2

4

)

d

d

(

2

5

)

d

d

(

2

6

)

and g

d

In the made, (d

two

fluid

x

*AA AA-P' >

Ξ

theory,

d

d

the f o l l o w i n g

*o< xA-P xA>

assumptions

(

are

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

7

)

256

g

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

( d

;

Z

BB BB P'*>

*o

( d

;

d

xB P xB>

(

2

8

)

(

2

9

)

and 8

AB< AB'P'»)

=

D

8 B B ] /

+

[*AA

2

where

=Σ

*h and d

xj

djA

(3°>

Σ xj

*jB

< >

i

xB

=

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

Finall^, g

i j

(

d

the

;

i j P ' *

)

31

one-fluid S

g

d

theory

i s simply

given

d

by (

o< x'P x>

3

2

)

where

d

x

2*i*J* iJ 3

-

33

< >

We ca examine the v a l i d i t y of a l l t h r e e m o d e l s by c o m p a r i n g the p r e s s u r e v a l u e s o b t a i n e d above w i t h known simulation r e s u l t s f o r hard spheres. For example, the c o n t a c t v a l u e s a r e a c c u r a t e l y g i v e n by the C a r n a h a n S t a r l i n g (C-S) f o r m u l a , g

+

=

4

2

~ y,, 4
(34)

y

3

where y = ^ p d . The r e s u l t s are l i s t e d i n T a b l e I. We choose t h r e e t y p i c a l m i x t u r e s f o r d i s c u s s i o n : ( i ) low d e n s i t y w i t h d i a m e t e r r a t i o ^ A A ^ B B ^ 1.9; ( ii) h i g h d e n s i t y w i t h d i a m e t e r r a t i o 1.1 and ( n i ) h i g h d e n s i t y w i t h a h i g h d i a m e t e r r a t i o of 3.3. At low d e n s i t i e s (pd^A = 0.3036 and x = 0.5) the c o m p r e s s i b i l i t y from the C-S e q u a t i o n i s 1.406. A l l t h r e e t h e o r i e s g i v e c o m p a r a b l e r e s u l t s w i t h the vdwl g i v i n g the b e s t p r e d i c t i o n . At h i g h e r d e n s i t y , (pd^A = 1.0825) w i t h d i a m e t e r r a t i o A A ^ B B I· * vdwl g i v e s a v a l u e f o r Ζ of 11.993 as compared t o the Monte C a r l o v a l u e of 12.3. vdw2 g i v e s 12.48 and vdw3 g i v e s 13.035. S i n c e the d i a m e t e r s a r e v e r y c l o s e , we can e x p e c t a l l t h r e e t h e o r i e s to be f a i r l y a c c u r a t e . (In the l i m i t A A B B ' three t h e o r i e s are e x a c t ) . At a higher diameter r a t i o d / d = 3.333, c o r r e s p o n d i n g t o a m o l e c u l a r volume r a t i o of ~37., MC g i v e s βΡ/ρ = 8.814, w h i l e vdwl g i v e s 6.027. Vdw2 and vdw3 t h e o r i e s are t o t a l l y inadequate at t h i s c o n d i t i o n , b e i n g 15.49 and 5208, r e s p e c t i v e l y . The vdw3 t h e o r y f a i l s to g i v e a r e a s o n a b l e v a l u e due to the m a g n i t u d e of the c o n t a c t value g (d^ )=5453. T h i s i s c a u s e d by the h i g h v a l u e of the r e d u c e d d e n s i t y p d ^ = i . 8 2 2 5 . Namely, the vdw3 A

d

d

= d

a

1

=

1

1

1

1

A A

A A

d

B B

A

A

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

12.

257

New Local Composition Model of Fluids

LI E T A L .

T a b l e I . The C o m p r e s s i b i l i t y F a c t o r , β Ρ / ρ , C o m p a r i s o n of the 1 - F l u i d , 2 - F l u i d and 3 - F l u i d T h e o r i e s w i t h E x a c t Re s u l t s f or H a r d Sph er e M i x t u r e s of S p e c i e s A and Β

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

Com pr e ss i b i 1 i ty Fa ct or : βΡ/ρ d

+ AA BB / d

p d

3 AA

X

MC

A

Vdwl

Vdw2

Vdw3

7 .5770 7 .5661 7 .5553 0 .01 7.5571 9.9620 9 .6602 9.437 8 9 .3 894 0.25 13 .035 12 .480 0 .50 11.993 12 .300* 16.709 16 .160 0 .75 15 .762 15.657 20 .801 20.728 20.765 0 .99 20 .736 8.3977 ii 1.111 1 .1266 0 .01 8.3720 8.3698 8 .3 837 11 .304 0 .25 10 .552 10 .905 10.577* 15 .113 14.363 0 .50 13.715 13 .835 19.718 18.964 0 .75 18 .280 18.417 24 .888 24 .786 24.837 0 .99 24 .7 97 1 .1023 1 .1020 0 .3036 0 .01 1.1018 1 .1017 iii 1.905 1 .2432 0 .25 1 .223 8 1 .2324 1 .2265 1.4396 1 .4170 0 .50 1 .3 988 1 .4061 1 .6 865 1 .642 0 1.6628 1 .6518 0 .75 1 .9710 0 .99 1 .9678 1 .9694 1 .9687 3 .2353 1 .1320 1 .1435 iv 3.333 1 .1328 1 .8225 0 .01 1303 .5 0 .25 3 .2066 2 .3269 2 .1117 5208 .6 8 .8140* 6.0273 15 .494 0 .50 +d^. = d i a m e t e r of l a r g e s p h e r e s and dgg = d i a m e t e r of s m a l l spheres. = ( d. + d „ ) / 2 •Monte C a r l o d a t a of L e e a n a L e v e s q u e ( 10); o t h e r s c a l c u l a t e d f r o m t h e f o r m u l a t i o n o f M a n s o o r i , e t a l . (11) V d w l , Vdw2 and Vdw3 a r e t h e v a n d e r W a a l s o n e - f l u i d , t w o - f l u i d and t h r e e - f l u i d t h e o r i e s . i

1.111

1 .0825

A

fi

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

258

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

t h e o r y p r e s u p p o s e s i n one o f i t s terms a h y p o t h e t i c a l pure f l u i d composed e n t i r e l y of l a r g e s p h e r e s . This p i c t u r e i s p h y s i c a l l y unsound due t o t h e f a c t t h a t h a l f of t h e m o l e c u l e s i n t h e m i x t u r e a r e s m a l l s p h e r e s w h i c h , b e i n g i n t e r s p e r s e d amongst t h e l a r g e ones, l e a v e a l a r g e r p o r t i o n o f 'empty s p a c e ' or ' f r e e v o l u m e ' a c c e s s i b l e to o t h e r m o l e c u l e s . Consequently, the t o t a l p r e s s u r e o f vdw3 i s c o n s i d e r a b l y h i g h e r than t h a t g i v e n by t h e r e a l f l u i d . The above c o m p a r i s o n c l e a r l y d e m o n s t r a t e s t h a t t h e vdwl t h e o r y g i v e s t h e b e s t r e s u l t s f o r h i g h d e n s i t i e s and l a r g e s i z e r a t i o s . The two o t h e r t h e o r i e s (vdw2 and vdw3) a r e i n a d e q u a t e by c o m p a r i s o n . M i x t u r e of L J Molecule^. R e c e n t l y , H o h e i s e l and L u c a s (12.) have made e x t e n s i v e s t u d i e s o f b i n a r y m i x t u r e s o f L e n n a r d - J o n e s m o l e c u l e s w i t h d i f f e r e n t s i z e r a t i o s (up to 2) and e n e r g y r a t i o s (up t o 5 ) . They compared t h e pcf o b t a i n e d from vdwl and t h e mean d e n s i t y approximation (NDA) w i t h s i m u l a t i o n d a t a . Their results showed t h a t b o t h t h e o r i e s a r e r e a s o n a b l e ; however, t h e MDA t h e o r y g i v e s b e t t e r J - i n t e g r a l s . The mean d e n s i t y a p p r o x i m a t i o n was p r o p o s e d by M a n s o o r i and L e l a n d (13)

4

i

and g

(

i j

r

Γ

g

- 2

ε

and 8

= o < /°ν*Τ/ t h e vdwl

χ χ

3

5

)

j

)

while σ

(

- E*i*j*ij

3

βίί:

3

},ρσ )

i nthis

case

< *>

i s g i v e n by E q u a t i o n

8

*i*j*ij ij

35 p l u s <

3 7 )

*J (

i j

r

)

=

3

3

«ο < * / σ ; * Τ / ε , ρ σ ) χ

< *>

χ

Thus t h e MDA d i f f e r s from t h e vdwl i n t h a t i n d i v i d u a l energy p a r a m e t e r s , * ± 2 , a r e used i n c h a r a c t e r i z i n g t h e temperature. The two t h e o r i e s c o i n c i d e when ε*^ = ΒΒ* ^ i * l pressure f o r Lennard-Jone s molecules can be w r i t t e n as ε

e

v

r

a

βΡ/ρ-1-1««2, , 4

where J

(

n

)

Λ

(

P e i j

the J - i n t e g r a l

) ( p . ^ ) [j«5>-2J<«> ]

(39)

i s d e f i n e d as

2

= J dr* r * g ( r * ) / r *

where r * i s t h e r e d u c e d presented i n Table I I .

n

distance.

(40) The r e s u l t s a r e

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

12. LI ET A L .

New Local Composition Model of Fluids

259

T a b l e I I . The C o m p r e s s i b i l i t y F a c t o r , β Ρ / ρ , f o r Equimolar Mixtures of Lennard-Jones Molecules of S p e c i e s A and Β ρ*=ρσ =0.8 COMPRESS IBIÎITY FACTOR: βΡ/ρ 3

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

Τ, κ

E

BB

/ 8

σ

AA

ΒΒ

/ σ

MDA**

ÀA MC*

Vdwl

Vdw3

Vdw2

270 1 .50 1 .00 3 .1563 3 .1346 3 .1680 270 3 .00 1 .00 3 .1483 3 .1431 3 .2772 270 5 .00 1 .00 2 .9402 3 .1211 2.8555 270 1.00 1.15 3 .1702 3 .0711 3 .0711 270 1.50 1.15 3 .0697 3 .0474 3 .1414 270 3 .00 1.15 3 .0760 3 .0196 3 .0417 270 5 .00 1.15 2 .8014 2 .8909 2 .9185 270 1.00 1.30 3 .1167 3 .1425 3 .1167 200 1.50 1.30 4.4724 3 .9323 4.0571 3 .9531 200 3 .00 1.30 4.2409 3 .9158 3 .9533 200 5 .00 1.30 3 .6644 3 .7173 3 .6924 200 1 .00 1.65 4.1416 3 .9395 3 .93 95 200 1.50 1.65 3 .9879 4 .3014 3 .9544 200 3 .00 1.65 4 .3526 3 .9320 4 .0978 200 5 .00 1.65 3 .6015 4.1050 3 .6296 200 1.00 2 .00 4 .1583 4.3519 4 .1583 200 1.50 2 .00 4.6298 3 .9738 4 .0277 200 3 .00 2 .00 4 .8513 3 .9341 3 .9303 200 5 .00 2.00 4.6499 3 .6627 3 .5321 200 0.33 1.30 3 .5412 3 .5952 3 .6183 200 0 .20 1.30 3 .4941 3 .3245 3 .4565 34 1.50 1.30 1 .1034 1 .097 4 0 .9648 * Monte C a r l o d a t a of H o h e i s e l and L u c a s (12) ** MDA=Mean D e n s i t y A p p r o x i m a t i o n Vdwl, Vdw2 and Vdw3 a r e t h e v a n d e r Waal s o n e - f l u i d , t w o - f l u i d and t h r e e - f l u i d t h e o r i e s

4.9951

σ

It shows t h a t f o r s m a l l size ratios, ^ Ββ/σ * 1.3), both MDA and vdwl give reasonable results (+ 0.7%). As /σ r e a c h e s 1.65, vdwl d e t e r i o r a t e s at large ε / (-12% a t ε / ε = 5 ) . The size disparity coupled with large energy difference p r o v e s t o be beyond t h e scope of b o t h theories. For example a t < * / σ - 2.00 and 8 /ε = 3.00, vdwl g i v e s an e r r o r of - 1 9 % , w h i l e MDA i s s i m i l a r l y - 1 9 % in error. There is little to choose between MDA and vdwl i n t h i s c a s e . Our conclusions a r e i n agreement with recent s t u d i e s on t h e L e n n a r d - J o n e s m i x t u r e s . In t h e f o l l o w i n g we s h a l l c o n s i d e r e.g. the MDA a p p r o x i m a t i o n f o r use i n our e q u a t i o n o f s t a t e . Α Α

σ

Β

β

Β

Β

Α

8

Β Β

i?_ix_t ure Given

Α

Α

Α

Β Β

Α Α

Α Α

β

Β

Α

Α

Ea.uatjlon o f Sjtajte an

equation

of

state

for a

pure

fluid,

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

the

260

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

f o l l o w i n g p r o c e d u r e can be used t o b u i l d an e q u a t i o n f o r m i x t u r e s. We f i r s t d e r i v e the H e l m h o l t z f r e e energy, A, from the g i v e n P-V-T relation. By i d e n t i f y i n g t h e p o t e n t i a l of mean f o r c e as the H e l m h o l t z free energy, we can immediately write down the free energy for mixtures a c c o r d i n g t o E q u a t i o n 19, i . e . - A' / ( NkT) +

x

+

(x

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

+

= ( x / a ) ln[x exp(-a Â' /kT) A

F

A

e

Â

A

A A

k T

B BA *P<-*A BA/ >] B

Χ

A

/<* ) l n [ x e x p ( - a Â ' / k T )

Α

B

Ρ

B

Α Β * * Ρ < - * Β

B

Α

Τ

Α Β / *

B B

(

> ]

4

1

)

In this e q u a t i o n the approximation i s made t h a t f o r mixtures the potentials of mean force are set proportional to the component H e l m h o l t z free energies (see Hill(i4)). To obtain the pressure, we d i f f e r e n t i a t e A' w i t h r e s p e c t to volume, a c c o r d i n g t o Ρ

-ΘΑ/aV

=

thus P

=

X

I

(42)

T

obtaining P

A<*AA AA

+

* Β Α

Ρ

Β Λ >

+

Χ

Β <

Χ

Β Β

Ρ

Β Β

+

X

P

AB AB

>

<

4 3 )

where p

ij

= -«iij/βν

I τ

(44)

Thus, the m i x t u r e e q u a t i o n of s t a t e . E q u a t i o n 43, depends on the composition in terms of the local compositions. We note t h a t E q u a t i o n 43 s a t i s f i e s the explicit composition dependence of E q u a t i o n 21 (N.B. the l o c a l c o m p o s i t i o n s , x^., c o n t a i n s the mole f r a c t i o n , x^). So f a r the development has been g e n e r a l . To make practical c a l c u l a t i o n s , we must adopt a f u n c t i o n form f o r the s p e c i f i c H e l m h o l t z f r e e e n e r g i e s . Mixture Parameters. Â' and P. i n Equations 41-44, b e i n g p r o p e r t i e s o f m o l é c u l e s w i t h j - i i n t e r a c t i o n s , are c a l c u l a t e d from an e q u a t i o n of s t a t e f o r pure s u b s t a n c e s d e v e l o p e d by Chung, et a l . (15.). The e q u a t i o n o f s t a t e has been t e s t e d f o r normal p a r a f f i n s ^2"" ^20^ ' ring compounds, and p o l a r compounds ( ^ S , a c e t o n e , e t c . ) , and a s s o c i a t i n g compounds (NH,, w a t e r , and a l c o h o l s , etc.). The pure f l u i d e q u a t i o n of s t a t e i s a c c u r a t e enough f o r engineering design c a l c u l a t i o n s . i

n

J^dujç__e d T e i i e r a t n r e and Dens i t y . For the e q u a t i o n of state used here, A'.. and P.. are f u n c t i o n s of the reduced temperature, T*^ , r e d u c e d d e n s i t y , p| j , and

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

12.

New Local Composition Model of Fluids

LI E T A L .

structure

used

parameter,

f o r T*^

λ.,

(see

(15.)).

261

The r e l a t i o n s

and p * j a r V

(45) p

ij

(46)

V

- P *x

where

2«-χ V*

V*

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

χ

(47)

mn f o r T* c o r r e s p o n d s t o t h e MDA model, w h i l e The r e l a t i o n f o r p * . c o r r e s p o n d s t o a one f l u i d model, the r e l a t i o n of E q u a t i o n s 45-47 w i t h E q u a t i o n 41 g i v e s a p o t ei non t i a l LCM model f o r f l u i d m i x t u r e s . Combi na mean The c h a r a c t e r i z a t i o n parameters The Comb_inj.JLg RuJ..e_j>. c a l c u l a t e d by the f o l l owing ^ij' *ij' i j combining r u l e s : V

a

n

d

+ λ

-JJL

E

a

r

e

i l

(48)

1J V*.

- *

( v

1/2

(49)

o*l/2

(50)

îi îj> v

ii°jj

11,

(51)

ij

8

/k

+k

ij - *V -f

2 5

<>

where the s u b s c r i p t s i i and j j r e f e r to component parameters and ξ and ζ a r e i n t e r a c t i o n parameters, BIPs. B e s u l t s jind

t h e pure binary

Discussion

The v a l u e s o f a. i E q u a t i o n 41 a r e r e l a t e d t o t h e c o o r d i n a t i o n numbers. We have c h o s e n a ' = a*A R = 0.5 i n t h i s work, i . e . , the v a l u e recommended by the W h i t i n g and P r a u s n i t z ( 5 ) . The q u a n t i t y F.,, i s r a t i o o f V.. to V . j , where V y i s the f i r s t - n e i g h b o r volume. To account: f o r V.. ( i£j ) , a b i n a r y i n t e r a c t i o n parameter 6 is introduced, where V^. = 6 (V v..) · Furthermore, the first-neighbor volume, ^.., i s a p p r o x i m a t e d as p r o p o r t i o n a l to t h e molar covolume V.}, thus n

A

=

a

1

3

i i

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

262

In the l o c a l c o m p o s i t i o n model used h e r e , three binary interaction parameters (ζ, Ç, and δ) are introduced. They are d e t e r m i n e d by f i t t i n g the m i x t u r e e q u a t i o n of s t a t e to m i x t u r e e x p e r i m e n t a l d a t a . For purpose of c o m p a r i s o n , v a p o r - l i q u i d e q u i l i b r i a calculations were also performed using conformai s o l u t i o n mixing rules, which correspond t o the vdwl model. The f o l l o w i n g c o n f o r m a i solution mixing rules are adopted to obtain mixture characterization param e t e r s V

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

8

- g*i*j îj V

x V

x x

<*4)

- S ^ j ' i j V i J

(55)

ij X

V

x x

V

S*i*j*iJ iJ 1

(56)

J

The s u b s c r i p t χ r e p r e s e n t s the m i x t u r e c h a r a c t e r i z a t i o n parameters. The c o m b i n i n g r u l e s used t o c a l c u l a t e the c r o s s i n t e r a c t i o n p a r a m e t e r s a r e the same as E q u a t i o n s 48-52. H e r e a f t e r , we r e f e r to the two m i x i n g r u l e s as LCM ( l o c a l c o m p o s i t i o n m i x i n g r u l e s ) and CSM ( c o n f o r m a i s o l u t i o n mixing r u l e s ) , r e s p e c t i v e l y . Unary. & 3 . W e have a p p l i e d the above e q u a t i o n of s t a t e to the c a l c u l a t i o n of m i x t u r e d e n s i t i e s and v a p o r liquid equilibria of binary systems. In these c a l c u l a t i o n s , the same c o m b i n i n g r u l e s . E q u a t i o n s 48-52, w i t h two b i n a r y i n t e r a c t i o n p a r a m e t e r s , ξ and ζ, were used i n b o t h LCM and CSM. For the LCM, a third p a r a m e t e r , δ, was i n t r o d u c e d t o c a l c u l a t e the v a l u e of Fij, i . e . , from E q u a t i o n 53. We have i n c r e a s e d the number of b i n a r y i n t e r a c t i o n p a r a m e t e r s i n the CSM to t h r e e and t h e n t o f o u r ( f o r D.. n d X . ) . When t e s t e d for the m e t h a n o l - c a r b o n d i o x i i e syste'm, no a p p r e c i a b l e improvement was obtained. Thus we retain only two b i n a r y p a r a m e t e r s f o r CSM. For systems w i t h b o t h v a p o r l i q u i d e q u i l i b r i u m and m i x t u r e d e n s i t y d a t a , the o p t i m a l v a l u e s of the b i n a r y i n t e r a c t i o n p a r a m e t e r s f o r each system were d e t e r m i n e d by u s i n g b o t h data s i m u l t a n e o u s l y in multiproperty regression analyses. The e x p e r i m e n t a l v a p o r - l i q u i d e q u i l i b r i u m and d e n s i t y ranges s t u d i e d a r e g i v e n i n T a b l e s I I I and IV. a

i

For m i x t u r e d e n s i t y c a l c u l a t i o n s , f i v e systems were selected f o r study. The results of m i x t u r e density c a l c u l a t i o n s u s i n g b o t h LCM and CSM are g i v e n i n T a b l e IV. The two m i x i n g r u l e s y i e l d s i m i l a r r e s u l t s e x c e p t for the system acetone-water. The LCM gives better r e s u l t s (1.7%) t h a n the CSM (7.9%) f o r f o r t y - o n e d e n s i t y d a t a p o i n t s i n the a c e t o n e - w a t e r system. The mixture acetone-water is a strongly nonideal system; its

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Me t h a n e Ethane Me t h a n e Pr opa ne Me t h a n e n-But ane Me t h a n e n-Pe ntane Me t h a n e n-Heza ne Me t h a n e n-Heptane Me t h a n e n-Oct ane Me t h a n e n-Nonane Me t h a n e n-De cane EthanePropane Ethanen-Butane Propanen-But ane Benz enen-Hexane Benz enen-Hept ane

Sy st em

2 .748 1.965 11131 7124 35198 7155 2071 10319 30126 1.47 3255 2138 .15.63 0.53

192200 172214 278378 224273 344

19 54 27 21 8 33 28 63 12 11 19 19 18 8

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

338394 3633 93 298328 334347

222255 298423 323423 543583 233

Pres. Range bar

Temp. Range Ε

f o r Binary

Systems

0 .8519 1.2147 1.2833 1.0884 1.0741 1.0144 1.0047 1.0006 0.9718 1.0085 0.9399

LCM LCM LCM LCM CSM LCM LCM CSM LCM LCM LCM

1.2203

LCM

1.2428

LCM 1.1267

1.1609

LCM

LCM

1.0581

LCM

0 .9738

0 .9730

0.9843 0.9992 1 .0074

0.5675 1.1136 0 .9904

0.3423

0 .4415

0.5295

0.8247

0.7 97 6

0 .7874

0 .8326

0.9268

l

0.8

0.5

1.7 1 .3 1 .7

4.0 4.9 1.2

6.5

1 .7

9.6

1.2

6.1

5.3

2.0

1 .5

K

2

1.6

1 .2

2 .3 2 .2 1.7

2.4 4.3 2 .5

66.

27 .

89.

20.

34.

8.4

9.8

1.6

K

A. A. D. % +

Continued on next page

0 .6139

1.0897

1 .0554

—

1.0283

0 .9983

—

0 .6281

0.7801

0.7908

0.4705

1.1926

1.1073

1 .1479

1.1152

1.0344

Binary Interaction Parameter s δ ξ ζ

Calculations

Mixing * Rules

Equilibrium

No. of Data Point s

I I I . Vapor-Liquid

Refer ence

Table

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

S

S

S

H

H

2 " η-Nona ne C0Me thane COEthane COPropane

2 ~ n-Hept ane

2 " Pr opa ne

H

2

HydrogenMe thane Hydrogen Ethane Hydroge η Propane N i t r o g e nMethane NitrogenEthane NitrogenPropane NitrogenHydrogen H SNe thane H SEthane

Sy st em

41 21 6 88 11 18 11 41 45 35 28 15 30 18 16

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(44)

No. of Data Points

(31)

Refere nee

Table I I I Continued

222273 9095 278344 200283 218344 311478 311478 123178 223273 273323

163183 15 8200 173348 122183 150

Temp. Range Ε 22160 7138 17207 2.850 3 .489 14138 8.446 14 121 0.630.5 1 .428 6.196 1 .4 28 3 .6 47 8.6117 13138

Pres. Range bar

1.2022

1.0658

LCM LCM

0 .9779

1.2537 LCM

LCM

1.1232

1.0508

LCM LCM

1.0698

LCM

1.0747

1.1376

LCM LCM

1.1730

1.0860

LCM LCM

1.0364

1.1970

LCM LCM

1.1420

1.1492

0.8552

0.8809

0.9707

0.5928

0.6975

0.8599

0.8486

0.8501

0.9611

0.8422

0.8356

0.9310

1 .3977

0.8926

1 .1462

1 .2255

1 .1196

1 .0032

0.8962

0.8873

1 .0130

1.0012

1.1650

1 .2797

1 .2053

1.2095

1.0296

1.3250

1.2572

1 .2528

Binary Interaction Parameters δ ξ ζ

LCM

LCM

Mixing Rules

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

l

19.

6 .0

4.4

2.5

6.0

6.7

8.7

3 .6

3 .8

16.

2.8

1.6

3 .4

23 .

14.

K

11 .

9.7

2 .7

9.2

9.8

4.3

3 .5

3 .0

2 .2

9.5

15 .

2.3

4.7

8.5

5 .2

Α. Α.D.%

V

2

S

2

_

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

n-Decane

CO -

n-Heptane

Co

n-Beiane

CO -

η-re nt a ne

CO -

2~ Propane

C 0

Eté ane

°2" Methane

C

CO

ch Hydr ogen-

H

C 0

Hydrogen 2~ co

C

NitrogenCO

2 Nitrogen-

C 0

Nitrogen-

Sy st em

38

(49)

(59)

(58)

(57)

(56)

16

62

10

48

9

55

(54)

(55)

20

19

37

(53)

(52)

(51)

34

37

(48)

(50)

8

24

19

No. of Data Point s

(47)

(46)

(45)

Refere nee

311477 462583

210219 223293 233273 278378 298

233361 2202 90 2232 83 2332 93 85

233273 300344 84

Temp. Range Κ 18139 35206 1 .42 .0 6.983 11203 8.3131 3 .5237 5 .2211 664 663 3 .427 2 .396 451 1.9 133 1451

Pres. Range bar

1.0019 1.2744

LCM LCM

LCM CSM

1.2433 1.0539

1.1736 1.0430 1.2521

1.193 0

LCM LCM CSM LCM

1.1081

1.0429 0.9867 1.1312

0 .7362

1.0019

LCM

LCM CSM LCM

LCM

LCM

1.1236

0 .7606

LCM

LCM

1.1679

1.1837

0.7511 0.9082

0.7819 0.8723 0.6703

0.6349

0.8647

0.8549 0.9718 0.8602

1.1018

0.9186

0.8460

1.0091

0.9186

1.1330

0.8154

0.8310

1

3 .7 8.6

4.3 8.9 9.1

5 .6

4.2

3 .2 5 .0 1.9

16.

29.

9.7

20.

1.7

2 .1

3 .3

13 .

Κ

Κ

2

12. 15 . 16.

11.

4.8

3 .5 3 .7 2.0

7.8

7 .1

5.1

5.9

5 .2

5.5

12.

4.5

Α. Α. D.%

2.6 5.4 Continued on next page —

1 .1255

0.9675

—

1 .1824

0 .9125

1.0564

1 .2824

—

1 .1320

1 .0833

1.0847

1.2171

1 .3834

1.0847

0.8331

1 .2988

1.2120

Binary Interaction Parameter s δ ξ ζ

LCM

LCM

Mixing Rul e s

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

ο

ο

S

? Iο

> ρ

m Η

r

to ON

Fluid

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Benz ene Ace t o n e Ethane Ace t o n e Propane Ace t o n e n-Pe nt ane Ace t o n e n-Hepta ne Ace t o n e Benz ene Me t h a n o l Eth ane Me th a n o l n-Pentane Me t h a n o l n-Hexane Me t h a n o l n-Heptane Me t h a n o l Benz ene EthanolPr opa ne Ethanoln-Hexa ne

2

co -

n - l e x a d e ca ne

Sy st em

8 8 11 8 11 5 21 18 8 18 16 17

(60)

(61)

(62)

(63)

(64)

(60)

(65)

(66)

(67)

(68)

(69)

(70)

17

303335 323336 332344 331349 400424 328

298

298

304322 338

350

462663 298313 298

16

(59)

(57)

Temp. Range Ε

No. of Data Points

Refere nee

Table III Continued

654 .46.9

1 .01

1.01

1 .01

.761.4 .150.3 1141 1 .00

1950 876 4.739 3 .427 1 .01

Pre s. Range bar

LCM CSM

LCM

LCM

LCM

LCM

LCM

LCM CSM LCM CSM LCM

LCM

LCM CSM LCM CSM LCM CSM LCM

Mixing Rul e s

0.9687 0.9089

1.1148

1.0170

0.9761

0.9356

0 .8623

1.2544 0 .9480 0 .9677 0 .9888 0.8746

1.1459

1.3504 1.0799 1.2044 1.0268 0 .8993 0 .9588 1.1227

0.8252 0.9497

0 .8659

0.8308

0.7873

0.8325

0.8302

0.6982 0.8919 0 .9820 0 .9767 0.8752

0.8048

0 .7200 0.9051 0.7870 0 .9405 0.8914 0 .9644 0.9218

—

0.8331

1 .0288

0 .8198

0.7879

0 .7982

0.7345

0.9739

—

0 .9886

—

0.8997

0 .9510

1 .0981

—

1 .0497

—

1.1369

—

1 .1274

Binary Interaction Parameters δ ξ ζ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

l

11.* 13.*

3 .7

6.9*

20.

23 .

8.1

2.8 6 .5 2.1 2 .1 2 .2

7.8

10. 13 . 1.6 7 .7 13 . 38. 5.8

K

Ε-

2.8 3 .5

4.9

5 .5

17 .

19.

26 .

5 .0 7.5 1 .6 1.9 19.

3 .4

9.8 8.9 13 . 15. 9.9 22 . 4.3

Α. Α.D.%

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

MethanolEthanol

Ethanoln-Heptane Ethanoln-Decane EthanolBenz ene 1-Propanoln-Hexa ne 1-Propanoln-De cane 1-PropanolBenz ene WaterMe thane WaterEthane WaterEthy1ene Ace t o n e He t h a n o l AcetoneEthanol Ace t o n e Water Me t h a n o l -

System

33 9362 348371 350370 411444 411444 378411 3 2 9338 330350 473 298 338350

11 9 8 10 50 36

12 12 9 25 13 12

(72)

(73)

(74)

(75)

(76)

(7778) (7 980) (81)

(82)

(82)

(83)

(84)

(82)

327336 353433 298

10

(71)

36

Temp. Range Κ

No. of Data Points

Reference

1.0848 0.9090 1.2883 1.1232 1.1644 0 .9955 1.0353 1.0737 0 .8864 0.8696 1.0527 0 .9777 1.0535 0.9711 0.8140

LCM LCM LCM CSM LCM LCM LCM LCM LCM LCM LCM CSM LCM CSM LCM

13689 13689 13344 1.01 1.01 1630 260 1 .01

1 .01

0.39

.11.16 1 .01

0.8972 1.0017 0 .8479 1 .1170 1.0302

0.9892

0 .9438

0.7249

0.6728

0 .6796

0.7440 0.8840 0 .8373

0.9305

0 .9669

0.7011

1.2490

LCM

1.01

2.8* 9.2* 4 .2 29. 1 .3

3 .7

1.4

4.2

3 .0

2.7

5 .4 6.9 7 .0

6 .6

7.8

12 .

16 .

K

1.3 2 .4 2.5 28. 9.7

3.8

2 .0

6 .4

9.8

9.2

9.0 11 . 7.8

11 .

2.9

4.3

29.

A.A.D.%

Continued on next page

1 .1264

—

1 .0728

—

1 .1327

0.9352

0.8794

1.2488

1 .2554

1 .2024

0.9327

—

0 .9935

0.9614

0.9175

0.9643

0.6861

0.8095

0 .8511

LCM

0.53

Binary Interaction Parameters ξ ζ °

Mixing Rules

Pres. Range bar

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

2

268

EQUATIONS O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

1 1

ι «< .

00

fH

CO

•

ON

TH

• •

·

•

•

•

Γ-

•

VO

•

cs

vo

CO

•

cs

•

cs vo

fH

O

·

tN

ON

ON

ο

VO

Γ-

<S

r-

es

CO

ON

TT

«O

ON

00

TI

Ttvo co

«η es

ON

es

vo

ON

00

ο

fH

-< M

cs

CO

«N co •M

cO

d ο •H •P

CO

CS

00 TJ-

00 ο

ο

τΗ

vo

T f «η t N ON fH ON "<-| CO ΤΓ

I •

d •H O

ON

ο ο

es

«n co Ο

ON

O

Ο

fH

fH

fH

Ο

fH

vo co

fH

Γ-

00

00

f*

es

ο ο

CO ON

fH ON

•

•

00 ο •

*n y* vo Ti c s vo

CO

co

ON

1 1

P4

03 •M

Ο (0 «

03

M

Ό

«0 -M (O

•
vo

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

•p

ΟΛ ON ON

ON

m u >> 03 u a* o3

M

ο

•

Ο

Ο

•

Ο

•

•

fH

Ο

Ο

d

Tf

00 00 ο

00 (0 d © •H rH H d •H

00

m TI T f r - CS Ι Λ vo es Ο ο

ON

•

Η

•

•

•

fH

oo M

ο d «β Wi * , ο 0* M

I vo τΗ fH «Ο

•

•

I CO

· CO «Λ

• «

S d * ο et

CO

CO

tN

M τ*

CO

CO

fH

Ο

00 •

•

•

fH

S

co es

CO

ο

fH Ο

fH Ο

•

•

r-i fH

fH

S

55

1 ο 1

Ο

U

c o c o oo 00 T f ON e s τι es

Tf TT

«Λ

•<*·

M C0

Ο

«β 4Λ

*» d • Οβ · Η Ο Α

ON

ON

es fH

ON

ON

©

fH fH

es

fH fH

vo

Γ

00

00

ΟΟ

00 00

ON

o

00

ON

00

co

H

1 1

d o

1 ο ι fH d r-4 Ο

β

to -Μ

C0 •> 09

O

a

o ο

4> U -ri

M to

a

M

d

-M d

o -ri

03 O

o

M 1

rd +*

P« M to

co

Ο

to

tH

Qi

u

>

M w Ζ

o o fH

αϊ

Ο

d Ρ< d at ο 03

U U U Μ Λ to « M 03 to 1 © -M P i -Μ -M 04 M +·> es*-» •M 1 •M 03 Ο o) to I ©

0k

o d

co co

fH

o •o

o

1

d d © CO

co

*d

•H d d κ o o C0 «M

«

Ο

1 d 1 «H OS fH Ο ο d ο d

d o o co co •4·» •M O O

fH

ι j

jQ '

a

O

+> CO

1 1 C1

j

fH M

fH «n 00

j

CD

00 oo d d •H •H H

•H 10 o o

a fH o 03 o a

Ο

•i-4

Ο

d o •ri •P

•M -M d •H fH CO O

I

n ο © ο Ό 1 «Η d
U M I M I M I

to

rd

-ri

H

• co CO to CO rH rH 0 d U

M •H •H

1

00

03

to

•

•

d

0k

G0

O 1 «o fH fH fH rj- m

•

o oo

li

·

fH

1 •

d

s ss

_)

ON ON CO ON VO f H

a 3

fH fH

fH

Ο

1

r - fH ON «ο vo «O vo CO c o c o c o

CO

•

•

•J

1

P< oo

0

U to

T f 00 m vo o es es o

CO

ΟΟ

as u

NJ

«r> fH

fH Γ

ο

fH fH

S S % CO O

55

• «

H

O

rO

•H

to

•

*M

00 ON • · O o

1

«j U 1 to •M cS * 0

a a «

© ο

U eS-J O

co

II

co «M

fH d co

• < •

04

+

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

12.

LI ET A L .

New Local Composition Model of Fluids

components d i f f e r a p p r e c i a b l y i n b o t h of i n t e r a c t i o n s ( p o l a r and a s s o c i a t i n g

269

s i z e and s t r e n g t h effects).

T a b l e IV. D e n s i t y C a l c u l a t i o n s U s i n g L o c a l C o m p o s i t i o n and C o n f o r m a i S o l u t i o n M i x i n g R u l e s

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

System

Ref. No of Data Points

Me t h a n e - m > n-de cane Ethane(92) n-buta ne Me t h a n o l - (93.) benz ene Ace t o n e - (94water 95) Me t h a n o l - (96) water

10 49 27 41 50

Τ oK

Ρ bar

Mixing Rules

310511 273413 293313 293353 298323

137620 5 .143 1 .013

LCM CSM LCM CSM LCM

1 .04 1 .07 1 .96 2 .15 1 .69

1 .013

LCM CSM LCM CSM

1 .70 7 .97 1 .71 3 .22

1 .013

Density A.A.D.%

For v a p o r - l i q u i d e q u i l i b r i u m c a l c u l a t i o n s , sixteen binary systems, ranging from nonpolar-nonpolar, nonpolar-polar, to p o l a r and a s s o c i a t i n g f l u i d s , were chosen f o r comparison. A summary of the r e s u l t s i s given i n Table I I I . The o v e r a l l average absolute percentage d e v i a t i o n (A.A.D.%) o f the e q u i l i b r i u m Kv a l u e ( K l ) f o r these f i f t e e n systems a r e 5% f o r the LCM and 10% f o r the CSM, r e s p e c t i v e l y . F o r n o n p o l a r systems e thane-n-butane and methane-n-de cane, b o t h the LCM and CSM f i t the v a p o r - l i q u i d e q u i l i b r i a d a t a w e l l . The LCM f i t s t h e d a t a b e t t e r than t h e CSM a t the t e m p e r a t u r e of 583 Κ f o r the system me thane-n-de cane (see F i g u r e 1 ) . F o r n o n p o l a r - p o l a r systems, CO^-methane, C 0 - n - h e x a n e , CO^-n-de cane , CC^-n-hexa de ca ne , CO*-benzene, acetoneethane, acetone-n-heptane, acetone-oenzene, ethanol-nhexane and 1 - p r o p a n o l - n - d e c a n e , the LCM y i e l d s better r e s u l t s t h a n t h e CSM ( s e e T a b l e I I I ) . F o r systems o f C 0 - n - h e x a n e and CO^-benzene, t h e CSM g i v e s poor r e s u l t s in liquid composition w h i l e the LCM y i e l d s improved results (see F i g u r e 2). For the C0 - -hexadecane system, the r a t i o of the covolume p a r a m e t e r s , V*, i s 1 : 7.6. The A.A.D.% f o r the e q u i l i b r i u m K-values (Kl) c a l c u l a t e d from the LCM i s 10% and from t h e CSM i s 13% f o r t e m p e r a t u r e s r a n g i n g from 462 Κ t o 663 Κ ( s e e T a b l e III). The a c e t o n e - n - h e p t a n e system forms a maximum p r e s s u r e a z e o t r o p e a t 33 8.15 K. The CSM g i v e s poor results near the azeotrope while the LCM gives reasonable results (see F i g u r e 3). For polar and associating fluids, acetone-water, methano1-C0 , methanol-water and a c e t o n e - C O ^ , CSM g i v e s poor r e s u l t s f o r the V L E c a l c u l a t i o n s . The improvement i n t h e f i t of the V L E d a t a by the LCM i s d r a m a t i c (see T a b l e I I I ) . O v e r a l l , f o r n o n p o l a r m i x t u r e s such as e t h a n e - n 2

2

2

n

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

270

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

v

n

i

1

1

1

1

0

.2

.4

.6

.8

Mole

F i g u r e 2. equilibria 298.15 Κ.

Fraction

Carbon

' I

Dioxide

E x p e r i m e n t a l and c a l c u l a t e d v a p o r - l i q u i d f o r the system c a r b o n d i o x i d e - n - h e x a n e a t

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

LI ET A L .

New Local Composition Model of Fluids

Λ

•

: M a r i p u r i and R a t c l i f f ( 1 9 7 2 ) : CSM

0

1

1

1

1

.2

.4

.β

.8

Mole F r a c t i o n

1

I

Acetone

Figure 3. Experimental and calculated vapor-liquid equilibrium f o r the system acetone-n-heptane.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

272

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

butane and me thane-n-de cane, b o t h LCM and CSM yield similar results. For s t r o n g l y n o n i d e a l systems such as m e t h a n o l - C 0 2 , a c e t o n e - w a t e r , n - h e x a n e - e t h a n o l and C 0 - n h e i a n e , the LCM i s c l e a r l y s u p e r i o r to the CSM. The applicability of the LCM has also been extensively tested for 1.) natural gas systems, including normal paraffins up t o n-decane, nitrogen, carbon dioxide, and hydrogen sulfide, and 2.) polar systems, i n c l u d i n g w a t e r , m e t h a n o l , e t h a n o l , 1 - p r o p a n o l , and a c e t o n e . The summary of the r e s u l t s f o r vaporl i q u i d equlibrium c a l c u l a t i o n s i s given i n Table I I I . We have c o v e r e d many m i x t u r e s w i t h methane ( a l l t o g e t h e r 15 s y s t e m s ) i . e . from ethane to n-decane and from C 0 to w a t e r . The a c c u r a c y of the c a l c u l a t i o n s f o r t h e s e systems a r e shown t o be s a t i s f a c t o r y . The systems hydrocarbons-alcohols are highly nonideal and are therefore difficult to f i t . The ethanol-n-heptane system forms minimum boiling t e m p e r a t u r e a z e o t r o p e a t 0.533 b a r as shown i n F i g u r e 4. The LCM d e s c r i b e s t h i s system r e a s o n a b l y w e l l over the entire concentration range. The LCM also gives satisfactory VLE results for o t h e r minimum boiling temperature azeotropes such as methanol-n-pentane, methanol-n-hexane, methanol-n-heptane, methanol-benzene and benzene-l-propanol (see Table III). It is d e m o n s t r a t e d t h a t the c o m p o s i t i o n dependence of the LCM is very e f f e c t i v e f o r VLE calculations for strongly n o n i d e a l systems, even near the r e g i o n of the a z e o t r o p e . The m i x t u r e s of methane-water, ethane-water and e t h y 1ene-water are of p r a c t i c a l i n t e r e s t . The LCM g i v e s adequate VLE results f o r methane-water system for p r e s s u r e s up to 700 b a r (see F i g u r e 5.) Even though m e t h a n o l and 1 - p r o p a n o l are b o t h p o l a r and a s s o c i a t i n g substances, the m e t h a n o l - l - p r o p a n o l m i x t u r e i s not a s t r o n g l y n o n i d e a l system. As shown i n F i g u r e 6, the LCM y i e l d s s a t i s f a c t o r y VLE r e s u l t s f o r t h i s s y s t e m . 2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

2

J J B r n a r y Sy._sjt.em_s. Ternary and mul t i c o m p o n e n t vaporliquid equilibria are obviously more important in industry than b i n a r y systems. Though the amount of binary vapor-liquid equilibria data a v a i l a b l e i n the literature is large, highly accurate data for multicomponent systems a r e r a t h e r s c a r c e . Thus, i t i s of g r e a t i m p o r t a n c e t h a t m i x i n g r u l e s can be a p p l i e d t o multicomponent systems using only binary parameters. The t e r n a r y systems methane-ethane-propane, acetonemethanol-ethanol and n-hexane-ethanol-benzene were selected for test. The r e s u l t s f o r these t h r e e t e r n a r y systems a r e g i v e n i n T a b l e V. The LCM p r e d i c t s the mole fractions i n both the liquid and the vapor phases r e a s o n a b l y w e l l f o r these three t e r n a r y systems.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

New Local Composition Model of Fluids

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

12. LI ET AL.

F i g u r e 5. equilibria

E x p e r i m e n t a l and c a l c u l a t e d v a p o r - l i q u i d f o r the system methane-water.

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EQUATIONS OF STATE: THEORIES AND APPLICATIONS

F i g u r e 6. equilibria 323 . 1 5 K .

ι

1

1

1

1

1

0

.2

.4

.6

.8

I

Mole F r a c t i o n M e t h a n o l E x p e r i m e n t a l and c a l c u l a t e d v a p o r - l i q u i d f o r the system m e t h a n o l - l - p r o p a n o l a t

T a b l e V. V a p o r - l i q u i d e q u i l i b r i a p r e d i c t i o n s u s i n g l o c a l c o m p o s i t i o n m i x i n g r u l e s f o r three t e r n a r y systems Sy st em

Me t h a n e ( 1) Ethane(2)Pr o p a n e ( 3 ) Ace t o n e ( 1 ) Methanol(2)Ethanol(3) n-Hexane(1)Ethanol(2)Benz e n e ( 3 )

Τ (Κ)

Ρ (bar)

Κ

No. of data ρο i nt s

(97)

33

158-214

2.2-55

3.08

A.A.D % 6.83

11.2

(82)

81

329-348

1 .0130

8 .04

8 .45

10 .3

(29)

43

328 .15

.54-.88

7 .75

9 .7 9

7 .72

χ

K

κ

Reference

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2

12.

LI ET AL.

275

New Local Composition Model of Fluids

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Çonjç_lu_s_i on_s A l o c a l c o m p o s i t i o n model has been d e v e l o p e d t o d e s c r i b e the composition dependence of m i x t u r e thermodynamic properties. The method employs a r e c e n t l y proposed e q u a t i o n of s t a t e . A m u l t i f l u i d model i s a d o p t e d f o r the r e d u c e d t e m p e r a t u r e and one f l u i d model a d o p t e d f o r the r e d u c e d d e n s i t y . For s t r o n g l y n o n i d e a l m i x t u r e s , such as m e t h a n o l C(>2, a c e t o n e - w a t e r , n-hexane-ethanol, and C i ^ - b e n z e n e , the LCM describes vapor-liquid equilibria behavior a c c u r a t e l y w h i l e the c o n f o r m a i s o l u t i o n m o d e l , which i s a one f l u i d model, does n o t work w e l l f o r these mixt ur e s. It should be noted that the p r e s e n t local c o m p o s i t i o n m i x i n g r u l e s c a n be used t o e x t e n d v i r t u a l l y any c o r r e s p o n d i n g - s t a t e s type equation of state to mixt ure s· In summary, the p r e s e n t f o r m u l a t i o n has the following useful features: 1) . The method works w e l l f o r n a t u r a l gas s y s t e m s . 2) . The method works f o r m i x t u r e s i n w h i c h components d i f f e r appreciably i n size. 3) . The method works w e l l f o r a wide v a r i e t y of f l u i d m i x t u r e s , r a n g i n g from n o n p o l a r to h i g h l y p o l a r fluids. 4) . The method c a n be a p p l i e d t o e x t e n d any c o r r e s p o n d i n g - s t a t e s type e q u a t i o n o f s t a t e to m i x t u r e s . A^^nowl^d_gm_en_t_s We thank t h e Gas R e s e a r c h I n s t i t u t e t h a t made t h i s r e s e a r c h p o s s i b l e .

for their

support

LJÊ.&Ê.MÂ o_£ Symb oi_s A Â'. ij

Ν

H e l m h o l t z f r e e energy c o n f i g u r a t i o n a l Helmholtz f r e e energy of molecules with i - j i n t e r a c t i o n s hard sphere diameter c h a r a c t e r i z a t i o n parameter f o r p o l a r and associating effects c o o r d i n a t i o n volume r a t i o radial distribution function between i j pairs Boltzmann's constant e q u i l i b r i u m constant range (distance) of f i r s t coordination shell f o r i molecules surrounding the center j molecule n e a r e s t n e i g h b o r number of i m o l e c u l e s i n the f i r s t c o o r d i n a t i o n s h e l l s u r r o u n d i n g a c e n t r a l j molecule t o t a l number of m o l e c u l e s i n t h e system global number of n e a r e s t n e i g h b o r pairs between i and j m o l e c u l e s

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

276

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Ρ

pressure

r

intermolecular distance measured from c e n t e r to c e n t e r gas c o n s t a n t temperature i n t e r n a l energy average p a i r i n t e r m o l e c u l a r energy volume m o l e c u l a r volume p a r a m e t e r c o o r d i n a t i o n volume p o t e n t i a l of mean f o r c e between s p e c i e s i and j m o l e c u l e s mole f r a c t i o n o f component i l o c a l c o m p o s i t i o n of i m o l e c u l e s around a j molecule c o o r d i n a t i o n number of f i r s t n e i g h b o r s compressibility factor

R Τ U Uj, V V* V\j '±i

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

z

£j

z Ζ

i

iLre_ek L_e._t_t.er_s α

'

β δ ε β Ç λ ρ

r e l a t e s t o t h e c o o r d i n a t i o n number, e q u a l to 0.5 i n t h i s work 1/kT b i n a r y i n t e r a c t i o n parameter energy parameter energy parameter f o r nonpolar c o n t r i b u t i o n b i n a r y i n t e r a c t i o n parameter s t r u c t u r e parameter density

ξ

binary

i n t e r a c t i o n parameter

Literature Cited 1. Wilson, G. M. J. Am. Chem. Soc., 1964, 86, 127. 2. Orye, R. V. and Prausnitz, J.M. Ind. Eng. Chem., 1965, 57, 18. 3. Abrams, D.S. and Prausnitz, J.M. AIChE J., 1975, 21, 116. 4. Benedict, Μ., Webb, G.B. and Rubin, L.C. J . Chem. Phys., 1940, 8, 334. 5. Whiting, W.B. and Prausnitz, J.M. Fluid Phase Equilibria, 1982, 9, 119. 6. Lee, L . L . , Chung, T.H. and Starling, K.E. Fluid Phase Equilibria, 1983, 12, 105. 7. Kirkwood, J.G. J . Chem. Phys., 1935, 3, 300. 8. Starling, K . E . , Lee, L . L . , Chung, T.H. and Li, M.H., 'Equation of state study for polar fluids,' 1983, third annual report to the Gas Research Institute, University of Oklahoma, Norman, OK. 9. Henderson, D. and Leonard, P . J . , 'Liquid Mixtures' in Physical Chemistry: An Advanced Treatise, Volume VIIIB (Academic Press, New York 1971), pp. 413-510. 10. Lee, L.L. and Levesque, D., Mol. Phys., 1973, 26, 1351.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

12. LI ET AL. New Local Composition Model of Fluids

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

11.

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Mansoori, G.Α., Carnahan, Ν.F., Starling, Κ.Ε., and Leland, T.W., J. Chem. Phys., 1971, 54, 1523. 12. Hoheisel, C. and Lucas, K. Mol. Phys., 1984, 53(1), 51. 13. Mansoori, G.A. and Leland, T.W. J . Chem. Soc. Faraday Trans. II, 1972, 68, 320. 14. H i l l , T . L . , 1956, Statistical Mechanics. McGrawH i l l , New York. 15. Chung, T.H., Khan, M.M.,Lee, L.L. and Starling, K.E. Fluid Phase Equilibria, 1984, 17, 351. 16. Guggenheim, E.A. 'Mixtures', 1952, Oxford University Press, London. 17. Wichterle, I. and Kobayashi, R.K. J . Chem. Eng. Data, 1972, 17(1), 9. 18. Wichterle, I. and Kobayashi, R.K. J . Chem. Eng. Data, 1972, 17(1), 4. 19. Roberts, L.R., Wang, R.Η., Azarnoosh, Α., and McKetta, J . J . J . Chem. Eng. Data, 1962, 7(4), 484. 20. Chu, T.-C., Chen, R.J.J., Chappelear, P.S., and Kobayashi, R.K. J . Chem. Eng. Data, 1976, 21(1), 41. 21. Poston, R.S. and McKetta, J . J . J . Chem. Eng. Data, 1966, 11(3), 362. 22. Chang, H.L., Hurt, L.J., and Kobayashi, R.K. AIChE J., 1966, 12(6), 1212. 23. Kohn, J.P. and Bradish, W.F., J . Chem. Eng. Data, 1964, 9, 15. 24. Shipman, L.M. and Kohn, J.P. J . Chem. Eng. Data, 1966, 11(2), 176. 25. Lin, H.M., Sebastian, H.M., Simnick, J . J . , and Chao, K.C. J. Chem. Eng. Data, 1979, 24(2), 146. 26. Haines, M.S. Master Thesis, University of Oklahoma, 1967. 27. Mehra, V.S. and Thodos, G. J . Chem. Eng. Data, 1965, 10(4), 307. 28. Kay, W.B. J . Chem. Eng. Data, 1970, 15(1), 48. 29. Ho, J.C.K. and Lu, B.C.Y. J . Chem. Eng. Data, 1963, 8(4), 549. 30. Nielsen, R.L. and Weber, J.H. J . Chem. Eng. Data, 1959, 4(2), 145. 31. Hong, J.H. and Kobayashi, R.K. J . Chem. Eng. Data, 1981, 26(2), 127. 32. Cohen, Α.Ε., Hopkins, H.G., and Koppany, C.R. Chem. Eng. Prog. Symp. Ser., 1967, 63-81, 10. 33. Trust, D.B. and Kurata, F. AIChE J., 1971, 17(1), 86. 34. Stryjek, R., Chappelear, P.S., and Kobayashi, R. J. Chem. Eng. Data, 1974, 19(4), 334. 35. Stryjek, R., Chappelear, P.S., and Kobayashi, R. J. Chem. Eng. Data, 1974, 19(4), 340. 36. Schindler, D.L., Swift, G.W., and Kurata, F. Hydrocarbon Processing, 1966, 45(11), 205. 37. Maimoni, A. AIChE J., 1961, 7, 371.

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Reamer, H. H., Sage, B.H., and Lacey, W.N. Ind. Eng. Chem., 1951, 43(4), 976. 39. Robinson, D.B., Kabra, H., Krishnan, T., and Miranda, R.D. GPA Report RR-15, 1975. 40. Brewer, J., Rodewald, N. and Kurata, F. AIChE J., 1961, 7(1), 13. 41. Ng, H . - J . , Kabra, Η., Robinson, D.B., and Kubota, H. J . Chem. Eng. Data, 1980, 25(1), 51. 42. Eakin, B.E. and DeVaney, W.E. AIChE Symposium Series, 1974, No. 140, Vol. 70, 80. 43. Christiansen, L . J . , Fredenslund, A. and Mollerup, J. Cryogenics, 1973, 13(7), 405. 44. Trust, D.B. and Kurata, F. AIChE J., 1971, 17-2, 415. 45. Zenner, G.H. and Dana, L . I . Chem. Eng. Prog. Symp. Ser., 1963, 59, 36. 46. Robinson, D.B. and Besserer, G.J. GPA Report RR-7, 1972. 47. Sprow, F.B. and Prausnitz, J.M. AIChE J., 1966, 12(4), 780. 48. Sobocinski, D.P. and Kurata, F. AIChE J., 1959, 5(4), 545. 49. Spano, J.O., Heck, C.K., and Barrick, P.L. J . Chem. Eng. Data, 1968, 13(2), 168. 50. Christiansen, L . J . , Fredenslund, Α., and Gardner, N. Adv. Cryog. Eng., 1974, 19, 309. 51. Fredenslund, A. and Mollerup, J . J . Chem. Thermo., 1975, 7, 677. 52. Tsang, C.Y. and Streett, W.B. Fluid Phase Equilibria, 1981, 6, 261. 53. Mraw, S.C., Hwang, S.C. and Kobayashi, R. J . Chem. Eng. Data, 1978, 23(2), 135. 54. Fredenslund, A. and Mollerup, J . J . Chem. Soc. Faraday Trans. I, 1974, 70( 9), 1653. 55. Akers, W.W., Kelley, R.Ε., and Lipscomb, T.G. Ind. Eng. Chem., 1954, 46(12), 2535. 56. Besserer, G.J. and Robinson, D.B. J . Chem. Eng. Data, 1973, 18(4), 416. 57. Ohgaki, K. and Katayama, T. J . Chem. Eng. Data, 1976, 21, 53. 58. Kalra, Η., Kubota, Η., Robinson, D.B., and Ng, H.-J. J . Chem. Eng. Data, 1978, 23(4), 317. 59. Sebastian, H.M., Simnick, J.J., Lin, Η.Μ., Chao, K.C. J . Chem. Eng. Data, 1980, 25(2), 138. 60. Ohgaki, K., Sano, F. and Katayama, Τ J . Chem. Eng. Data, 1976, 21(1), 55. 61. Gomez-Nieto, M. and Thodos, G. Chem. Eng. Sci., 1978, 33, 1589. 62. Lo, T . C . , Bieber, Η. Η., and Karr, A.E. J . Chem. Eng. Data, 1962, 7(3), 327. 63. Maripuri, V.O. and Ratcliff, G.A. J . Chem. Eng. Data, 1972, 17(3), 366. 64. Tasic, Α., Djordjević, Β., Grozdanic, D., Afgan, Ν., and Malic, D. Chem. Eng. Sci., 1978, 33, 189.

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Tenn, F.G. and Missen, R.W. Can. J . Chem. Eng., 1963, 41, 12. Raal, J.D., Code, R.K., and Best, D.A. J . Chem. Eng. Data, 1972, 17, 211. Benedict, M., Johnson, C.A., Solomon, Ε., and Rubin, L.C. Trans. AIChE, 1945, 41, 371. Nagata, I. J . Chem. Eng. Data, 1969, 14, 418. Gómez-Nieto, M. and Thodos, G. AIChE J., 1968, 24(4), 672. Ho, J.C.K. and Lu, B.C.-Y. J . Chem. Eng. Data, 1963, 8(4), 549. Katz, K. and Newman, M. Ind. Eng. Chem., 1955, 48(1), 137. Ellis, S.R.M. and Spurr, M.J. Brit. Chem. Eng., 1961, 6, 92. Smith, V.C. and Robinson, R.L. J . Chem. Eng. Data, 1970, 15(3), 391. Prabhu, P.S. and Van Winkle, M. J . Chem. Eng. Data, 1963, 8, 210. Ellis, S.R.M., McDermott, C., Williams, J.C.L. Proc. of the Intern. Symp. on D i s t . , 1960, Inst. Chem. Engrs., London. Ccon, J., Tojo, G., Bao, M., and Arce, A. An. Real. Soc. Espan. De Fis.Y Quim., 1973, 69, 1177. Culberson, O.L. and McKetta, J.J., Jr. Petro. Trans., AIME, 1951, 192, 223. Olds, R.H., Sage, B.H., and Lacey, W.N. Ind. Eng. Chem., 1942, 34(10), 1223. Culberson, O.L. and McKetta, J.J., Jr. Petro. Trans., AIME, 1950, 189, 319. Reamer, H.H., Olds, R.H., Sage, B.H., and Lacey, W.N. Ind. Eng. Chem., 1943, 35(7), 790. Anthony, R.G. and McKetta, J . J . J . Chem. Eng. Data, 1967, 12(1), 17. Amer, H.H., Paxton, R.R., and Van Winkle, M. Ind. Eng. Chem., 1956, 48(1), 142. Griswold, J . and Wong, S.Y. Chem. Eng. Prog. Sympos. Series, 1952, 48(3), 18. Katayama, T. Ohgaki, Κ., Maekawa, G., Goto, M., and Nagano, T. J . Chem. Eng. Japan, 1975, 8(2), 89. Schmidt, G.C. Ζ. Phys. Chem., 1926, 121, 221. Ochi, K. and Kojima, K. Kagaku Kogaku, 1969, 33, 352. Paul, R.N. J . Chem. Eng. Data, 1976, 21(2), 165. Takenouchi, S. and Kennedy, G. C. Am. J . of Sci., 1961, 262, 1055. Selleck, F . T . , Carmichael, L . T . , and Sage, B.H. Ind. Eng. Chem., 1952, 44, 2219. Clifford, I.L. and Hunter, E. J . Phys. Chem., 1933, 37, 101. Reamer, H.H., Olds, R.H., Sage, B.H., and Lacey, W.N. Ind. Eng. Chem., 1942, 34(12), 1526. Kay, W.B. Ind. Eng. Chem., 1940, 32(3), 353. Sumer, K.M. and Thompson, A.R. J . Chem. Eng. Data, 1967, 12(4), 489.

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EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Noda, Κ., Ohashi, Μ., and Ishida, K. J . Chem. Eng. Data, 1982, 27, 326. Thomas, K.T., McAllister, R. A. AIChE J., 1957, 3(2), 161. Mikhail, S.Z. and Kimel, W.R. J . Chem. Eng. Data, 1961, 6(4), 533. Wichterle, I. and Kobayashi, R. 'Low Temperature Vapor-Liquid Equilibria in the Methane-EthanePropane Ternary and Associated Binary Methane Systems with Special Consideration of the Equilibria in the Vicinity of the Critical Temperature of Methane', 1970, Monogram (Rice University, Houston, Texas).

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012

RECEIVED November 5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

13 Equation of State of Ionic Fluids 1

2

3

Douglas Henderson , Lesser Blum , and Alessandro Tani 1

IBM Research Laboratory, San Jose, CA 95193 Department of Physics, University of Puerto Rico, Rio Piedras, PR 00931 Institute di Chimica Fisica, Université di Pisa, via Risorgimento 35, 56100 Pisa, Italy

2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

3

An ionic fluid is modelled as a mixture of dipolar hard spheres (the solvent) and charged hard spheres (the ions). The free energy is expanded in a power series in the inverse temperature. Some of the terms in this expansion are infinite. However, these terms can be resummed to give a finite result. This expansion reduces to previously known results for pure dipolar hard spheres and pure charged hard spheres. As is the case for the pure dipolar and charged hard sphere systems, the convergence of the expansion can be enhanced by means of a Padé approximant. Simple expressions for the integrals appearing in the expansion are given. Most theoretical studies of the equation of state of ionic fluids have been based upon the primitive model in which the solvent is modelled as a dielectric continuum and the ions are modelled as charged hard spheres. It is clearly desirable to model the solvent as a collection of molecules. A simple model of the solvent would be the dipolar hard sphere system. Although deficient as a model of water, the dipolar hard sphere system may be a fairly reasonable model for many organic solvents. Even for water, the dipolar hard sphere model is a first step and is clearly an improvement over a dielectric continuum. Using this model of the solvent, an ionic fluid can be modelled as a mixture of dipolar hard spheres and charged hard spheres. Such a model would include the pure solvent and the molten salt as limiting cases and could, in principle, be applied to a continuously miscible system. This model was first discussed by Blum and colleagues and Adelman and Deutch who solved analytically the mean spherical approximation (MSA) for the restricted model in which all the spheres have the same diameter. Extensions and generalizations have also been g i v e n . Patey and colleagues - have considered simplified versions of the hypernetted chain ( H N C ) approximation for this fluid. There is also the modified Poisson-Boltzmann (MPB) approximation considered by Outhwaite and the study of Adelman and C h e n . Unfortunately, there are few exact results or computer simulations which can be used to determine the accuracy of the theory. There is the simulation of Patey and Valleau for very dilute solutions. Very recently, the one-dimensional mixture of charged and dipolar hard spheres was solved exactly by Vericat and B l u m . In the high temperature limit, the one-dimensional M S A is recovered. 1

2

3-5

7

5

6

8

9

1 0

0097-6156/ 86/ 0300-0281 $06.00/ 0 © 1986 American Chemical Society

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The solution of the M S A is analytic but implicit. Explicit results can be obtained only by some expansion. The M P B results are resonably simple but are restricted to small dielectric constants or low densities. The H N C results are numerical. Clearly, there is a need for a simple explicit theory. Perturbation theory is a very promising candidate for such a simple explicit theory. In perturbation theory, the free energy is expanded in powers of the inverse temperature using a hard sphere fluid as a reference system. This theory, which we have reviewed in previous A C S symposia, is a very satisfactory theory of simple fluids, such as the inert gases and the pure dipolar and pure charged hard sphere fluids. Prior to this work, it has not been applied to a mixture of dipolar and charged hard spheres. However, it should be a good approximation for this system also. 11

12

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

In this publication, we apply perturbation theory to this mixture. For simplicity, we restrict ourselves to the case where the dipolar and charged hard spheres have an equal diameter, σ. The general case of differing diameters would be rather complex. However, we give some plausible expressions for the case where the dipolar diameter differs from the ion diameter σ{. Perturbation Expansion 11

13

Using perturbation t h e o r y , - the Helmboltz free energy can be expanded in powers of 0 = l / k T , where k is Boltzmann's constant and Τ is the temperature. To third order, the result is Ρ

(

Λ

~

Λ

θ

)

=

- \

Ρβ Σ^{ 2

8° (12)<υ^.(12) ;

> ( 1

Γ

2

ij -

}

Wk/

1

g° ( 23)dr dr k

i j

j k

i k

2

3

,

(1)

ijk where An is the free energy of the reference hard sphere system and includes the entropy of mixing term, Σ x^nxj, p = N / V , the number of hard spheres per unit volume, X . S I N J / N , the fraction bf hard spheres of species i, gjj( 12) and g (123) are the reference pair and triplet correlation functions for a pair or triplet of species i and j or i, j, and k, respectively, and the terms in angular brackets are orientationally averaged pair interactions, i.e., ijk

= J u^{\2)aQ dn / v

J άΩ^Ω

2

(2)

2

= J u ( 1 2 ) u ( 2 3 ) u ( 1 3 ) d f i d « d f i / J dti^^d^ ij

jk

ik

i j

j k

i k

1

2

3

.

(3)

In principle, other terms are present in E q . (1). However, for the system considered here, they vanish either because for the dipolar hard spheres 3

< ( 1 2 ) > = = 0 υυ

(4)

j

or because of the charge neutrality of the charged hard spheres, x

2

Σ i i = j

0

5

<>

where zj is the valence (including sign) of the ion of species i. For simplicity, we assume that the salt is binary and symmetric and that all the hard spheres have the same diameter, a. Thus, x = x = x / 2 , x = 1-x and υ ^ ( Γ ) = ο ο for ra, 1

2

3

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

13.

283

Equation of State of Ionic Fluids

HENDERSON ET A L .

2 2

u ( r ) = u (r) = n

= - u (r)

22

u (r)=

(6)

12

- i ^ t f - ^ ) r

1 3

u (r) = ϊ ψ r

(7)

(? · £ )

23

(8)

2

and 2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

r where species 1 and 2 are the charged hard spheres and species 3 is a dipolar hard sphere. The quantities e and μ are the electronic charge and the magnitude of the dipole moment of the dipolar hard spheres, respectively, and z= | z\ \. The caret above a vector indicates that it is a unit vector. Finally, D(12) = 3(M! ' r ) ( £ 1 2

10

' 1 2 ) - 0*1 * M )

( >

f

2

2

Thus, 2

= 2/3

(11)

2

(12)

and <(r-

/ t l

) > = I

The triplet orientation averages have been worked out by Rasaiah and Stell cos# >ccd = - T ^

14

and are

13

< >

>cdd = I {cos(0 - 0 ) + cos0 cos0 }

(14)

1 + 3cos0,cos0<,cos0, ^ — "

05)

2

3

2

3

and

<

>ddd =

2

where #3 is the interior angle at the vertex 3 in a triangle formed by spheres 1, 2, 3, etc., and ccd, cdd, and ddd mean that the three members of the triplet are a charge-charge-dipole, a charge-dipole-dipole, and a dipole-dipole-dipole, respectively. The subscript ccc will be used to denote three charges. Defining 2

2

K*Q = 4 π β ζ ε ρ χ

(16)

and y =

βμ (ΐ 2

Ρ

- χ)

Eq. (1) becomes

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(Π)

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

284 fl(A-A )

_

0

_

3vpy

ICQ çoo

Ν

8

Ιόττρ J

0

Γ

27y

2

r

/·« g ( )

-<» g (r)dr 0

8up J

0

0

r

d r

0

8ττρ

J

r

0

4 g (123)dr dr 0

3,c

oy

Γ J 3

2(4τ7) ρ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

9

+

*0

2

Γ

6(4π)'ρ

+

2

12 13 23

o

r I

r

1 2

3

r

3

3

g (123)[cos(0 - 0 ) +cos0 cos0 ] 0

r

o „ ,3

dr dr

2

2

J

3

2

3

dr^dr^ r

M

3

3

2

r

2

3 J (477) ρ r ^a

27y

3

Γ

g (123)cos0

r^o

γ 2

Γ

^ 2 3 Î2 13 23 r

~

g (123)[l + 3cos0 cos0 cos0 ] 0

1

3

3

>

2(4ττ) ρ

~

r

1

Γ

Γ ^ σ

Γ

2

3

(

}

Γ

12 13 23

where go(r) and gn(123) are the pair and triplet correlation functions of a pure fluid of hard spheres of diameter σ. The first of the pair integrals and the first and second of the triplet integrals diverge. The divergence of the pair integrals arises because of the asymptotic value of unity for gn(r) at large r. The divergence of the second triplet integral also arises because gn(123) tends to unity when all three hard spheres are far apart. The divergence of the first triplet integral arises because of the asymptotic form gn(123) where even two spheres are far apart. Let us consider first the divergence arising from the asymptotic character of the go's when all spheres are far apart. Thus, «A-A ) Ν

d

6

4

0

κ *° ) p

r

0

f d r + 16*p JQ

2(4,7 ) ρ

r

3

6 ( 4 w

r,jiO r

1 2

r r n

f \*0

£2 £3 d

r

r

r

12 13 23

2 3

(r)dr

- é~ J> P

+

_ ^ _ 3

«.(«3)-l J

r

r

6(4.) p r ^ 0 12 13'23 3KQY 3

2(4ττ) ρ where ho(r)=go(r)~l. same as in E q . (18).

»

2

~

3

cos0

f Γ,^Ο r

~

j-V fe O23) - l]dr d£3 + ». 0

1 2

r

1 3

r

2

(19)

2 3

The missing integrals in E q . (19) indicated by " + ···" are the

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

13.

285

Equation of State of Ionic Fluids

HENDERSON E T A L .

The first three integrals can be summed. It is convenient to consider the energy rather than the free energy. Thus, considering only the first three terms of E q . (19) and differentiating with respect to β,

Ν

8ττρ J

Γ

+ 2

r

9Κ·ΛΥ

+

Γ

Γ

Γ

Γ

12 13 23

cosO-χ

?L- Γ 2(4ττ) ρ Γ,^Ο r

8ττρ J . >

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

(4,r) p Λ^Ο 3

0

__L_ d r d r r r 2

1 2

1 3

r

( l2

0

r

4TT J

Γ

3

+

-

(20)

2 3

r

13 23

J

13 23

Now f

cosfl^ ^ u l

r

d

3

(22)

= ^

Therefore

dr. Ν

r

8ττρ J . . ^ o ( 1 2

7

r r J

4,7 J

r

1 3

12

2 3

12

4

U

4

=

- ^ _ 0 - 9 y ) f

JJ

4TT

(23)

U -

r rr

1 2

dr

1 2

+

r

13 23 13 3

The first and second forms of E q . (23) are equivalent to order 0 . Defining the ring or chain sum

(R

J

2)

^> - ÏÏÏ

+

(24)

and taking the Fourier transform we have

*(k) =

isk

= 4*

2

!

-

Hence,

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(25)

286

EQUATIONS O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

expj-k/l-l (r) =

yVJ J

\

(26)

Thus, P(U-Up)

l

Ν

(

_

1

9

y

) /

" « p { - .

o

0

( l - | y ) r } d r

*0

(27)

8ττρ and /?(A - Λ )

κ

0

Ν

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

3

f ι - - y")

—

12ττρ V 2 The expansion of the dielectric constant is just = 1 + 3y +

ε

(28)

/

···

(29)

Therefore, (30) and E q . (28) can be rewritten as Ν

12ττρ

+ -

(31)

so that the Debye-Hiickel result is obtained. Resummation has transformed the series from an expansion in powers of β to an expansion in power of β / . Returning to Eqs. (18) and (19) 1

2

4

]3(A-A )

3

Ν

12πρ (

0

•

2 ^0

ι

6 0

0

dr dr 2

6 ( 4 π ) Dρ r ^ 0

Γ

; i

+

Γ

cos0, _2_

r ^0 r 5j

3

Γ

12 13 23

3*iy /» 5L_ Γ 2(4ττ) ρ

°

g (123)-l

r 1

J

3

16ττρ

1 2

r

1 3

r

[ g 0

( l 2 3 ) - l]dr dr + ·· 2

3

2 3

The three body integrals in E q . (32) can be simplified by introducing h (123) = g (123) - 1 - h (12) - h (13) - h (23) 0

0

(32)

0

0

(33)

Q

since f J

r Z0 i}

-^^-dr dr 2

Γ

Γ

Γ

12 13 23

~

2

3

= 8 r J2 7

~

C

r

3

r

2

-

f"r h (r)dr| 0

J

Q

o

ϋ*°

f°°ds J

0

h (12)cos0 ° '

f Γ

Trh^Odr J

0

d r d r = 167Γ 2

3

(34)

3

2

f

h (r)dr 0

r

12 13 23

and

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(35)

13.

HENDERSON ETAL.

Equation of State of Ionic Fluids

Γ

3

_°:

287

dr,dr = 8ττ Γ h (r)dr 3

(36)

0

The first integral on the RHS of Eq. (34) contains a divergent term. However, it is resummable. In analogy to Eqs. (24)-(27), the divergent integral is of order K g " . Unfortunately, it is difficult to perform the resummations with only two terms (of order β and j3 ) in the expansion. As a result, we adopt an ad hoc procedure. We assume that a truncated perturbation expansion correctly identifies the terms which appear but does not give the correct values for some of the coefficients because of resummed contributions from higher order terms. For example, comparison with the Stell-Lebowitz expansion for pure charged hard spheres, 1

2

3

15

β ( Α - Α

0

)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

Ν

- ~P~ \ L0

KQ

K-Q

12πρ

16πρ

pM^dr

f"rh (r)dr

0

0

8ip

I

ο -Ό

J

8πρ

f"h (r)dr + pJQ

fdr 0

1 2

J

JQ

12

fdr

f' "

1 3

13

lr -r l 1 2

n

h (123)dr ] + ··· -J 0

(37)

23

shows that third order perturbation theory correctly identifies all the terms appearing in the expansion but does not give the coefficients correctly for the term of order K or the pair integral in the *Q term. Note that the pair term of order κ (β ) and the triplet term of order *g(j3 ) are given correctly. The procedure we adopt here is to choose the coefficients of the pair integrals of order K and KQ from the Stell-Lebowitz series and the pair terms of order K* y and KQY from the expansion of the MSA free energy for the charged hard sphere/dipolar hard sphere mixture in powers of KQ. When the g's are set equal to their low density limit of unity for nonoverlapping spheres, the perturbation expansion must give the corresponding MSA result. We emphasize that perturbation theory, if taken to high enough order, would give the correct coefficients for all the various integrals. Our ad hoc procedure is adopted only to simplify the derivation of the final expression for the free energy. Q

0

3

Q

0

Q

Numerical Results for Perturbation Theory Integrals To evaluate the various integrals appearing in perturbation, we assume that the hard sphere g (r) is given adequately by the Percus-Yevick (PY) theory. Two integrals can be obtained immediately and are 16

0

τ' 1 r°°u ( \A I = rh (r)dr = cc

10-2η + η

2

0

c c

2

„ Jn

. (38)

20Π+2ΤΟ

ϋ

and τ"

1 f°° u ( \A a Jo ° 2

C C

3

2

(η-4)(η + 2) 24(1+ 2 ) V

where η=πρσ /6. The other pair integrals can be obtained numerically from G(s), the Laplace transform of go(r). Thus,

J"[G(S)-

J°°h (r)dr= 0

JO

J

0

L

i]ds z

s

J

and

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(40)

288

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Γ

n

Mil dr = -1 r G ( s ) d s ,

2

s

(41)

where

G

(s)=

^

(42) s

127,[L(s) + e S(s)] L ( s ) = 12η[1 + 2 η + 5(1 + η / 2 ) ]

(43)

12ΤΪ(1 + 2η) + 18T, S + 6 η ( 1 - T?)S + (1 - i | ) V

S(s) = -

2

2

(44)

We have calculated these three integrals and have found them to be well represented by 1 f°°i_ , χ ι

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

!

cc = 7 J

h

0 ^

d

1 +0.97743ρσ- + 0.05257p σ =

r

(45)

Ϊ

J

3

0

1 + 1.43613ρσ +0.41580p σ

g (O r^&OVf

,.

0

led = a J 0

r

J

3

2

6

1 ++0.79576ρσ 1 0 . 7 9 5 7 6 ρ σ ~ ++ϋ0.104556ρ . 1 0 4 5 5 6 ρ σσ

ar =

2

3

(4oJ

2

6

1 + 0.486704ρσ - 0.0222903ρ σ

and Γ

3 Γ°°8θ( ) r

dd =

Ι

σ

3

,

2

1+0.18158ρσ -0.11467ρ σ

6

=

(

4

7

)

— r Î ΤΊΓ « r 3 ( 1 - 0 . 4 9 3 0 3 ρ σ + 0.06293ρ V ) We have also calculated the three body integrals both by M C simulation and by numerical integration using the superposition approximation d

r

J

3

g (123) = g (12)g (13)g (23) 0

0

0

(48)

0

or equivalently, h (123) = h (12)h (13)h (23) + h (12)h (13) + h (13)h (23) 0

0

0

0

0

0

0

0

+ h (13)h (23) 0

(49)

0

with go(r) or hn(r) given by the P Y approximation, and have found that the r e l è v e n t three body integrals are well represented by !

-c = Τ σ

Γ 3

ά Γ

J

12 f 0

d r

13

J

J

1 2

1 3

Γ |r - ,| 1 0

h (123)dr 0

23

r i

3

2

6

3

2

2

3(1 - 1 . 0 5 5 6 0 ρ σ 4- 0 . 2 6 5 9 1 ρ σ ) 2(1 + 0 . 5 3 8 9 2 ρ σ - 0 . 9 4 2 3 6 ρ σ ) « σ

f dr -Ό

/.r / · *' d ar r, , / · ΐ 2 ++rη f -pli f ·>0 13 | r - r | Γ

«οο

Iced = j

12

Γ

cos0, cost/. h (123) - p d r23 23

n

n

1 2

Γ

3

0

J

1 2

Γ

2

Γ

1 3

3

2

6

3

2

6

11(1 + 2 . 2 5 6 4 2 ρ σ + 0 . 0 5 6 7 9 ρ σ ) 6( 1 + 2 . 6 4 1 7 8 ρ σ + 0 . 7 9 7 8 3 ρ σ )

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

13.

/.« d r

*

Λ« d r

1 2

° L ΊνΓ J

cdd =

J

1 2

0

J

= 0.94685

289

Equation of State of Ionic Fluids

HENDERSON E T A L .

ΛΓΙ2+ Ι3

cos(0 — #3) + cos0 cos0

γ

1 3

2

Τ7Γ J 1 3

0

J

123

go<

^^7323pa

2

r

1 3

3

3

dr

)

lr -r | 1 2

2

23

2 3

+ 3.11931pV 3

2

(

5

2

)

6

(1 + 2 . 7 0 1 8 6 ρ σ + 1 . 2 2 9 8 9 ρ σ ) and Iddd

3

a

r

«>dr

1 2

r

ocdr

ΛΓ +Γ

1 3

~ 1 — 1 — J, 0

-

5

r

0

12

r

13

, r

12

13

1 + 3cos0!cos0 cos0 2

,

i2- nl

0

( 1 2 3 )

2

23

*23

r

3

3 dr

g

6

Π + 1.12754ρσ + 0.56192ρ σ ) 3

^

2

5 3 )

6

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

24(1 - 0 . 0 5 4 9 5 ρ σ + 0 . 1 3 3 3 2 ρ σ ) 1 7

1 8

Previous e x p é r i e n c e » with the evaluation of similar triplet integrals indicates that the use of the P Y go's and the superposition approximation does not introduce appreciable error. The leading term in Eqs. (50) and (53) of 3/2 and 5/24, respectively, were obtained previously by Larsen et α / . and Rushbrooke et α / . and have been reobtained by ourselves. The leading term of 11/6 in E q . (51) is new. Unfortunately, we have not yet obtained an analytic result for the leading term in E q . (52). Although none of these integrals is independent of p, the variation of the integrals I , 1^, I £ I , I , I , I , I , and I is small compared to that of the prefactor ρ for the pair integrals or p for the triplet integrals. Because of this, it may not be too bad an approximation to assume these integrals to be a constant, provided the density does not vary overly much. Thus, at low densities, 1 9

c c

c>

c d

d d

c c c

c c d

2 0

c d d

d d d

2

I

- 1

c c

1

4

- /

C

-1/3

led

(54a) 2

1

Idd

54b

( > (54c)

<

5 4 d

)

1/3

(54e)

3/2

(55a)

and I

c c c

I

c c d

11/6

(55b)

I

c d d

0.94685

(55c)

and 5

W /

24

while at high densities

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

55d)

<

EQUATIONS O F STATE: T H E O R I E S A N D APPLICATIONS

290

I

cc

- 0.75

(56a)

I

cc

- 0.25

(56b)

I

cc

-0.10

(56c)

I

cd

1.24

(56d)

I

d d

0.55

(56e)

c c c

0.59

(57a)

and I

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

(57b) (57c) and I

0.45

d d d

(57d)

Expansion of MSA Results 1

According to Blum et al., in the MSA for the charged hard sphere/dipolar hard sphere model for equal size spheres, WJ-UQ)

=

K

gb -2(3 y) 0

1 / 2

K o

b -6yb 1

2

4πρ

Ν

where bn.bi. and 02 are parameters whose expansions in powers of y and KQ are b = £ (1 Q

κ

+ i

bl = | 4 3 y )

b 2

2

1

/

γκσ)

+ -

(59)

Vl (l-^^) +

2

= ^ 2

(60)

2

41y _J3_y^ 64 384

+

...

(61)

0

and j3 = 1 + b /3

(62)

j3 = 1 - b /6

(63)

0

(64)

3

6

12

2

2

= 1 + b /12 2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

13.

Equation of State of Ionic Fluids

HENDERSON E T A L .

291

At KQ=0, the parameter D2 is related to the dielectric

In Eqs. (59)-(61), κ = κ / / ε . constant by 0

(65)

βΐ If κ is expanded as (66)

«= «0(1-fy + - ) Equations (58)-(65) may be combined to give the expansion

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

Ν

4πρ

(

2

\

V

2

2

\

4

/

-3* y 0

which may be integrated to give

Ν

12wp \

2

/

16ττρ V

6

/

(68) 83 π ρ·σ

° ' ( - γ ' ) - ^ ( - ^ )

Equation (68) bears a close resemblance to perturbation theory. It lacks the terms in K Q and K Q This is not surprising since E q . (68) was constructed from a low concentration (KQ small) expansion and is not a complete β expansion. If desired, E q . (68) can be rewritten using ε - l = 3y(l + y) + ··· to obtain

4

KA-A-)

3

Ν

12πρ 9

r

1 6 π ρ LL

„

π

18

J J

..2/ L

88 π ρ σο

J

8

2

y [i-JL ] ... y

8πρσ

(69)

+

3

1

If we recognize that 1 + 3y/8 is just the expansion of 2 / ( 1 + λ * ) , where λ = β / β , then E q . (69) becomes 3

6

fl(A-A ) _^ 0 =

Ν

Λ Γ

+

\6πρ \6πρ

12™ \2πρ

1

+

2 3

L

18 18

(

e

_

t

)

1 _ JJ

2

κ (ε-1) 1

4ττρ (1+λ· ) σ

--^ b-ji Λ y 2

8ττρσ

3

L

1

6

(70)

J

When written in the above form, the third term gives exactly the free energy of solvation as calculated by Garisto et α / . from the M S A . We refrain from expressing y in the last term of E q . (70) in terms of ε because the free energy of the pure dipolar fluid is conventionally written in terms of a series in y rather than in ε. 6

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

292

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Free Energy of an Ionic Fluid Our procedure is to combine Eqs. (18), (32), (34), (35), and (36) with the coefficients of the integrals in Eqs. (34)-(36) chosen so as to reproduce E q . (37) when y=0 and E q . (68) when the integrals I , I , I ' , I I , I I , I , and I have the values given in Eqs. (54) and (55). While ad hoc, this procedure is suggested by the form of perturbation theory and reduces to all previously known expansions in the appropriate limits. A n extension to the case where a j i a is desirable. To do this without further approximation would require calculating the integral I , etc., for every concentration. This would be acceptable for the pair integrals but would be far too time consuming for the triplet integrals. Instead, we assume that Eqs. (38)-(39), (45)-(47), (50)-(53) can be used with ρ σ replaced by ρ < σ > , where c c

c c

c

c

c d

d d

c c c

c c d

c d d

d d d

{

d

cc

3

3

3

<σ > = χ σ

3

+ (1 - χ ) σ

3

(71)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

21

This type of averaging has proven useful in theories of nonelectrolyte solutions and should be useful for this sytem provided σ{ and a are not too different. Thus, for d

P(A-A ) _ 0

Ν (52

ο 3

^

+ ο

+

2

8πρ

2

£[('-»'»[£])<-•»-(£)] 3

+ 16πρσ

ά

Iddd} + )

('2)

where £ = 1 is a parameter whose power denotes the order in β of the term which follows that power of ξ and

1

",d =

-v

(73)

Equation (72) reduces to the Stell-Lebowitz series for charged hard spheres in a vacuum when y=0 (or ε = 1 ) and reduces to the perturbation series of Rushbrooke et α / . for dipolar hard spheres when KQ=0. When σ[=σ^, E q . (72) reduces to the M S A expansion, E q . (68), plus the terms in *Q and KQ, if Eqs. (54) and (55) are used for the integrals. When o^a^, the M S A expansion corresponding to E q . (68) has a complex dependence on and a . However, the form of E q . (72) is suggested by that result. The combination ya<j ensures that the free energy, in excess of the free energy of the solvent and the free energy of solvation, reduces to the Stell-Lebowitz expansion with 2 0

x

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

13.

293

Equation of State of ionic Fluids

HENDERSON E T A L .

ε expanded in powers of y for charged hard spheres in a dielectric continuum when a tends to zero. The free energy of solvation becomes the Born result for the free energy of solvation of a charged hard sphere in a dielectric continuum. d

It might be preferable to eliminate y from part of E q . (72) by using ε = 1 + 3y + ···

l^i

B 5

3y(l-2y)

(74)

+ ...

(75)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

and

a

n

d

s

o

t

n

a

t

n e

The factors of y are retained in the coefficients of Idd Iddd t series reduces to that of Rushbrooke et α / . when κ = 0 . For E q . (77) to be useful, the dielectric constant, ε, must be calculated as a function of concentration. The fact that this information is not necessary for E q . (72) may be an advantage for this equation. To the order in which we are working, perturbation theory yields the result 2 0

ε = 1 + 3y + · ·

(29)

For the pure dipolar hard sphere fluid

e

=

l

+

3

y

+

3y

2 +

3 y Y ^ - A

(78) 1 8

where I is an integral calculated by Tani et α / . , seems to be a good approximation. Comparison of Eqs. (29) and (78) suggests that the same series might be used for the dielectric constant of the mixture. In this approximation, ε would decrease with increasing concentration as y decreased because 1-x is decreasing. Such a decrease does not seem large enough compared with experiment. * Empirically, d d A

22

23

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

294

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

9 I

2/

ddA

, \

i+y + y ί — γ - i l ε = 1 + 3y

ii|iL L (79) 1 + CK aj where c is an empirically adjusted parameter and I is assumed to be the function of Tani et al. with ρσ replaced by ρ<σ > if σ ^σ , seems reasonable. For an aqueous solution of N a C l , c~0.1 gives fairly good agreement with experiment. For pure charged spheres and pure dipolar hard spheres, the studies of Larsen et α/. and Rushbrooke et al., respectively, suggest that Eqs. (72) or (77) will converge slowly for large κ or y but that good results can be obtained from a P a d é approximant. Thus, we write the series d d A

3

3

{

1 9

2Q

« ^ o )

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

ά

=

n

e

Ay2

^

+

+

{

»/2

^

A j / a +

+

...

(

8

0

)

in the form

P(A-A ) 0

=

n

3 / 2

Α3/2» A

the 2

n

2

l + M ^ + drf

N

where

1 / 2

|

1 +

coefficients

n 2 , dj,

η,

d2

and

are

chosen

to

reproduce

, A ^ t and A 3 . Hence, 5

2

nj = A A

,

3 / 2

(82)

A

3/2 3 ~ A

d,

A

=

3/* A

A

2

-

2

3~ A

2 A

A

A

A

2

A

3/2 2 5/2 +

A

2

(83)

A

3/2 5/2

A

S/2

(

8

4

)

A

3/2 5/2

and

d = -

A

z

A

3

2

A

2 ~

A

" ' A

/

2

(85)

A

3/2 5/2

For y=0, all the A are, in general, nonvanishing and E q . (81) is the Padé approximant of Larsen et al. For κ = 0 , A / = A / = 0 so that n

η

3 /

2

n

1

= 0

n

2

= A

5

: 2

(86) (87)

2

ά = 0

(88)

χ

A, d

2

=

and E q . (81) is the Padé approximant of Rushbrooke et al. appropriate limiting forms.

(89) Thus, E q . (80) has the

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

13.

Equation of State of Ionic Fluids

HENDERSON E T A L .

295

T £ e r e is a little ambiguity as to whether the terms κ ( ε - 1 ) , κ ε - l ) , and κ ( ε - l ) should be considered to be of order £ , £ , and | , respectively, as in Eq. (77), or of order £, £ , and £, respectively. The question is purely academic in Eq. (79) since £ = 1 . However, it is a question of importance when a Padé is formed. For the moment, we leave E q . (77) in the above form. However, experience may cause us to rethink this issue. 2

3

3

2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

Summary We have given two promising forms for the free energy of an ionic fluid. These two expressions are derived from perturbation theory with plausibility arguments used to determine the coefficients of some of third order terms. These expressions reduce to the previously known perturbation expression for a pure fluid of charged hard spheres when dipole moment or the diameter of the dipolar hard spheres vanishes and when the concentration of dipoles is zero. They also reduce to the previously known perturbation expression for a pure fluid of dipolar hard spheres when the charge or concentration of the ions is zero. The expressions developed in this paper should be of value in electrochemical studies, including corrosion, and in studies of biological systems. The perturbation expansion is straightforward but slowly convergent for most practical applications. Unfortunately, there is a great deal of flexibility in the summation procedures used to obtain practical expressions. Equations (80)-(85) give just one possibility. Many more are possible and equally plausible. Before a final choice is made, application to experimental data will be required. Unfortunately, simulated data are not available. Simulation studies of ion-dipole mixtures are clearly required. We hope that the chemical engineering community will join us in generating such data and in making calculations based on the expressions presented here and in comparing the results with experimental and simulated data.

Literature Cited 1. Blum, L. Chem. Phys. Lett. 1974, 26, 200; J. Chem. Phys. 1974, 61, 2129; J. Stat. Phys. 1978, 18,451;Vericat, F.; Blum, L. J. Stat. Phys. 1980, 22, 593; Peréz-Hernández, W.; Blum, L. ibid. 1981, 24, 451. 2. Adelman, S. Α.; Deutch, J. M. J. Çhem. Phys. 1974, 60, 3935. 3. Vericat, F.; Blum, L. Mol. Phys. 1982, 45, 1067. 4. Chan., D. Y.C.;Mitchell, D. J.; Ninham, B. W.; Pailthorpe, B. J. Chem. Phys. 1978, 69, 691; 1979, 70, 1578; Chan, D. Y. C.; Mitchell, D. J.; Ninham, B. W. ibid. 1979, 70, 2946. 5. Garisto, F.; Kusalik, P. G.; Patey, G. N. J. Chem. Phys. 1983, 79, 6294. 6. Levesque, D.; Weis, J. J.; Patey, G. N. Phys. Lett. 1978, 66A, 115; J. Chem. Phys. 1980, 72, 1887; Patey, G. N.; Carnie,S.L.ibid., 1983, 78, 5183. Kusalik, P. G.; Patey, G. N. J. Chem. Phys. 1983, 79, 4468. 7. Outhwaite, C. W. Mol. Phys. 1976,31,1345; 1977, 33, 1229. 8. Adelman, S. Α.; Chen, J.-H. J. Chem. Phys. 1979, 70, 4291. 9. Patey, G. N.; Valleau, J. P. J. Chem. Phys. 1975, 63, 2334. 10. Vericat, F.; Blum, L. J. Chem. Phys. 1985, 82, 1492. 11. Barker, J. Α.; Henderson, D.J.Chem. Phys. 1967, 47, 2856; 1967, 47, 4714; Rev. Mod. Phys. 1976, 48, 587. 12. Henderson, D. Adv. in Chem. 1979, 182, 1; 1983, 204, 47. 13. Henderson, D.; Barojas, J.; Blum, L. Revista Mexicana de Fisica 1984, 30, 139. 14. Rasaiah, J.C.;Stell, G. Chem. Phys. Lett. 1974, 25, 519. 15. Stell, G.; Lebowitz, J. L. J. Chem. Phys. 1968, 49, 3796.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

296

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

16. 17. 18. 19. 20. 21.

Percus, J. K.; Yevick, G. J. Phys. Rev. 1958, 110, 1. Barker, J. Α.; Henderson, D.; Smith, W. R. Mol. Phys. 1969, 17, 579. Tani, Α.; Henderson, D.; Barker, J. Α.; Hecht, C. E. Mol. Phys. 1983, 48, 863. Larsen, B.; Rasaiah, J.C.;Stell, G. Mol. Phys. 1977, 33, 987. Rushbrooke, G. S.; Stell, G.; Hoye, J.S.Mol. Phys. 1973, 26, 1199. Henderson, D.; Leonard, P. J. In "Physical Chemistry - An Advanced Treatise"; Eyring, H.; Henderson, D.; Jost, P. W., Eds.; Academic Press: New York, 1971; Vol. 8B, Chap. 7. 22. Halsted, J. B.; Ritson, D. M.; Collie, C. H. J. Chem. Phys. 1948, 16, 1. 23. Hubbard, J. B.; Onsager, L.; van Beek, W. M.; Mandel, M. Proc. Natl. Acad. Sci., U.S.A., 1977, 74, 401.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch013

RECEIVED November 8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

14 Thermodynamics of Multipolar Molecules The Perturbed-Anisotropic-Chain Theory P. Vimalchand, Marc D. Donohue, and Ilga Celmins

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

Department of Chemical Engineering, The Johns Hopkins University, Baltimore, MD 21218

The Perturbed-Hard-Chain theory (PHCT) is modified to treat rigorously dipolar and quadrupolarfluidsand their mixtures. The multi-polar interactions, which are treated explicitly, are calculated by extending the perturbation expansion of Gubbins and Twu to chain-like molecules. Moreover, the square-well potential used to characterize the spherically symmetric interactions in the PHCT has been replaced by a soft-core (Lennard-Jones) potential. Theoretical calculations and data reduction on a number of purefluidsand mixtures indicate that the above changes result in physically more meaningful pure component parameters, and the properties of even highly non-ideal mixtures can be predicted accurately without the use of a binary interaction parameter. Although considerable phase equilibrium data are available for hydrocarbon systems of low molecular weight, data for high molecular weight hydrocarbons, especially aromatic hydrocarbons, are scarce. Unfortunately, none of the theories that have been developed for the properties of lower molecular weight aliphatic compounds encountered in natural gas and oil are accurate for heavy hydrocarbons such as coal derivatives. They require large values of an adjustable binary interaction parameter, and therefore, the predictive ability of these theories becomes poor for multicomponent mixtures and for systems for which there are no experimental data. There are two main reasons why these theories fail for high molecular weight multipolar compounds. First, these theories are valid only for compounds with molecular weight below 150 because they ignore the effect of rotational and vibrational motions on thermodynamic properties. While this deficiency was overcome, in part, by the polymer theories of Flory (1) and Prigogine (2L), their theories are valid only at high density. The second deficiency in all the theories that were developed for oil and gas-refining operations concerns the nature of intermolecular forces. All these theories tacitly assume that molecules interact with London dispersion forces (also referred to as van der Waals forces). Since coal and its derivatives are generally high molecular 0097-6156/ 86/ 0300-0297S06.00/ 0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

298

weight (150 to 1000) compounds that have numerous benzene rings and functional groups which have strong anisotropic (dipolar and quadrupolar) potential functions, any theory for coal compounds must reflect this difference. These two deficiencies are taken into account in developing the Perturbed-Anisotropic-Chain theory (PACT) which enables calcula tion of thermodynamic properties for a wide variety of pure fluids and mixtures. The P A C T equation is based on the Perturbed-So ft-Chain theory (PSCT) and on improvements to Pople's perturbation expansion (2.) by Gubbins and Twu (!). The PSCT equation is essentially the PerturbedHard-Chain theory (PHCT) of Donohue and Prausnitz (5_), but the potential energy function used to characterize the interaction between the molecules is different. The original PHCT equation uses a squarewell intermolecular potential, while in the PSCT equation, interactions are calculated with the Lennard-Jones potential. Multipolar interactions in the P A C T are treated by combining the perturbation expansion of Gubbins and Twu for anisotropic molecules with lattice theory for chain-like molecules. The PHCT and PSCT equations have been shown to predict accu rately the properties of pure fluids and mixtures of non-polar molecules of varying size and shape jBeret and Prausnitz (£); Donohue and Prausnitz (u); Kaul et al. (Ζ) and Morris (S.)], including polymeric sys tems [Beret and Prausnitz (&) and Liu and Prausnitz (£.)]. For mixtures containing quadrupolar molecules (such as carbon dioxide - ethylene sys tem; carbon dioxide - ethane system; benzene - 1-methyl naphthalene system), the P A C T equation predicts properties better than either the PHCT or PSCT equations [Vimalchand and Donohue (ID)]. In this paper, we extend the P A C T to include dipolar interactions. The theorem ical results show that the P A C T equation also can be used for predicting accurately the properties of highly non-ideal mixtures (without the use of a binary interaction parameter) containing strongly dipolar molecules such as acetone. The Perturbed-Anisotropic- Chain Theory The canonical ensemble partition function used in the PerturbedAnisotropic-Chain theory is of form:

««™ -MM

m hw)

hw) «

where Ν is the number of molecules at temperature Τ and volume V; A is the thermal deBroglie wavelength, and k is Boltzmann's constant. The partition function is based on the generalized van der Waals theory where molecular translational motions are governed by intermolecular attractions and repulsions. The molecular repulsions are written in terms of free volume, V}, which is the volume available to the center of mass of a single molecule as it moves about the system holding the positions of all other molecules fixed. The molecular attractions are generalized in terms of a potential field, φ/2, which is the intermolecular potential energy of one molecule due to the presence of all other molecules. For non-central force molecules (with dipolar or quadrupolar forces), the intermolecular potential energy function can be written as a sum of iso tropic and anisotropic interactions. The isotropic pair potential, which depends only on the distance between the molecules, is taken as an

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

14.

VIMALCHAND ET AL.

The Perturbed-Anisotropic-Chain Theory

299

unweighted average over all orientations. Summing the intermolecular interactions, φ/2 is given by the sum of isotropic and anisotropic interac tions of one molecule with all other molecules in the system, φ /2 and φ /2, respectively. The rotational and vibrational motions, which are strongly densitydependent, are affected by intermolecular interactions. To account for deviations in equation of state due to rotational and vibrational motions, a parameter c is defined such that 3c is the total number of densitydependent degrees of freedom. This includes the three densitydependent translational degrees of freedom. Further, the non-idealities in the equation of state (i.e., due to attractive and repulsive forces) caused by each of the 3c- 3 density-de pendent rotational and vibrational degrees is approximated as equivalent to the non-idealities caused by each of translational degree of freedom. The inclusion of the rotational and vibrational motions in an approximate way through the parameter c improves the simultaneous prediction of various thermodynamic proper ties. The partition function in equation 1 approaches the correct ideal-gas limit as density approaches zero and a Prigogine- and Flory-type partition function at liquid densities. Moreover, this partition function is valid for both non-polar and anisotropic molecules. For non-polar molecules, with φ = 0, the partition function in equation 1 reduces to a form analogous to that used in the PHCT. The isotropic interactions, φ , are calculated using a Lennard-Jones potential energy function, and following Gubbins and Twu (4), the iso tropic dipole-induced dipole interactions are calculated assuming an aver age polari ζ ability for the molecule. In this work, for anisotropic interac tions, φ , we are considering only systems in which the dipolar forces are predominant, and all other anisotropic interactions are assumed to be negligible. The P A C T equation of state, obtained by differentiating the parti tion function in equation 1, is given by 180

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

αηι

αη%

100

αηι

Ρ

( 1 + c z** + c

=

ani

+ c z }

(2)

In the P A C T repulsions due to hard-chains are calculated using the parameter c and the equation of Carn ah an-Starling (11) for hard-sphere molecules. The attractive Lennard-Jones isotropic interactions are calcu lated using the perturbation expansion of Barker and Henderson (12). Higher-order terms in the perturbation expansion are accounted for by using a Pade' approximation for Helmholtz free energy. The perturba tion expansion results for spherical molecules are extended to chain-like molecules with the following reduced quantities: f

=

^ T*

=

— eq

(3)

and ~ Vi

_

~

_ J i _ _

vl ~

(A)

V

N rd*/V2

1

A

where N is Avagadro's number. The equation of state for pure nonpolar molecules contains three characteristic parameters. Besides c, the parameters are molecular soft-core size, ν (or the related hard-core size, v£), and characteristic temperature, T*. Values for these three A

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

'

300

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

parameters are obtained from data reduction, using experimental vapor pressure and liquid-density data. The other parameters e/k, q, r, and d are only necessary for mixture calculations. The Parameter q is propor tional to the molecular surface area, e is the energy per unit external sur face area of a molecule and r is the number of segments of diameter d (or σ) in a chain-like molecule. For pure fluids, the parameters e and q and parameters r and σ always appear as a product. For mixtures, σ and e are determined by correlating v* and cT for a large number of similar fluids. In PACT, both the anisotropic multipolar interactions and isotropic dipole-induced dipole interactions are calculated using the perturbation expansion of Gubbins and Twu (4.) assuming the molecules to be effectively linear. The anisotropic interactions are calculated by treating these forces as a perturbation over isotropic molecules. Higher-order terms in the perturbation expansion are accounted for by using a Pade' approximation for Helmholtz free energy. Gubbins and Twu obtained the isotropic dipole-induced dipole interactions by treating these forces as a perturbation over Lennard-Jones molecules and truncating the series after the first perturbation term. Their results for small molecules are extended to chain-like molecules by use of reduced quantities in equa tions 3 and 4 and by defining the following characteristic reduced tem peratures. For dipolar interactions,

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

3

and for isotropic dipole-induced dipole interactions

™

Τ x

ν* ckT

,

αλ

αμ

where ε characterizes the segment-segment dipolar interactions and a is the average polari ζ ability. For pure fluids, parameters ε and q always appear as a product which can be evaluated from known values of dipole moment, μ, and using the relation μ

μ

For mixtures, the segmental anisotropic dipolar interaction energy can be obtained from q [ = cT /(e/&)] and equation 7. Complete equations for the configurational Helmholtz energy and the equation of state involving dipolar and isotropic dipole-induced dipole interactions are given by Vimalchand (12.). Mixtures. The pure-component partition function is extended to mix tures using a one-fluid approximation, however, without the usual random-mixing assumption. Following Donohue and Prausnitz, the mixing rules for both the isotropic and anisotropic terms are derived using a Lattice theory model. The pure component partition function given by equation 1 is extended to mixtures of chain molecules satisfying the following conditions: (i) For mixtures of both small and large (polymeric) molecules, mixture properties should be based on surface and volume fractions rather than on mole fractions, (ii) Both the isotro pic and anisotropic part of the second virial coefficient should have a quadratic dependence on mole fraction, (iii) The pressure and other

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

14.

VIMALCHAND ET AL.

The Perturbed-Anisotropic-Chain Theory

301

excess thermodynamic properties must remain finite when the chain length becomes infinitely large, (iv) The athermal entropy of mixing must reduce to the well-known Flory-Huggins entropy of mixing when the reduced volume of the pure components and solution are identical, (v) For correct prediction of Henry's constant, the residual chemical potential should be proportional to the product q^j rather than g-r-. (vi) For molecules with significantly different intermolecular potential ener gies, nonrandom-mixing due to molecular clustering becomes important. For mixtures of spherical molecules, the mixing rules should agree with the nonrandom-mixing theory of Henderson (14.). Details of the derivation of mixing rules satisfying the above condi tions are given by Vimalchand (12.). Complete expression for the configurational Helmholtz free energy is given in the Appendix. t

t

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

Results and Discussion Pure Fluids. The Perturbed-Anisotropic-Chain theory has been tested with a wide variety of fluids in which molecules interact with substantial quadrupolar [Vimalchand and Donohue (lu)] and dipolar forces. The properties of pure fluids were predicted with parameters which are independent of temperature and pressure. The pure-component partition function was fitted to available experimental data and then purecomponent parameters were correlated to ensure reliability of data and data reduction. Pure-component parameters have been obtained for eleven fluids with appreciable dipofe moments and these are tabulated in Table I. Table I., Pure-Component Parameters

T*K

lOOv mol

c

6

Τ* Κ

Τ

D ipolar Fluids Carbon monoxide Diethyl ether Chloroform Methyl acetate Dimethyl ether Hydrogen sulfide Sulfur dioxide Methyl iodide Methyl chloride Acetone Acetonitrile

101.9 297.2 351.9 324.0 278.6 271.9 279.1 417.3 280.5 325.3 272.1

2.0274 5.9382 4.6651 4.5508 3.6650 2.1695 2.4765 3.8332 3.2858 4.4500 3.4482

1.0275 1.8054 1.6158 1.8314 1.3214 1.1181 1.5037 1.0002 1.1194 1.3768 1.0013

105 115 140 145 115 150 105 160 178 145 180

225.1 311.4 305.8 326.4 371.3 439.9 377.8

3.1947 6.2948 6.4009 7.3984 6.3798 3.7143 5.7982

1.1636 1.6501 1.6078 1.8224 1.5507 1.0045 1.4883

105 105 105 105 118 142 118

2.5 42.8 49.5 105.7 106.0 127.2 214.7 216.4 315.7 411.9 1094.3

Nonpolar fluids Ethane Pen tan e Isopentane Hexane Cyclohexane Carbon disulfide Carbon tetrachloride

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

302

Parameters for eight non-polar fluids are also given in Table I. For dipo lar fluids, the product ε ς was determined using the dipole moments reported by McCellan (UI). The remaining three pure-component parameters ( T*, v*, and c) for each compound were found by fitting the partition function to experimental liquid-density and vapor pressure data. For mixture calculations, either parameter e or q need to be determined independently. The segmental energy parameter, e, was determined by examining the predicted binary mixture data where one of the com ponents is a C or higher alkane. Following Kaul et al. (Ζ.), e for normal and branched alkanes was assigned a value of 105 Κ. With the parameters given in Table I, errors in calculated vapor pressure and liquid-density are typically within 2% over a wide range of temperature and pressure. Figure 1 shows the experimental and calcu lated vapor pressures for hydrogen sulfide, sulfur dioxide and acetone from triple point to critical point. The average error is less than 2%. Liquid densities were also calculated over a wide range of temperature and présure with similar accuracy. Also, the saturated liquid and vapor volumes usually were predicted within 2% error (up to 0.95 of the critical temperature) as shown in Figure 2 for hexane and for the dipolar fluids, sulfur dioxide, acetone and methyl acetate. Since both acetone and acetonitrile have large dipole moments, we included the isotropic dipole-induced dipole interactions assuming an average polari ζ ability [LandoltrBornstein (lu.)] for these molecules. The parameters obtained by fitting the experimental vapor pressure and liquid-density data are given in Table II. The average errors in vapor pressure and liquid-density are similar to those reported above. The inclusion of isotropic induction forces yields a characteristic energy parameter T* of 303.4 Κ compared to T* = 325.3 Κ obtained without induction forces while the other parameters (v and c) are nearly identi cal in both cases. Similarly for acetonitrile Τ decreases to 226.4 Κ from a high value of 272.1 obtained without induction forces. The values of characteristic dispersive energy ( = cT*) of acetone and acetonitrile, when induction forces are included may be the more realistic and as a result parameters for these fluids can be expected to correlate well with other fluids. μ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

5

Mixtures. Fluid-mixture properties are calculated with pure-component parameters and a single binary interaction parameter defined by

tu = v ^ 7 ^ ( i - % )

(8)

Experimental K-factor (ratio of vapor phase mole fraction to liquid phase mole fraction) data are used to determine the binary interaction parame ter, kjj, which is independent of temperature, density, and composition. Table II. Parameters including average polarizability for compounds with large dipole moments Dipolar fluids Acetone Acetonitrile

Τ* Κ 303.4 226.4

lOOv mo I 4.5515 3.7124

c 1.3628 1.0000

e

Τ 145 180

τ* κ μ

à

406.8 1017.9

0.0592 0.0514

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

VIMALCHAND ET AL.

The Perturbed-Anisotropic-Chain Theory

303

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

14.

Figure 1. Comparison of experimental and predicted vapor pressures of dipolar fluids from their triple point to critical point.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

304

points are expt.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

PACT

0.021

I 200

.

I

300

.

1

400

1

1

500

TEMPERATURE , 'K

Figure 2. Comparison of calculated and experimental saturated liquid molar volumes (up to 0.95 of the critical temperature).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

14.

VIMALCHAND ET AL.

The Perturbed-Anisotropic-Chain Theory

The binary interaction parameter corrects for inadequacies in theory and in mixing rules. To predict properties of multicomponent mixtures with ease and reasonable accuracy, it is desirable to develop an equation to predict mixture properties without the use of k - or at least, with small values of & ·. The PHCT, compared to many other equations of state, predicts accurately the properties of non-polar mixtures with small values of kjj (less than 0.05). The PHCT, however, still requires large values of for fluid mixtures containing a dipolar and a quadrupolar sub stance. Consideration of dipolar and quadrupolar forces in the P A C T allows fairly accurate prediction of mixture properties from purecomponent parameters alone. Usually, the P A C T requires small values of to predict mixture properties with accuracy. Binary interaction parameters for 9 dipolar fluid mixtures and 13 nonpolar - dipolar fluid mixtures are given in Table III. The inclusion of isotropic dipole-induced dipole interaction does not change k^ significantly and therefore it is neglected except for the system acetone pentane and acetone - cyclohexane. For these two systems, the acetone parameters given in Table II were used. Like many widely used equations of state [such as the Peng - Robin son equation (11.)], both the PHCT and P A C T have (three) adjustable parameters for pure-components which are fitted to experimental data. As a result, each fits the pure-component properties rather well. How ever, a much more stringent test of the theory is the prediction of mix ture properties from pure-component parameters alone. Such a predic tion is made in Figures 3 to 6. The K-factors calculated from the P A C T with hij = 0 are compared with experimental values for mixtures con taining dipolar fluids. Also for comparison, K-factors calculated with k^ = 0 using the PHCT and the Peng - Robinson equation of state (PR) are shown. The inclusion of dipolar interactions improves the prediction of Kfactors significantly for the systems sulfur dioxide - acetone (Figure 3), hydrogen sulfide - pentane (Figure 4), and methyl iodide - acetone (Fig ure 5). Sulfur dioxide, acetone, hydrogen sulfide, and methyl iodide are all dipolar fluids. Acetone forms a weak complex with sulfur dioxide and yet the P A C T gives a good prediction of K-factors. In the hydrogen sulfide - pentane system, where molecular sizes differ considerably, both the P A C T and the PHCT predict better K-factors than the Peng - Robin son equation. In all these figures, it can be seen that the P A C T predicts the experimental data much more closely than either the PHCT or the Peng - Robinson equation. The y-x phase diagram for the systems and acetone - cyclohexane (Figure 6) show the effect of including, explicitly, the dipole interactions of acetone, rather than the use of the equivalent attractive dispersion interaction as was done in the PHCT. The P A C T predicts the azeotrope and closely follows the experimental data, however, the PHCT fails to predict the azeotrope and in addition, poorly follows the experimental values. This may be explained as follows: In the alkane-rich liquid phase each acetone molecule is surrounded by alkane molecules and therefore there are no ace to ne-ace to ne (dipole-dipole) interactions. However, when the dipole interaction between pure acetone molecules is empirically replaced by an equivalent dispersion interaction (as was done in the PHCT), the equation predicts an erroneously large attractive energy between acetone and the surrounding alkane molecules. This additional attractive energy incorrectly lowers the predicted mole fraction {j

ι;

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

305

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

306

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

of acetone in the vapor phase as shown in the Figure 6. Similar arguments indicate that the PHCT when compared to the P A C T would predict an increase in mole fraction of acetone in the vapor phase in equilibrium with an acetone-rich liquid phase where alkane molecules are surrounded by acetone molecules. Mixture predictions by the PACT with pure-component parameters alone are quantitatively correct but usually not within experimental error. Hov/ever, mixture properties usually can be fit with small errors using small values of a binary interaction parameter given in Table III. Phase equilibria of even complex hydrogen-bonding systems like chloroform acetone and chloroform - diethyl ether are predicted well with small values of interaction parameters. Conclusion. A new theoretical equation of state, the PerturbedAnisotropic-Chain theory has been developed. This equation takes into account the effects of differences in molecular size, shape, and intermolecular forces including anisotropic dipolar and quadrupolar forces. While prediction of properties of pure fluids is no better for the P A C T than many other equations of state, the prediction of mixture properties, especially highly non-ideal mixtures is improved significantly. The P A C T accurately predicts mixture properties from pure-component parameters alone or with small values of a binary interaction parameter. Table III. Binary Interaction Parameters Binary mixtures

100%

Dipolar fluid mixtures Sulfur dioxide - Acetone Sulfur dioxide - Methyl acetate Sulfur dioxide - Chloroform Acetone - Methyl acetate Acetone - Diethyl ether Acetone - Chloroform Acetone - Methyl iodide Methyl acetate - Chloroform Diethyl ether - Chloroform

-0.75 -1.50 1.74 -1.24 -1.00 -5.00 0.96 -2.55 -4.18

Non- polar and Dipolar fluid mixtures Carbon monoxide - Ethane Ethane - Hydrogen sulfide Ethane - Methyl acetate Ethane - Diethyl ether Hydrogen sulfide - Pentane Acetone - Pentane Acetone - Carbon disulfide Acetone - Cyclohexane Acetone - Carbon tetrachloride Isopentane - Diethyl ether Hexane - Chloroform Chloroform - Carbon tetrachloride Methyl iodide - Carbon tetrachloride

0.20 3.40 3.78 0.82 2.90 0.62 1.60 0.11 0.55 0.60 0.50 0.21 0.96

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

VIMALCHAND ET AL.

The Perturbed-Anisotropic-Chain Theory

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

14.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

307

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

308

Figure 4. Comparison of experimental K-factors for hydrogen sulfide pentane system and calculated values (with k^ = 0).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

The Perturbed-Anisotropic-Chain Theory

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

VIMALCHANDETAL.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

310

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

X acetone

Figure 6. Comparison of experimental y - χ phase equilibrium data of acetone in acetone - cyclohexane system and calculated values (with

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

14.

The Perturbed-Anisotropic-Chain Theory

VIMALCHANDET AL.

311

Because of this success, the P A C T may prove very valuable in predict ing, to a fair degree of accuracy, mixture properties of systems contain ing intermediate molecular weight compounds for which no experimental data exist. We are currently extending the P A C T for systems involving a dipolar and a quadrupolar molecule as well as for complex systems involving polymeric and hydrogen-bonding molecules. Acknowledgments Support of this research by the Chemical Sciences division of the Office of Basic Energy Sciences, U . S. Department of Energy under contract number DE-AC02-81ER10982-A004 is gratefully acknowledged.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

Appendix A summary of various terms in the Helmholtz free energy expression used for calculating the mixture properties with the PerturbedAnisotropic-Chain theory is given below. IG

A - A (T, V)

Γ

= Α

Ϊ

Η ^

Λ

Α

+

μ

+ Α

μ

Repulsions: 2

{r

r

d

d

re

A *> = NkT 1

r d

where < · · · > represents a mixture property, r =0.7405 and c

< >

=

Σ

-^J-

χ

;

= Σί

The ratio of segmental diameters, d (hard-core) and σ (soft-core), is evaluated as a function of temperature and fitted to a polynomial in reduced temperature [Vimalchand and Donohue (lu)]. Lennard-Jones Attractions: -1

u

A

J

A\ jNkT

NkT

J

NkT

A[ /NkT

wher ,4

l m

m _ 1

<«/>

NkT

!

C

l m

'"-

1

NkT t

t

,

m

C 2m

i

L

„m+l

i

t

,

,

C W
i

2

V

m+2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

312

The results obtained for pure-component are generalized to multicomponent mixtures by extending the van der Waals one-fluid theory to seg mental interactions. For example, < C T

#

1

% ; >

x x

N/2

c

Σ ij i *j

j jt

r

C:k

a

The one-fluid theory is applied differently to each term of the perturba tion expansion. This eliminates the approximation that the mixture is completely random. Details of the derivation, expressions for and ( \ and the constants A and C are given by Donohue and Prausnitz (5.) and Vimalchand and Donohue (1H). r

2

lm

nm

Dipole-Induced Dipole Interactions

,

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

^μιηάμ

~, νΤ

= -8.886 Τ {ν, Τ) μμ

NkT

μ

where

<οΤ* ν*>

x x

c

Σ ij i

αμ

ησ% c k {

c: kV ji

+ γ.a .

7

3

Dipolar interactions:

A^

μμ

Α NkT

NkT

-1

INkT

μ

Α£ /NkT

where

NkT

f)

= -.2.962

7Ί* m

NkT

43.596 Κ Λν,

f)

μβ

2

3

υ T

with

x

Ei σ

ϋ

+

σ

3 ι ^ it

V2

jj

and

Τ = <Γ*>

„

5

e ijli

'*-ΖΓ

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

14.

The Perturbed-Anisotropic-Chain Theory

VIMALCHAND ETAL.

313

Expressions for <εΤ ν*> and <εΤ ν > can be obtained from the mixing rules given by Vimalchand and Donohue (10) for quadrupolar interactions by replacing TQ and 6Q with Τ* and € respectively. The cross-term, β .. is given by &)

μ

2

μ

μ

μ

μ

9

The terms J and Κ involve integrals over the radial distribution function for Lennard-Jones molecules. Gubbins and Twu (£) evaluated the integrals as a function of reduced volume and reduced temperature. fifi

μμμ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch014

Literature Cited 1. Flory, P. J. J.Am.Chem.Soc. 1965, 87, 1833. 2. Prigogine, I. "The Molecular Theory of Solutions", North-Holland, Amsterdam, 1957. 3. Pople, J. A. Proc.Roy.Soc. 1954, A221, 498. 4. Gubbins, Κ. E.; Twu, C. H. Chem.Eng.Sci. 1978, 33, 863; ibid., 1978, 33, 879. 5. Donohue, M. D.; Prausnitz, J. M. AIChEJ. 1978, 24, 849. 6. Beret, S.; Prausnitz, J. M. AIChEJ, 1975, 21, 1123; Macromolecules 1975, 8, 878. 7. Kaul, B.; Donohue, M. D.; Prausnitz, J. M. FluidPhaseEquilib. 1980, 4, 171. 8. Morris, W. O. M.S. Thesis, The Johns Hopkins University, Bal timore, 1985. 9. Liu, D. D.; Prausnitz, J. M. Macromolecules 1979, 12, 454; J.Appl.Polym.Sci. 1979, 24, 725; Ind.Eng.Chem.,Prod.Des.Dev. 1980, 19, 205. 10. Vimalchand, P.; Donohue, M. D. Ind.Eng.Chem.Fundam. 1985, 24, 246. 11. Carnahan, N. F.; Starling, Κ. E. AIChEJ. 1972, 18, 1184. 12. Barker, J. Α.; Henderson, D. J.Chem.Phys. 1967, 47, 4714. 13. Vimalchand, P. Ph.D. Thesis, The Johns Hopkins University, Bal timore, 1985. 14. Henderson, D. J.Chem.Phys. 1974, 61, 926. 15. McCellan, A. L. "Tables of Experimental Dipole Moments", W. H. Freeman and Co., London, 1963. 16. Landolt-Bornstein, "Zahlenwerte und Funktionen", Springer, 6th ed., Vol. 1, part 3, p. 510, Berlin, 1951. 17. Peng, D.; Robinson, D. B. Ind.Eng.Chem.Fundam. 1976, 15, 59. RECEIVED

December 2, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

15 Mixing Rules for Cubic Equations of State G. Ali Mansoori

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

Department of Chemical Engineering, University of Illinois, Chicago, IL 60680

Through the application of conformal solution theory of statistical mechanics a coherent theory for the development of mixing rules is produced. This theory allows us to use different approximations for the mixture radial distribution functions for derivation of a variety of sets of conformal solution mixing rules some of which are density and temperature dependent. The resulting mixing rules are applied to the van der Waals, Redlich-Kwong, and Peng-Robinson equations of state as the three representative cubic equations of state.

T h e r e e x i s t s a wealth of i n f o r m a t i o n in t h e l i t e r a t u r e about cubic equations of s t a t e a p p l i c a b l e t o v a r i e t i e s of f l u i d s of chemical and engineering interestAlthough cubic equations of s t a t e a r e generally e m p i r i c a l m o d i f i c a t i o n s of the v a n d e r Waals equation of s t a t e , they have found widespread applications in p r o c e s s design calculations because of t h e i r s i m p l i c i t y - Extension of their a p p l i c a b i l i t y to m i x t u r e s i s generally acieved by introduction of m i x i n g rules f o r their parameters. Mixing rules are expressions r e l a t i n g p a r a m e t e r s of a m i x t u r e equation of s t a t e t o pure fluid parameters through, usually, some composition dependent expressions. Except f o r the van der Waals equation of s t a t e the m i x i n g r u l e s f o r cubic equations of state are empirical expressions. In t h e p r e s e n t r e p o r t w e introduce a statistical mechanical conformal solution technique through which we can d e r i v e v a r i e t i e s of s e t s of m i x i n g r u l e s a p p l i c a b l e t o cubic equations of s t a t e . This pressure, energy, and c o m p r e s s i b i l i t y equations of s t a t i s t i c a l m e c h a n i c s . In P a r t II o f t h e p r e s e n t r e p o r t w e introduce the conformal s o l u t i o n t h e o r y of polar fluid m i x t u r e s ( 1 ) and i t s r e l a t i o n s h i p t o t h e i d e a of m i x i n g r u l e s . In P a r t III we introduce the concept of the c o n f o r m a l 0097-6156/ 86/ 0300-0314506.00/ 0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Mixing Rules for Cubic Equations of State

MANSOORI

15.

315

s o l u t i o n m i x i n g r u l e s and we p r o d u c e d i f f e r e n t s e t s of m i x i n g r u l e s b a s e d on d i f f e r e n t a p p r o x i m a t i o n s f o r the mixture radial distribution functions. In P a r t IV w e r e v i e w the e x i s t i n g f o r m s of t h e c u b i c e q u a t i o n s of s t a t e f o r m i x t u r e s and the d e f i c i e n c i e s of t h e i r m i x i n g r u l e s and c o m b i n i n g r u l e s . F i n a l l y , i n P a r t IV w e i n t r o d u c e g u i d e l i n e s for t h e use of c o n f o r m a l solution mixing rules and c o m b i n i n g r u l e s in equations of s t a t e and we d e m o n s t r a t e a p p l i c a t i o n of such m i x i n g r u l e s and combining r u l e s f o r three r e p r e s e n t a t i v e cubic equations of s t a t e .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

II. C o n f o r m a l S o l u t i o n T h e o r y o f M i x t u r e s Conformal solutions refer to substances whose intermolecular potential energy function, are related to each other and t o those of a r e f e r e n c e fluid, designated by s u b - s c r i p t (oo), according to (1,2) *ij = ' i j ' o o f r / h i j " )

usually

(1)

3

For substances whose intermolecular potential energy f u n c t i o n c a n b e r e p r e s e n t e d b y an e q u a t i o n o f t h e f o r m *ij = i j [ ( i j E

L

/ r

>

n

"(l-ij/r) ]

(2)

m

and f o r w h i c h e x p o n e n t s m a n d n a r e t h e s a m e a s f o r t h e r e f e r e n c e s u b s t a n c e , c o n f o r m a l p a r a m e t e r s f^- a n d h|j w i l l be d e f i n e d b y t h e f o l l o w i n g r e l a t i o n s w i t h r e s p e c t t o t h e i n t e r m o l e c u l a r p o t e n t i a l e n e r g y p a r a m e t e r s E j j a n d L^y f..

hj

= F--/F ij oo» I 1

/ l l

n

= (I L --/i / L

h--

ij

^ ij oo^

(3)

}3

Thus t h e c o n f i g u r a t i o n a l thermodynamic properties of a pure s u b s t a n c e of t y p e (a) a r e r e l a t e d to those of t h e reference substance according to the following relations: F

a

( V , T) = f

P (V, a

a a

T) = ( f

F (V/h 0

a a

/h

a a

S ( V , T) = S ( V / h a

c

G ( P , T) = f a

a a

a a

, T/f

)P (V/h 0

,

T/f

a a

a a

a a

) - NkT£nh

, T/f

) + Nkenh

a a

G (Ph

a a

/f

a a

,T/f

a a

H (Ph

a a

/f

a a

, S )

0

a a

a a

a

(4)

a

) a

(5) (6)

a

) - NkTfcnh

a

a

(7)

and H ( P , S) = f a

w h e r e F, pressure,

0

0

(8)

P, S , G, a n d H a r e t h e H e l m h o l t z f r e e energy, entropy, Gibbs free energy, and e n t h a l p y ,

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

316

respectively. A c c o r d i n g to the above equations, a l l the thermodynamic properties of substance (a) c a n be e x p r e s s e d i n t e r m s o f t h e p r o p e r t i e s of a r e f e r e n c e p u r e substance (o) through the conformal p a r a m e t e r s f and h . The c o n f o r m a l s o l u t i o n t r e a t m e n t of f l u i d s c o m p o s e d of p o l a r m o l e c u l e s i s m o r e c o m p l i c a t e d than f o r n o n - p o l a r fluids. This i s m a i n l y due t o e l e c t r o s t a t i c i n t e r a c t i o n s which cause a d e p a r t u r e of the i n t e r m o l e c u l a r potential from spherical symmetry. The e l e c t r o s t a t i c potential between two otherwise neutral molecules arises from permanent a s y m m e t r y in the charge distribution within the molecules. For any p a i r of l o c a l i z e d charge d i s t r i b u t i o n , the mutual e l e c t r o s t a t i c i n t e r a c t i o n energy can be w r i t t e n in t e r m s o f an i n f i n i t e s e r i e s of i n v e r s e powers of s e p a r a t i o n of any t w o p o i n t s . F o r no o v e r l a p b e t w e e n t h e charge distributions the s e r i e s c o n v e r g e s ( l ) . Thus t h e t r u e p a i l — p o t e n t i a l of p o l a r m o l e c u l e s i s o r i e n t a t i o n - d e p e n d e n t and i s t h e s u m o f d i s p e r s i o n f o r c e a s w e l l a s e l e c t r o s t a t i c interactions. In o r d e r t o e x t e n t u t i l i t y of t h e above f o r m u l a t i o n of the c o n f o r m a l s o l u t i o n t h e o r y t o p o l a r f l u i d s we have p r o p o s e d the f o l l o w i n g a n g l e - a v e r a g e d potential function f o r polar molecular interactions which r e p r e s e n t s the f i r s t o r d e r c o n t r i b u t i o n to the anisotropic f o r c e s ( 1 ) a a

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

a a

* (r,T) = K€ [(o t j

i j

1 J

/ r ) - (o^/r)™]

+ 7|i Mj /[450(kT) r i

4

4

3

- Q| Qj /(1.4kTr 2

-

n

2

1 0

1 2

] -

n

t

2

Hj /(3kTr ) 2

6

(M Qj +vij Q i

2

2

2

) - (oc |ij +oc |i )/r 1

2

j

2

1

1

2

)/(2kTr ) 8

(9)

6

where K = [n/(n-m and w h e r e | i p Q j , a n d OCJ a r e the dipole moment, quadrupole moment, and p o l a r i z a b i l i t y of m o l e c u l e i , r e s p e c t i v e l y . For a polar fluid, whose intermolecular potential energy function c a n be r e p r e s e n t e d by e q . 9 the c o n f o r m a l p a r a m e t e r s f and h a a

a

a

w i l l have the following f o r m s : f

aa = Eaa< > > oo< > >

where

T

r

/ E

T

r

E,j(T,r) = K e A i j

LijCT.r) =

0

h

1 j

a a = a a< > >

(T,r)[H (T,r)] i j

ij[ ij< > >]~ H

T

r

i j

i j

7M

i

4

ii

j

4

r

/ L

oo( > > J T

r

n / m

, / m

HtjCT.r) = [ C ( T , r ) / A ( T , r ) ] Ajj(T,r) = 1 +

T

a

m /

^ n

m )

/[1800(kT) r 3

, 2

- o m

i j

n

Ke ] j j

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

< > 10

15.

MANSOORI

and

Mixing Rules for Cubic Equations of State

C|j(T r) = 1 + ii Mj /[12lcTr f

1

2

2

6

"" o m

m

1 j

317

K€ j] 1

+ (7/20)Q Q /[kTr ^ " ' " o y ^ C y i

+ (n +

Q

j

(oc ii

j

1

2

1

2

2

j

2

+y

2

+oc |i

j

Q

2

i

j

2

1

)/[8kTr - o 8

2

)/[4r - a 6

m

m

m

1 j

1 J

m

K6| ] j

Ke ] i j

The b a s i c c o n c e p t o f t h e CST of m i x t u r e s i s t h e s a m e a s f o r pure fluids, except that f and h i n e q s . 4 - 8 s h o u l d be a a

replaced

with

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

parameters, f

x x = x x < ij> f

f

f

and

x x

a a

h

,

x x

the

mixture

conformal

as g i v e n b e l o w h

ij' i x

)

h

xx = xx( h

Uy

h

ij» i>

< >

x

11

Eqs.ll are c a l l e d the conformal solution mixing rules. F u n c t i o n a l f o r m s of t h e s e m i x i n g r u l e s w i l l be d i f f e r e n t f o r different theories of m i x t u r e s as it will be demonstrated l a t e r in this r e p o r t . In t h e f o r m u l a t i o n of a m i x t u r e t h e o r y w e a l s o n e e d t o know t h e c o m b i n i n g r u l e s for unlike-interaction potential parameters which are u s u a l l y e x p r e s s e d by the f o l l o w i n g e x p r e s s i o n s ffj = ( 1 - k i j ) ( f i t f j j >

1 / Z

;

h i j = C l - *tj)[( h

t i

1 / 3

+hjj

1 / 3

)/2]

3

(12)

w h e r e k j j and fcjj a r e a d j u s t a b l e p a r a m e t e r s . III. S t a t i s t i c a l M e c h a n i c a l T h e o r y of l i i x i n a R u l e s The m o s t i m p o r t a n t r e q u i r e m e n t i n t h e d e v e l o p m e n t o f t h e CST o f m i x t u r e s are mixing rules. In t h e d i s c u s s i o n p r e s e n t e d h e r e we have i n t r o d u c e d a new t e c h n i q u e to r e - d e r i v e t h e e x i s t i n g m i x i n g r u l e s and d e r i v e a n u m b e r of new mixing rules some of which are densityand temperature-dependent. A c c o r d i n g to s t a t i s t i c a l mechanics the m a c r o s c o p i c t h e r m o d y n a m i c p r o p e r t i e s of a p u r e f l u i d a r e r e l a t e d to i t s m i c r o s c o p i c m o l e c u l a r c h a r a c t e r i s t i c s by the f o l l o w i n g t h r e e equations (5,4) oo

(13)

u = Ujg + 2 n p j 0 ( r ) g ( r ) r d r 0 2

oo

P = pRT + ( 2 / 3 ) n p J r * r ( r ) g ( r ) r d r 0 2

(14)

oo (15) (4TT/RT)J[g(r)-l]r dr 0 w h e r e u i s t h e i n t e r n a l e n e r g y , P i s t h e p r e s s u r e and KJ i s the isothermal compressibility, 0(r) is the pair intermolecular potential energy f u n c t i o n , and g ( r ) i s t h e

k

t

= 1/pRT -

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

318

radial (or pair) distribution function. Eqs.13-15 are commonly called the energy equation, the virial (or pressure) equation, and the c o m p r e s s i b i l i t y equation, respectivelyFor a multicomponent mixture these equations assume the following forms ( 5 - 5 ) oo

u = u

1 g

+ 2np2iSjX Xjj0 j(r)g (r)r dr 0 i

i

(16)

2

1 j

oo

P = pRT + ( 2 / 3 ) n p 2 2 j X x J r ^ ( r ) g 0 ic = ( l / p R D l B l / S i S j X i X j l B ^ j i

i

j

1 j

| J

(r)r dr

(17)

2

(18)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

T

In t h e a b o v e e q u a t i o n s s u m m a t i o n s a r e o v e r a l l t h e ( c ) c o m p o n e n t s o f t h e m i x t u r e , Xj a n d xj a r e t h e m o l e f r a c t i o n s , and |B| i s a c x c d e t e r m i n a n t w i t h i t s r e p r e s e n t a t i v e t e r m s i n the f o l l o w i n g f o r m oo

ij

B

=

x

i ij 6

+

x

i jP ij x

G

G

ij

=

4nJ[g j(r)-1 ] r d r 0 2

1

w h e r e 6,-j i s t h e K r o n e e k e r d e l t a , a n d IBIJJ i s t h e c o f a c t o r o f t e r m B y i n d e t e r m i n a n t |B|. E q s . 1 3 - 1 8 c a n b e u s e d i n t h e manner p r e s e n t e d below in o r d e r t o d e r i v e m i x i n g r u l e s b a s e d on d i f f e r e n t m i x t u r e t h e o r y a p p r o x i m a t i o n s : III.1. O n e - F l u i d T h e o r y o f M i x i n g R u l e s : F o r t h e d e v e l o p m e n t of o n e - f l u i d m i x i n g r u l e s we i n t r o d u c e a p s e u d o - p u r e f l u i d which can r e p r e s e n t the configurational p r o p e r t i e s of a m i x t u r e p r o v i d e d that the p s e u d o - p u r e f l u i d and the m i x t u r e m o l e c u l a r i n t e r a c t i o n s o b e y e q . 1. By r e p l a c i n g eq.1 in e q s . 1 3 , 1 4 , 1 7 , a n d 18 a n d t h e n e q u a t i n g c o n f i g u r a t i o n a l i n t e r n a l e n e r g y , p r e s s u r e , and i s o t h e r m a l c o m p r e s s i b i l i t y of t h e p s e u d o - p u r e f l u i d and t h e m i x t u r e w e w i l l o b t a i n t h e following equations f

xx xx^oo(y)%o(y)y

f

xx xx/y^ oo(y)goo(y>Y

h

h

2 d

y 2i2jX x f h jj0 =

,

1

2 d

j

i j

1

o o

( y ) g ( y ) y d y (19)

y 2i2jX x f h Jy0' =

1

j

i j

j j

2

i j

o o

(y)g (y)y dy i j

2

(20) {1-4nph J[g x x

0 0

(y)-1]y dy}-1=2i2 x x |B| /|B| 2

j

i

j

1 J

(21)

It s h o u l d b e p o i n t e d o u t t h a t f o r t h e c a s e o f t h e h a r d - s p h e r e fluid eq.19 vanishes, eq.21 remains the same, while eq.20 reduces to the following form h

x x 9 o o < ) = 2 i 2 j X x h g ( 1) 1

i

j

i j

i j

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(22)

MANSOORI

15.

Mixing Rules for Cubic Equations of State

Solution of eqs.19-21 should produce the t w o n e c e s s a r y expressions (mixing rules) relating f and h of t h e p s e u d o - p u r e f l u i d t o f j j a n d h|j o f c o m p o n e n t s o f t h e mixture. F o r t h i s p u r p o s e w e s h o u l d u s e an a p p r o x i m a t i o n technique r e l a t i n g t h e r a d i a l d i s t r i b u t i o n f u n c t i o n s (RDF) i n t h e m i x t u r e t o t h e p u r e r e f e r e n c e f l u i d RDF. H o w e v e r , a t a f i r s t g l a n c e i t s e e m s t h a t w e h a v e i n o u r hand t h r e e e q u a t i o n s and t w o unknowns. A s i t w i l l be d e m o n s t r a t e d b e l o w f o r m o s t o f t h e a p p r o x i m a t i o n s o f t h e m i x t u r e RDFs which a r e used here these t h r e e equations produce t w o mixing rules. In t h e p r e v i o u s i n v e s t i g a t i o n s for the d e v e l o p m e n t of m i x i n g r u l e s (5-11) a l l t h e i n v e s t i g a t o r s have used only eq.19 and/or eq.20. Our s t u d i e s i n d i c a t e t h a t w h i l e e q s . 1 9 and 20 a r e e s s e n t i a l i n t h e d e v e l o p m e n t o f m i x i n g r u l e s , eq.21 can add a new dimension which could be s i g n i f i c a n t i n t h e c a l c u l a t i o n o f p r o p e r t i e s o f m i x t u r e s . In what f o l l o w s d i f f e r e n t a p p r o x i m a t i o n s w i l l be used f o r relating gy to g in order to derive different sets of mixing rules. x

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

319

O

x

x

x

Q

III. 1 . i . Random M i x i n g A p p r o x i m a t i o n ( R M A ) f o r M i x t u r e RDFs: In t h i s a p p r o x i m a t i o n i t i s a s s u m e d t h a t t h e n o n - s c a l e d RDF o f a l l t h e c o m p o n e n t s o f t h e m i x t u r e a n d t h e i n t e r a c t i o n RDFs a r e i d e n t i c a l (5_), i . e . Q\](r)

=g

2 2

(r) = ... = g^(r)= ...

(23)

When t h i s a p p r o x i m a t i o n i s r e p l a c e d in e q s . 1 9 - 2 1 , eq.21 w i l l v a n i s h a n d e q . 1 9 a n d 20 w i l l p r o d u c e t h e f o l l o w i n g m i x i n g rules *xx< > = 2 i S j X x ^ ( r ) r

i

*'xx< > r

=

j

(24)

i j

SiSjXiXjf'ijCr)

(25)

For e x a m p l e , in the case of the L e n n a r d - J o n e s (12-6) intermolecular potential function we will derive the f o l l o w i n g m i x i n g r u l e s (12) f r o m e q s . 1 3 and 14. 'xx^xx f

xx xx h

2

= SiSjXiXjfijhij

4

= SiSjXiXjfjjhjj

2

4

(

2

6

)

(27)

For a h a r d - s p h e r e potential we will d e r i v e only one m i x i n g r u l e t h r o u g h the RMA and that i s d e r i v e d by r e p l a c i n g e q . 2 3 in 2 2 . The r e s u l t i n g m i x i n g r u l e w i l l be

h

x x

, / 3

= ZiZjXiXjh^'S

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

( 2 8

)

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

320

l l l . l . i i . Conformal Solution Approximation (CSA) f o r Mixture RPFs: This a p p r o x i m a t i o n technique seems m o r e logical f o r use in the d e v e l o p m e n t of m i x i n g r u l e s than R M A . A c c o r d i n g t o t h i s a p p r o x i m a t i o n t h e s c a l e d RDFs i n a m i x t u r e a r e a l l identical (5J, i-egii(y)=

9zz(y) = - - - = g j j ( y ) = . . .

(29)

When we u s e t h i s a p p r o x i m a t i o n in e q s . 1 9 and 20 t h e y both produce the same m i x i n g r u l e which i s f

XX XX h

= SlSjXiXjf^hy

(

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

N o w , b y r e p l a c i n g e q . 2 9 i n 21 an a d d i t i o n a l m i x i n g r u l e be p r o d u c e d w h i c h i s t h e f o l l o w i n g |B*|/pRTic

Txx

= SiSjX|Xj|B*||j

3

0

)

will

(31)

where |B*| j = x [ 6 j + X j ( h j / h ) ( p R T i c - 1 ) ] . Eq.30 is actually the second van d e r Waals mixing rule which i s well known, but e q . 3 1 i s a new m i x i n g r u l e f o r h which i s replacing the f i r s t van d e r Waals mixing rule. This new mixing rule, in p r i n c i p l e , i s a c o m p o s i t i o n - , t e m p e r a t u r e - , and d e n s i t y - d e p e n d e n t m i x i n g r u l e . This i s because K y w h i c h a p p e a r s i n t h e r i g h t a n d l e f t hand s i d e s o f t h i s equation i s g e n e r a l l y t e m p e r a t u r e - and d e n s i t y - d e p e n d e n t . For example, f o r a binary m i x t u r e eq.31 can be w r i t t e n in t h e f o l l o w i n g f o r m (5.) t

i

1

1

x x

T x x

x

x

x

h

x

x

= {liSjXjXjhjj + x x ( h 1

{1+x

2

1 1

h

2 2

1 2

2

)(pRTic

By u s i n g t h e h a r d - s p h e r e p o t e n t i a l ( b y r e p l a c i n g 22) w e w i l l d e r i v e t h e f o l l o w i n g m i x i n g r u l e

e q . 2 9 in

h

x

x

2 2

1 2

)(pRTK

- 1 ) }/ (31-1)

n

- 2h

T x x

-1)}

l X 2

(h +h

-h

x

T x x

= SiSjXtXjhjj

(32)

This mixing rule i s the f i r s t van d e r Waals m i x i n g rule which, in conjunction with eq.30 is usually used f o r calculation of mixture thermodynamic properties ( 7 , 8 . 1 0 , 1 1 ). It s h o u l d b e p o i n t e d o u t t h a t e q . 3 2 c o n s t i t u t e s another mixing rule f o r hard-sphere m i x t u r e s . A s a result, while the CSA approximation produces two mixing rules f o r potential functions with two parameters, it also produces two mixing rules f o r a h a r d - s p h e r e potential which i s a one-parameter potential function.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

MANSOORI

15.

Mixing Rules for Cubic Equations of State

321

MLI.iii, H a r d - S p h e r e Expansion (HSE) A p p r o x i m a t i o n f o r M i x t u r e RDFs: It i s d e m o n s t r a t e d t h a t t h e RDF o f a p u r e f l u i d ( x ) can be expanded around t h e h a r d - s p h e r e ( h s ) RDF in t h e f o r m ( 3 ) g

x x

(y)= g

(y)+ (f x

h s

X

/ T

o*^l

+

< xx o*> 92 f

/ T

2

+

—

< > 33

Let us a l s o a s s u m e that we could make a s i m i l a r e x p a n s i o n f o r RDFs i n a m i x t u r e a r o u n d t h e h a r d - s p h e r e m i x t u r e RDFs as t h e f o l l o w i n g

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

gij= 9 i j

h s

(y)

(fij/T *)g!(y)

+

0

+

(fij/To*) g (y) 2

2

+

—

(

3 4

>

The j u s t i f i c a t i o n b e h i n d t h i s e x p a n s i o n i s g i v e n e l s e w h e r e ( 6 9 ) . Now b y r e p l a c i n g e q s . 3 3 a n d 3 4 i n e i t h e r o f e q s . 1 9 o r 20 w e w i l l b e a b l e t o d e r i v e t h e f o l l o w i n g t w o m i x i n g r u l e s by e q u a t i n g the coefficients of the s e c o n d and t h i r d o r d e r i n v e r s e t e m p e r a t u r e t e r m s of the r e s u l t i n g e x p r e s s i o n . T

f

XX XX = h

SiSjX^jfyhy

' x x ^ x x = 2i2jX x f 2h l

J

1 J

(

3

5

)

(36)

1 J

These m i x i n g r u l e s a r e used f o r c a l c u l a t i o n of e x c e s s p r o p e r t i e s of a m i x t u r e o v e r the h a r d - s p h e r e m i x t u r e (13) at t h e s a m e t h e r m o d y n a m i c c o n d i t i o n s ( 9 ) . A p p l i c a t i o n o f t h e HSE a p p r o x i m a t i o n in eq.21 w i l l not produce any additional mixing rule. I l l . l . i v . D e n s i t y E x p a n s i o n (DEX) A p p r o x i m a t i o n f o r M i x t u r e R D F s : i t h a s b e e n d e m o n s t r a t e d t h a t t h e RDF o f a p u r e f l u i d can b e e x p a n d e d a r o u n d t h e d i l u t e g a s RDF, e x p [ - 0 ( r ) / k T ] , i n the f o r m (14) g

x x

( y ) = [1 + F

x x

( y ) ] exp[-0 (r)/kT]

(37)

x x

Let us a l s o a s s u m e that we could make a s i m i l a r e x p a n s i o n f o r RDFs i n a m i x t u r e a r o u n d t h e d i l u t e g a s m i x t u r e RDFs a s the f o l l o w i n g g i j ( y ) = [1 + F

( y ) ] exp[-0 (r)/kT]

x x

(38)

1 j

Now b y r e p l a c i n g e q s . 3 7 a n d 3 8 i n e q . 1 9 a n d a f t e r a n u m b e r of a l g e b r a i c m a n i p u l a t i o n s w e w i l l d e r i v e t h e f o l l o w i n g mixing rule fxx xx=2 2 x x f h

1

j

1

j

1 j

h {1-(f /f i j

i j

x x

-1)[u-u

+T(C -C v

v i g

1 g

)/kT

)/(u-u

1 g

)]}

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(39)

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

322

The l a t t e r m i x i n g r u l e c a n b e u s e d , j o i n e d w i t h a n o t h e r m i x i n g r u l e , f o r c a l c u l a t i o n of m i x t u r e p r o p e r t i e s . Similar a p p r o x i m a t i o n s c a n be u s e d i n o r d e r t o d e r i v e o t h e r m i x i n g r u l e s f r o m the v i r i a l and c o m p r e s s i b i l i t y e q u a t i o n s .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

III.2. M u l t i - F l u i d T h e o r y o f M i x i n g R u l e s : The b a s i c a s s u m p t i o n i n d e v e l o p i n g t h e m u l t i - f l u i d m i x i n g r u l e s i s the same as the o n e - f l u i d a p p r o a c h e x c e p t that in this case we w i l l s e a r c h f o r a h y p o t h e t i c a l multicomponent ideal m i x t u r e which could r e p r e s e n t the configurational p r o p e r t i e s of a multicomponent r e a l m i x t u r e , both w i t h the s a m e number of c o m p o n e n t s and at the same t h e r m o d y n a m i c conditions. In t h i s c a s e e q s . 1 9 - 2 1 w i l l be r e p l a c e d b y t h e f o l l o w i n g s e t of e q u a t i o n s f

xi xi^oo(y)9oo(y^V dy=2jXjf h jj0

f

h

x 1

2

h

x 1

Jy0-

o o

(y)g

o o

i j

1

(y)y dy=2jX f h 2

j

i j

1 j

o o

(y)g j(y)y dy

Jy^'

o o

(40)

2

i

(y)g (y)y dy i j

2

{l-4nph J[g (y)-1]y2dy}-1=2 X3|B| /|B| x i

0 0

j

i j

(41) (42)

E x p r e s s i o n s f o r Bjj and G|j w i l l b e t h e s a m e a s i n e q . 1 8 . In the c a s e of the h a r d - s p h e r e f l u i d e q . 4 0 w i l l r e d u c e to the following form h

xigoo

eq.41

h s

( )=2jX h g 1

j

i j

j j

h s

(1),

(40-1)

w i l l v a n i s h and e q . 4 3 w i l l r e m a i n t h e s a m e .

Ml.2.i. A v e r a g e Potential Model ( A P M ) f o r M i x t u r e RDFs: t h i s a p p r o x i m a t i o n i t i s a s s u m e d t h a t (5_), g ( r ) = [ g ^ r ) + gjj(r)]/2 l j

Q (r) * gjj(r)

In (43)

t 1

When this a p p r o x i m a t i o n i s r e p l a c e d in e q s . 4 0 - 4 2 , eq.42 w i l l v a n i s h a n d e q . 4 0 and 41 w i l l p r o d u c e t h e f o l l o w i n g m i x i n g rules *xi< > = 2 j X j 0 ( r )

(44)

*'xi(r) = SjXj^'i^r)

(45)

r

i j

For e x a m p l e , in the c a s e of t h e intermolecular potential function f o l l o w i n g m i x i n g rules(JL2) . f

h

2

= 7-x-f-h-

Lennard-Jones (12-6) we w i l l derive the

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(

4

6

)

15.

MANSOORI

Mixing Rules for Cubic Equations of State

323

'xihxi = SjXjfjjh^

(47)

4

For a h a r d - s p h e r e rule hx i

1

/

= 2jXjhij

3

p o t e n t i a l we w i l l d e r i v e only one m i x i n g

(*»>

1 / 3

Ml.2.ii. Multi-fluid CSA Approximation for Mixture A c c o r d i n g to this approximation t h e s c a l e d RDFs m i x t u r e a r e r e l a t e d as the following

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

gij(y) = tgij(y) + gjj(y)]/2

RDFs: in a

g (r) * gjj(y)

(49)

t 1

W h e n w e u s e t h i s a p p r o x i m a t i o n i n e q s . 4 0 a n d 41 t h e y b o t h produce the same m i x i n g rule which is f -h =

Y x f - h -

( °) 5

N o w , b y r e p l a c i n g e q . 4 9 i n 42 an a d d i t i o n a l m i x i n g r u l e be p r o d u c e d w h i c h i s t h e f o l l o w i n g

will

|B*|/ RTK

(51)

P

= SjXjlB*^

T x 1

where |B*|

tj

= x {6 i

i j

+ (x h /2)[(pRTK j

1 j

T x i

-1)/h +(pRTic j-1)/h j]} x 1

T x

X

Eq.50 i s actually the second van d e r Waals m u l t i - f l u i d m i x i n g r u l e , but eq.51 i s a new m i x i n g r u l e f o r h j . By using the h a r d - s p h e r e p o t e n t i a l (by r e p l a c i n g e q . 4 9 in 4 0 - 1 ) we will derive the following mixing rule X

h

= ZjXjhfj

x j

(52)

This mixing rule is the f i r s t m u l t i - f l u i d m i x i n g rule which, in conjunction with eq.50 for calculation of m i x t u r e thermodynamic should be p o i n t e d out that eq.52 c o n s t i t u t e s rule for hard-sphere mixtures. I l l . 2 . i i i . M u l t i - F l u i d HSE M i x i n g R u l e s : as t h e o n e - f l u i d c a s e w e c a n d e r i v e rules f

van der Waals i s usually used properties. It another mixing

In a s i m i l a r m a n n e r the following mixing

h •= Y x f - h -

(53)

f' x i - "h x i• = Y xJ ifj hi j 2

These

2

mixing

rules

(

are

used

for

calculation

of

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

5

4

excess

)

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

324

p r o p e r t i e s of a m i x t u r e o v e r the h a r d - s p h e r e

mixture.

III. 2 . i v , M u l t i - F l u i d DEX M i x i n g R u l e s : In a s i m i l a r m a n n e r a s the o n e - f l u i d case we can d e r i v e the f o l l o w i n g m i x i n g rule f

xi xi h

=

SjXjfijh^d-Cf^/fxi-DEupUig)/^ +T(C

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

This mixing rule can be used, rule, f o r calculation of m i x t u r e

v 1

-C

v 1 g

)/(u u

joined with properties.

r

1 g

)]}

another

(55) mixing

IV. A p p l i c a t i o n o f M i x i n g R u l e s f o r C u b i c E o u a t i o n s o f S t a t e In o r d e r t o a p p l y t h e v a r i e t i e s o f t h e c o n f o r m a l solution mixing rules which a r e introduced here f o r cubic and other equations of s t a t e t h e f o l l o w i n g c o n s i d e r a t i o n s should be taken into account: (i) Conformal solution mixing rules a r e f o r the molecular conformal volume parameter, h, and the m o l e c u l a r c o n f o r m a l e n e r g y p a r a m e t e r , f. (ii) Conformal solution mixing rules a r e applicable for c o n s t a n t s o f an e q u a t i o n o f s t a t e o n l y . B e f o r e u s i n g a s e t o f m i x i n g r u l e s f o r an e q u a t i o n o f s t a t e one h a s t o e x p r e s s t h e p a r a m e t e r s of t h e equation of s t a t e with r e s p e c t t o the m o l e c u l a r c o n f o r m a l p a r a m e t e r s h and f. This w i l l then make i t p o s s i b l e t o w r i t e t h e c o m b i n i n g r u l e s a n d m i x i n g r u l e s f o r t h e e q u a t i o n o f s t a t e . In w h a t f o l l o w s m i x i n g r u l e s and combining rules f o r three representative cubic e q u a t i o n s of s t a t e a r e d e r i v e d and t a b u l a t e d . IV.1. M i x i n g R u l e s f o r t h e v a n d e r W a a l s E q u a t i o n o f S t a t e : The v a n d e r W a a l s e q u a t i o n o f s t a t e ( 1 5 ) c a n b e w r i t t e n i n the f o l l o w i n g f o r m Z = Pv/RT = v / ( v - b ) - a/vRT

(56)

P a r a m e t e r b of this equation of s t a t e i s p r o p o r t i o n a l to m o l e c u l a r v o l u m e ( b « h ) and p a r a m e t e r a i s p r o p o r t i o n a l t o (molecular volume)(molecular energy (aocfh). Then, in order to apply the mixing rules introduced in this r e p o r t f o r the van d e r W a a l s equation of s t a t e we must r e p l a c e h with b and f w i t h a/b i n a l l t h e m i x i n g r u l e s . In T a b l e I m i x i n g r u l e s f o r t h e v a n d e r W a a l s e q u a t i o n o f s t a t e b a s e d on d i f f e r e n t t h e o r i e s of m i x t u r e s a r e r e p o r t e d . The combining rules f o r a ^ and b y (i*j) of this equation of s t a t e , c o n s i s t e n t w i t h eqs.12 w i l l be

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

MANSOORI

15.

Mixing Rules for Cubic Equations of State

325

T a b l e I: M i x i n g R u l e s f o r t h e v a n d e r W a a l s E q u a t i o n o f S t a t e One-Fluid

Mixing

Rules

a RMA

[2i2jX Xja jb j]^/2 i

i

i

2 2jX x a b 3]1/2

/ [

i

i

j

i j

i j

Theory{

^ , Theory{

vdW

^ i ^ i ^ ^ j ^ i j l i i . ^ i ^ j ^ i i ^iJ_

]

a = SiSiXiXiQii

b = SiSjXiXjbij

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

HSE

a = SiSjXiXja^j

Theory{

b =

[2i2jX Xja j] /2 2jX Xja j /b j i

a = [a DEX

v d W

1

2

i

+(b/vRT)

2

i

i

i

2i2jX Xja j /b i

2

i

]/[ 1+a

i j

v

d

W

/vRT]

Theory{ b = SiSjXiXjbij a = SjSjXjXjaij

CSA

Theory{ 1+A

-|B*|/ ZiSjXiXjlB*^;

X X

A PuMl t i -TFhl u e iodr y {M i x i n g M b

B*

i j

=x (6 i

i j

+x A J

x x

b j/b) i

Rules

= SjXjaij vdW

Theory{ b

=

SjXjbij

= SjXjay HSE

Theory{ b

a

DEX

i

=

la j

=

A

X

X

I

V C

i

j

j

i

2

i

j V Y " K i / v R T ) SjXjajj^/b^j]/[ 1 +a i

bi

= SjXjbij

b

d W

V

/

v

R

T

^

ai

= SjXjaij

Theory{ 1+A

X

j

Theory{

CSA

A

[SjX a j]2/2 x a j /b j

=

PRTK

= pRTK

T

T

x

x

x

i

-1 -l

x i

« | B * | / 2jXj|B*lij;

B*i =x [6 j+x b j(A i/bi+A j/b )/2] j

i

i

= [2a(v-b)^-RTb(2v-b)]/[RTv

j

i

X

X

^-2a(v-b)^l

= [2ai(v-bi) -RTbi(2v-bi)]/[RTv -2ai(v-bi) l 2

2

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

j

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

326

a

ij = < - ij) ij( ii jj/t> 1

k

b

a

a

1 i

b )

1 / 2

;

)/2]

3

j j

(57) b

t j

= (1-£

1 j

)[(b

1 1

1 / 3

+b

j j

1 / 3

IV.2. M i x i n g R u l e s f o r the R e d l i c h - K w o n g E q u a t i o n of S t a t e : The R e d l i c h - K w o n g equation of s t a t e (16) which i s an e m p i r i c a l m o d i f i c a t i o n o f t h e v a n d e r W a a l s e q u a t i o n c a n be w r i t t e n in the f o l l o w i n g f o r m

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

Z = Pv/RT = v / ( v - b ) -

a/[RT

3 / 2

(v+b)]

(58)

P a r a m e t e r b of t h i s e q u a t i o n o f s t a t e i s p r o p o r t i o n a l t o m o l e c u l a r v o l u m e (boch) and p a r a m e t e r a i s p r o p o r t i o n a l t o (molecular volume) (molecular energy) or (a<xf.h ). Then, in o r d e r to apply the m i x i n g r u l e s i n t r o d u c e d in this r e p o r t f o r the R e d l i c h - K w o n g equation of s t a t e we must r e p l a c e h w i t h b and f w i t h ( a / b ) in a l l the mixing r u l e s . In T a b l e II m i x i n g r u l e s f o r t h e R e d l i c h - K w o n g e q u a t i o n o f s t a t e b a s e d on d i f f e r e n t t h e o r i e s o f m i x t u r e s a r e r e p o r t e d . The c o m b i n i n g r u l e s f o r a^j a n d b j j ( i * j ) o f t h i s e q u a t i o n o f 3 / 2

3 / 2

2 / 3

s t a t e , c o n s i s t e n t w i t h eqs.12 w i l l be t h e s a m e as e q s . 5 7 . IV.5. M i x i n g Rules f o r the P e n g - R o b i n s o n Equation of S t a t e : The P e n g - R o b i n s o n e q u a t i o n of s t a t e ( 1 7 ) w h i c h i s a n o t h e r e m p i r i c a l m o d i f i c a t i o n of the van d e r W a a l s e q u a t i o n can be w r i t t e n in the f o l l o w i n g f o r m

Z = Pv/RT = v / ( v - b ) - a ( T ) v / { R T [ v ( v + b ) + b ( v - b ) ] }

(59)

P a r a m e t e r b of t h i s equation of s t a t e i s a c o n s t a n t which i s p r o p o r t i o n a l to the m o l e c u l a r volume (boch). However, p a r a m e t e r a of t h e P e n g - R o b i n s o n e q u a t i o n of s t a t e i s not a constant and i t i s a function of t e m p e r a t u r e as the following.

a(T) = a { l + 8 [ l - T / T ) c

c

1 / 2

]}

2

(60)

where a

c

= 0.45724R T 2

C

2

/P

C

9 = 0.37464+1 . 5 4 2 2 6 C J - 0 . 2 6 9 9 2 ( A )

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2

15.

MANSOORI

Mixing Rules for Cubic Equations of State

327

T a b l e II: M i x i n g R u l e s f o r t h e R e d l i c h - K w o n a E q u a t i o n o f S t a t e One-Fluid

RMA

Mixing

Rules

___________

Theorvl

b = [2 2jX x a

V

1

1

j

j

j

2/3

b

l

j

l0/3

/

2

i

2

j

X

j

X

j

a

i

2/3 . 4/3 1/

j

b

j

]

2

b = SjSjXiXjbjj

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

HSE

Theory! b -

[SiZ x x a j

1

j

2/3b

i j

i j

1/3]2

a = _>i2jX x a j(a/b)^3( _ i

DEX

j

i

1

.

/ 2 i 2 j X

/

[ ( a i j

4/3

X j a i j

p2/3(

b i

b / a

b i r

l/3

)273_|]^

Theory{ b = SiSjXjXjbij _ _ _ _ _ _ _ ^

CSA

Theory{ 1+A =|B*|/ S i S j X i X j l B * ^ ;

B*ij=

X X

Multi-Fluid

Mixing ai -

APM

XiCeij+xjAxxbij/b)

Rules [2jX a j

1 j

2/3b

1 j

4/3]5/2

/ 2 j X j a i j

2/3

b l j

10/3

Theorvl

,

ai =

[2jX

vdW Theory{

J

J

j

a

J

i

j

2/3

b

i

j

1/3]3/2

/

[

2

j

X

j

b

|

j

]

l/2

1 J

bi = SjXjbij

_______ HSE

Theory{

^

^ ^ ^ . z / S b y ^ S j Z ^ x j ^ ^ / S b y - l / S

=

ai = _ > j X a i j ( a / b ) ^ 3 { - ( j

DEX

1

[

a i j

/

)2"/3(

b i j

/

b i

a i

)2/3_

1 ] C i

}

Theory{ bi = SjXjbij

CSA

Theory{ 1+__

C Ci

x i

= |B*|/ _>jXj|B*lij ;

B*ij= x ( 6 j + x b j [ A i

i

j

i

x i

/b +A I

x j

/bj])

= (3/2)(a/bR)T "V/Untv/Cv+b)]-l/2; = C3/2)(ai/biR)T- £n[v/(v+b i ) ] - l / 2 3 / 2

A

X

A

X

X

I

= =

PRTK PRTK

T

x

T x 1

x

-1

-1

=-l+RT /2( 2_ 2)2/[ 3

=

v

b

-l+RT3/2( 2v

b l

2)2/[

R T

R T

{ ( v

{ ( v

v + b

))

v + b i

2_ (

)}2-

a

a i

2 v

+b)(v-b)

2

1

(2v+bi)(v-bi) ]

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

328

and T and P a r e the c r i t i c a l point t e m p e r a t u r e and p r e s s u r e , r e s p e c t i v e l y ; and u> i s t h e a c e n t r i c f a c t o r . In order to utilize the statistical mechanical mixing rules f o r the P e n g - R o b i n s o n equation of s t a t e we must f i r s t s e p a r a t e t h e r m o d y n a m i c v a r i a b l e s f r o m constants of this equation of s t a t e . F o r t h i s p u r p o s e we m a y w r i t e t h i s e q u a t i o n of s t a t e in t h e f o l l o w i n g f o r m c

c

Z = Pv/RT = v / ( v - b ) -

[(A/RT+C-2(AC/RT)

1 / 2

]/

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

[(v+b)+(b/v)(v-b)]

(61)

w h e r e A = a ( l + 8 ) and C = a 6 / R T . This new f o r m of the P e n g - R o b i n s o n equation of s t a t e i n d i c a t e s that t h e r e e x i s t three independent constant p a r a m e t e r s in this equation w h i c h a r e A , b, a n d C. P a r a m e t e r s b and C a r e p r o p o r t i o n a l t o t h e m o l e c u l a r v o l u m e (boch a n d C « h ) w h i l e p a r a m e t e r A i s proportional to (molecular volume)(molecular energy) or (Aocfh). B a s e d on d i f f e r e n t t h e o r i e s o f m i x t u r e s m i x i n g r u l e s f o r this new f o r m of t h e Peng-Robinson equation of s t a t e a r e r e p o r t e d i n T a b l e III. The combining r u l e s f o r t h e unlike i n t e r a c t i o n p a r a m e t e r s of this equation of s t a t e a r e as t h e f o l l o w i n g 2

c

A

ij = ^- ij) ij( ii jj k

b

A

b

ij = (1-^ij)[( ii

c

ij =

b

A

c

/ b

1 / 3 + b

2

c

(62)

ii jj>

jj

b

1 / 3

)/2]

3

^ - ^ n i c ^ ^ c ^ ^ / z ] ^

(63)

(64)

Similar procedures t o those d e m o n s t r a t e d above can be used f o r d e r i v a t i o n of conformal solution mixing r u l e s f o r o t h e r cubic equations of s t a t e .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

15.

MANSOORI

Mixing Rules for Cubic Equations of State

T a b l e HI: One-Fluid

329

M i x i n g Rules f o r the P e n a - R o b i n s o n Eg. of S t a t e Mixing Rules

A = [SiSjXiXjAijb^/^^jXiXjA^byS]!/? RMA Theory{ b = t Z i S j X i X j A i j b i j f / S i S j X i X j A i j b i j l J ^ C = [_> iSjXiXjAijCij^/SiSjXiXjAijCij]' ^ 7

A = 2i2jXiXjAjj _Ei_EjXiXjbij vdW Theory! b = 2i_£jXjXjCij C =

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

A HSE Theory!

2i_!iXiXj Ajj - CSiSjXiXjAij^/SiSjXiXjA^/bij = [S i_>j x i Xj Aj j ] ^ /2 i2j x i Xj Ai j - /C i j C =

DEX Theory!

A - SiSjXiXjAijd-lCAij/bijXb/A)-!!^ = Si-SjX-jXjb-jj C SiSjXjXjCij

=

b

b

=

CSA Theory!

A = SiSiXjXjAjj xx=lB*I/ 2i_>jX |B*hj; C = 2i_!jXiXjCij 1+A

B* =x (6 j+x A b /b);

lXj

ij

i

i

j

xx

ij

Multi-Fluid Mixing Rules A APM Theory! b - [SjXjAijbijJ/^jXjAijbij]Ml C = C__ x A jC 3/2: x A C ]1/2 j

i

ij

j

j

ij

ij

SjXjAtj SjXjbij ZjXjCij

A vdW Theory! b C HSE Theory!

j

Ai = SjXjAij i 2j j ij] /_>jXjAij /bij Ci [ZjXjAylZ/SjXjAijZ/Cy

b

=

[

x

A

2

2

Ai = 2jXjAij{1-[CAij/bij)(b/A)-1]^ i) DEX Theory! i = 2 j j i j Ci - SjXjCij b

x

b

Ai = SjXjAij CSA Theory { 1+A = |B*[V __jX |B*| ; Ci = SjXjCij xi

j

B* =Xi[6ij+(x b /2)(A /bi+A /bj)]

ij

ij

j

ij

xi

A = P R T K - 1 = -HRT/{RTv v-b)^-2Av^/( v ^ + b ^ } A = p R T < - l = -HRT/{RTv /( v - b i ) - 2 A i V / ( v + b i ) } I = {[A-Y(ACRT)]/(2bRTv 2)}en[CY+b-b^2)/(v+b+bv 2)] + ^(ACRT)/!2[^(ACRT)-A]) Ci= {[Ai-v^(AiCiRT)]/(2biRTV2)}£n[(v+b bjV2)/(v+b i+bjV-)] +V(AiCiRT)/{2[V(A iCiRT)-Ail} x x

x i

T x x

2

T x i

2

3

r

2

2

2

r

r

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

xj

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

330

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

Acknowledgment: The a u t h o r t h a n k s P r o f e s s o r C a r o l H a l l f o r h e r h e l p f u l c o m m e n t s and c o r r e c t i o n s . This r e s e a r c h i s s u p p o r t e d b y t h e U.S. D e p a r t m e n t of E n e r g y G r a n t N o . DE-FG02-84ER13229.

Literature Cited 1. Massih, A.R.; Mansoori, G.A. Fluid Phase Equilibria 1983, 10, 57. 2. Brown, W.B. Proc. Roy. Soc. London Series A, 1957, 240;Phil. Trans. Roy. Soc. London Series A, 1957, 250. 3. Hill, T.L. "Statistical Mechanics" McGraw-Hill, New York, N.Y. 1956. 4. Kirkwood, J.G.; Buff, F. J. Chem. Phys. 1951, 19, 774. 5. Mansoori, G.A.; Ely, J. F. Fluid Phase Equilibria 1985, 22, 253. 6. Lan, S.S.; Mansoori, G.A. Int. J. Eng. Science 1977, 15, 323. 7. Leach, J.W.; Chappelear, P.S.; Leland, T.W. AIChE J. 1968, 14, 568; Proc. Am. Petrol. Inst. Series III, 1966, 46, 223. 8. Leland, T.W. Adv. Cryogenic Eng. 1976, 21, 466. 9. Mansoori, G.A.; Leland, T.W. J. Chem. Soc., Faraday Trans.II 1972, 68, 320. 10. Mansoori, G.A. J. Chem. Phys. 1972, 57, 198. 11. Rowlinson, J.S.; Swinton, F.L. "Liquids and Liquid Mixtures" 3rd Ed., Butterworths, Wolborn, Mass. 1982. 12. Scott, R.L. J. Chem. Phys. 1956, 25, 193. 13. Mansoori, G.A.; Carnahan, N.F.; Starling, K.E.; Leland, T.W. J. Chem. Phys. 1971, 54, 1523. 14. Mansoori, G.A.; Ely, J.F. J. Chem. Phys. 1985, 82, 406. 15. Van der Waals, J.D. "Over de continuiteit van den gasen vloeistoftoestand" Leiden, 1873. 16. Redlich. O.; Kwong, J.N.S. Chem. Rev. 1949, 44, 233. 17. Peng, D.Y.; Robinson, D.B. Ind. Eng. Chem. Fundam. 1976, 15, 59. RECEIVED November

5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

16 Improved Mixing Rules for One-Fluid Conformal Solution Calculations James F. Ely

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

Chemical Engineering Science Division, Center for Chemical Engineering, National Bureau of Standards, Boulder, CO 80303

During the past few years there has been great interest in improving equation of state mixing rules for fluid modeling. In this report new one-fluid mixing rules are proposed which explicitly take size difference effects into account. The resulting rules give the hard sphere mixture compressibility factor exactly. Comparisons of predicted excess properties for Lennard-Jones mixtures of varying size and energy ratios are presented. The results of the new mixing rules are superior to the van der Waals one-fluid model, especially for the excess volume. D u r i n g the p a s t few y e a r s t h e r e has been i n c r e a s e d i n t e r e s t t h e deveopment o f improved m i x i n g r u l e s f o r use w i t h e n g i n e e r i n g e q u a t i o n o f s t a t e models o f f l u i d s . T h i s a c t i v i t y has been f o c u s e d i n two a r e a s — m i x i n g r u l e s developed f o r the parameters o f s i m p l e ( c u b i c ) e q u a t i o n s o f s t a t e (1-5) and t h e o r e t i c a l l y d e r i v e d m i x i n g r u l e s which a r e used i n h i g h a c c u r a c y pure f l u i d e q u a t i o n s o f s t a t e v i a c o n f o r m a l s o l u t i o n arguments ( 6 - 1 0 ) . Examples o f the l a t t e r approach a r e the extended c o r r e s p o n d i n g s t a t e s and hard sphere e x p a n s i o n models proposed by L e l a n d and coworkers (10-12) w h i l e the former i n c l u d e s a p p l i c a t i o n s o f m o d i f i e d Redlich-Kwong type e q u a t i o n s o f s t a t e (1_3JJ*) t o complex m i x t u r e s . A somewhat s u r p r i s i n g r e s u l t o b t a i n e d by comparing t h e s e two approaches i s t h a t the m o d i f i e d c u b i c e q u a t i o n s o f s t a t e u s i n g van der Waals m i x i n g r u l e s t y p i c a l l y g i v e as good i f not b e t t e r phase e q u i l i b r i u m r e s u l t s than the c o n f o r m a l s o l u t i o n model u s i n g the same m i x i n g r u l e s (1_5). T h i s r e s u l t becomes much more pronounced as t h e s i z e d i f f e r e n c e s i n the m i x t u r e become l a r g e , much more so than f o r systems which d i s p l a y l a r g e energy d i f f e r e n c e s . In addition to t h e s e o b s e r v a t i o n s based on d a t a f o r r e a l f l u i d m i x t u r e s , r e c e n t computer s i m u l a t i o n s t u d i e s (16-18) on model Lennard-Jones m i x t u r e s have d r a m a t i c a l l y p o i n t e d out the f a i l u r e s o f the v a r i o u s c o n f o r m a l s o l u t i o n models as a f u n c t i o n o f s i z e d i f f e r e n c e . I n t h i s r e p o r t , a new s e t o f m i x i n g r u l e s i s proposed f o r use i n c o n f o r m a l s o l u t i o n m i x t u r e c a l c u l a t i o n s . These new m i x i n g r u l e s This chapter not subject to U.S. copyright. Published 1986, American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

332

a r e d e r i v e d by making a new "mean d e n s i t y " a p p r o x i m a t i o n and then expanding t h e r a d i a l d i s t r i b u t i o n f u n c t i o n o f a b i n a r y p a i r about t h a t o f a hard s p h e r e . The new r u l e s reduce t o t h e van d e r Waals o n e - f l u i d model i n t h e l i m i t o f low d e n s i t y and a r e e x a c t f o r t h e c o m p r e s s i b i l i t y f a c t o r o f hard sphere m i x t u r e s . Some p r e l i m i n a r y r e s u l t s o b t a i n e d w i t h t h e m i x i n g r u l e s a r e compared w i t h computer s i m m u l a t i o n s f o r Lennard-Jones m i x t u r e s w i t h v a r y i n g s i z e differences.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

Derivation The d e r i v a t i o n o f t h e new m i x i n g r u l e s c l o s e l y f o l l o w s t h e d e r i v a t i o n o f t h e h a r d sphere e x p a n s i o n (HSE) m i x i n g r u l e s p r e s e n t e d by Mansoori and L e l a n d (19_). One f i r s t assumes t h a t t h e N-body c o n f i g u r a t i o n a l energy o f a m i x t u r e o r pure f l u i d i s p a i r w i s e a d d i t i v e and t h a t each p a i r w i s e p o t e n t i a l can be w r i t t e n as a sum o f an e f f e c t i v e h a r d sphere r e p u l s i o n and l o n g range a t t r a c t i o n which i s a u n i v e r s a l f u n c t i o n o f reduced i n t e r m o l e c u l a r s e p a r a t i o n r / o j j r

v

)

u

-

S

(

r

) +

e

ij

u

u

(

r

/

)

°

i

j

I t i s a l s o g e n e r a l l y assumed t h a t t h e p a i r w i s e p o t e n t i a l parameters may be temperature and/or d e n s i t y dependent ( 1 2 ) . The n e x t s t e p i s t o make an a p p r o x i m a t i o n c o n c e r n i n g t h e r a d i a l d i s t r i b u t i o n f u n c t i o n s i n t h e m i x t u r e . I n t h e HSE, Mansoori and L e l a n d i n t r o d u c e d t h e mean d e n s i t y a p p r o x i m a t i o n (MDA) g

ij

(

r

:

C

p

]

a '

T

:

C

e

aB

L

C

a6

a

]

)

"

g

(

o

r

/

o

ij

;

k

T

/

e

i j ' ^

I n t h i s e q u a t i o n [ p ] denotes t h e s e t o f m o l e c u l a r number d e n s i t i e s i n t h e m i x t u r e and [ e $ ] and [ a g ] i n d i c a t e t h a t i n t h e m i x t u r e , t h e r a d i a l d i s t r i b u t i o n f u n c t i o n depends upon t h e complete s e t o f i n t e r m o l e u c l a r p o t e n t i a l parameters. The s u b s c r i p t o denotes a pure f l u i d r a d i a l d i s t r i b u t i o n f u n c t i o n . I n words, t h e mean d e n s i t y a p p r o x i m a t i o n says t h a t t h e d i s t r i b u t i o n o f m o l e c u l e s i n a m i x t u r e as a f u n c t i o n o f reduced d i s t a n c e and temperature i s t h e same as t h a t i n a pure f l u i d e v a l u a t e d a t an e f f e c t i v e o r mean reduced number d e n s i t y , po3. The mean d e n s i t y a p p r o x i m a t i o n i s a g r e a t improvement over t h e more common van d e r Waals o n e - f l u i d a p p r o x i m a t i o n which assumes t h a t a

a

g i J

(r;[p ],T;[ a

e c i 8

],[o

a 8

])

=

a

3

g ^ r / a ^ ;KT/e,po )

a l t h o u g h t h e mean d e n s i t y a p p r o x i m a t i o n and van d e r Waals (VDW1) a p p r o x i m a t i o n s a r e i d e n t i c a l i n t h e case o f e q u a l energy parameters. The s h o r t c o m i n g s o f t h e VDW1 a p p r o x i m a t i o n a r e i l l u s t r a t e d i n F i g u r e 1 which has been redrawn from ( 2 0 ) . T h i s f i g u r e i l l u s t r a t e s the r a d i a l d i s t r i b u t i o n f u n c t i o n s f o r a b i n a r y m i x t u r e and h y p o t h e t i c a l pure f l u i d (denoted by " o " ) . N o n e t h e l e s s , a problem s t i l l e x i s t s w i t h t h e MDA i n t h a t as t h e s i z e d i f f e r e n c e s i n t h e m i x t u r e i n c r e a s e , t h e mean d e n s i t y no l o n g e r s u p p l i e s an adequate e s t i m a t e o f t h e environment encountered by t h e i - i o r j - j p a i r s , a l t h o u g h i t a l m o s t always p r o v i d e s a good e s t i m a t e o f t h e i - j d i s t r i b u t i o n (30).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ELY

16.

333

One-Fluid Conformal Solution Calculations

In o r d e r t o a t l e a s t p a r t i a l l y improve t h i s s i t u a t i o n we have developed a s i m p l e m o d i f i c a t i o n t o the MDA g i v e n by t h e f o l l o w i n g g

ij

( r , ; C p

]

T ; [ e

a '

]

C

aB ' °aB

] )

R

=

i J«o

( r /

°U

s k T / e

ij'

p ;

'

3 )

(

1

)

where

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

R

ij

=

[

8 i > i j '

^

[

^

]

)

/

C

(

°

!

P

°

3

)

(

2

)

We s h a l l r e f e r t o t h i s as t h e " m o d i f i e d " mean d e n s i t y a p p r o x i m a t i o n (MMDA). One way o f v i e w i n g t h i s a p p r o x i m a t i o n i s i s t h a t we not o n l y must s c a l e t h e e f f e c t i v e pure d i s t r i b u t i o n f u n c t i o n i n t h e r a d i a l d i r e c t i o n but a l s o i n t h e p r o b a b i l i t y d i r e c t i o n , thereby a v o i d i n g t h e problems demonstrated i n F i g u r e 1. To i l l u s t r a t e t h e e f f e c t s o f t h i s a p p r o x i m a t i o n more c l e a r l y , F i g u r e 2 compares t h e MDA and MMDA r a d i a l d i s t r i b u t i o n f u n c t i o n s t o t h e s i m u l a t e d argon-argon d i s t r i b u t i o n i n an a r g o n - k r y p t o n m i x t u r e r e p o r t e d by Mo, e t a l . (21_). As i s shown i n t h i s f i g u r e , t h e r e i s a d e f i n i t e improvement a t s m a l l i n t e r m o l e c u l a r s e p a r a t i o n s even f o r a system where t h e s i z e d i f f e r e n c e i s s m a l l ( 0 2 / 0 1 = 1.07). An o b v i o u s s h o r t c o m i n g o f t h i s a p p r o x i m a t i o n i s t h a t MMDA g j j ( r ) does not approach 1 as r -» °°. I n p r a c t i c e t h i s i s not a problem s i n c e we always d e a l w i t h i n t e g r a l s o f g j j ( r ) and g ( r ) does approach the correct l i m i t . Thus, one s h o u l d view t h e MMDA as a s c a l i n g o f integrals of g j j ( r ) . C o n t i n u i n g w i t h t h e d e r i v a t i o n o f t h e m i x i n g r u l e s , t h e pure f l u i d d i s t r i b u t i o n f u n c t i o n a p p e a r i n g i n E q u a t i o n 1 i s expanded i n a power s e r i e s i n 1/T t o o b t a i n 0

Sij

(

^ ' ; W'

r

]

T

[

[

a

]

a3

)

R

"

ij

[ g

( r / a

o

; P

ij

^

3 )

( 3 )

J

n=1

J

A s i m i l a r e x p a n s i o n f o r a h y p o t h e t i c a l pure f l u i d w i t h parameters 0 ' and e y i e l d s 00

g(r;p,T;e,o)

=

g (r/o;po ) + ^ ^ n=1 3

Q

/

k

^

U

Y ( °;po )] r /

3

F i n a l l y , t o complete t h e d e r i v a t i o n , one t a k e s t h e v i r i a l e q u a t i o n (22) f o r a m i x t u r e N z

mix "

1

-

2

N

r

2*&P i JK ^ E E E E V Vj i-1 j - 1 J

p

0

3

^ i / ^

g

i j

(

r

)

d

r

and h y p o t h e t i c a l pure f l u i d a t t h e same number d e n s i t y

o

- 1

=

2ir6p J o

r

3

(du / d r ) g ( r ) d r 0

0

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

W

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

334

g

(r)

g

(r)

F i g u r e 1. R a d i a l d i s t r i b u t i o n f u n c t i o n s f o r a m i x t u r e o f s o f t s p h e r e s w i t h a s i z e r a t i o o f a p p r o x i m a t e l y two ( 2 0 ) . The van der Waals a p p r o x i m a t i o n (denoted by o) y i e l d s a d i s t r i b u t i o n f u n c t i o n w h i c h i s v e r y r e a s o n a b l e f o r t h e 1-2 p a i r but u n s a t i s f a c t o r y f o r the 1-1 and 2-2 p a i r s i n t h e m i x t u r e .

g(r)

F i g u r e 2. Comparison o f t h e mean d e n s i t y and m o d i f i e d mean density approximations to the r a d i a l d i s t r i b u t i o n f u n c t i o n f o r the k r y p t o n - k r y p t o n p a i r i n an e q u i m o l a r a r g o n - k r y p t o n m i x t u r e ( 2 1 ) . The open c i r c l e s a r e from t h e computer s i m u l a t i o n , s o l i d l i n e i s the MDA and dashed l i n e i s t h e MMDA.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ELY

16.

One-Fluid Conformal Solution Calculations

335

s u b s t i t u t e s E q u a t i o n s 3 and 4 f o r t h e r a d i a l d i s t r i b u t i o n f u n c t i o n s and s u b t r a c t s t o f i n d N

N

• if g | » +

E

K ^ ^ U

-

n+1

N

(PF)

I

,

<*>

N

E

[x.x.a .R..s t 3

n

1

3

- a c

n + 1

]

In t h i s e q u a t i o n J and I a r e i n t e g r a l s over t h e h a r d sphere d i s t r i b u t i o n f u n c t i o n g and t h e ¥ , r e s p e c t i v e l y , and t h e d e r i v a t i v e o f t h e i n t e r m o l e c u l a r p o t e n t i a l . By s e t t i n g t h e z e r o t h and f i r s t o r d e r terms o f t h e e x p a n s i o n g i v e n i n E q u a t i o n 5 e q u a l t o z e r o , one o b t a i n s t h e d e s i r e d m i x i n g r u l e s n

n

0

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

E

J

-

o

o

N

N

T

=

5

n

V

ft

R. .

x.x. a . i J U 5

fa

(6)

U

and N

sa

3

N

3 x.x. e. . o. . R.. i J i j l j i j

E E

(7)

where R j j i s d e f i n e d i n E q u a t i o n 2 . Note t h a t as w r i t t e n , a appears on both s i d e s o f these e q u a t i o n s , and t h a t t h e h a r d sphere d i s t r i b u t i o n f u n c t i o n s which d e f i n e R j j depend upon d e n s i t y and c o m p o s i t i o n . A l s o s i n c e lim p+o

HS g.. 11

=

J

_. HS lim g o p-»o

= 1

the m i x i n g r u l e s reduce t o t h e common van d e r Waals o n e - f l u i d Q

N

N

rules

Q

and N

S d w \dw

"

N

E E

x . x . e.. o..

i n t h e l i m i t o f low d e n s i t y . Hard Spheres The f i r s t n o t a b l e a s p e c t o f t h e new m i x i n g r u l e s (MMDA r u l e s ) i s t h a t they a r e e x a c t f o r t h e c o m p r e s s i b i l i t y f a c t o r o f h a r d spheres

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

336 =

°)« T h i s i s o b v i o u s from t h e w e l l known r e l a t i o n s f o r t h e e q u a t i o n s o f s t a t e f o r pure hard spheres and hard sphere m i x t u r e s H S

Z

1

=

4

+

H S

g

n

O

(9)

(a )

0 0

0

and HS Z

N

1 +

=

N

E

mix

E

x.x.n..g

£j

l j

I J

&

H S

(10)

(a..)

I J

I J

where n = T P a and n, . = 5 p a?.. E q u a t i n g 9 and 10 r e s u l t s i n . 'o_ 6 . . o_ i j 6 i j ° e x a c t l y E q u a t i o n 5. One can s o l v e f o r a f o r hard spheres a l g e b r a i c a l l y g i v e n v a l u e s f o r t h e c o n t a c t d i s t r i b u t i o n f u n c t i o n s . We s h a l l c o n s i d e r two c a s e s , t h e P e r c u s - Y e v i c k and t h a t g i v e n by Carnahan and S t a r l i n g . From t h e Wertheim (23) and T h i e l e (2*1) s o l u t i o n f o r t h e P e r c u s - Y e v i c k e q u a t i o n one has 3

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

K

1

HS,PY, ^ *ij i j (

0

M

X

3

+

)

m

m —72

< 1

d

O

. i j

Ml

x

m and g

HS,PY, (o

>

0

N

^

-

0

^o

-

(1

^

/( I ON 1 2 )

n) o

where 3

K

6

m

m

6

K

2

N S

n

=

T

1

1*1

x.a ? I l

and d

ij

•

2

V j

/

(

o

i

+

a

J

)

n i s d e f i n e d above. I n t h e C a r n a h a n - S t a r l i n g (25) and c o r r e s p o n d i n g MCSL (26) case f o r m i x t u r e s one f i n d s 0

HS.MCSL,

,

_

_1

3 c%

n

m

d

ij m (13)

2

c S 3 f__2>! 0

0

2 ,2 n d.. m i j m

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ELY

16.

337

One-Fluid Conformal Solution Calculations

and

'

°

(O °

=

«1

-

^

0*0

3

S o l v i n g E q u a t i o n 5 f o r n (= -g- p o ) u s i n g t h e c o n t a c t u t i o n f u n c t i o n s g i v e n by E q u a t i o n s 11 and 12 one f i n d s n

distrib-

(1 + 2G) 1 + 6G rr^2G)

=

Ml

,x

( 1 5 )

where

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

n G

=

V* V* >, >, ft j f t

HS,PY, . x.x. n. . g. . ( o . .) i J iJ iJ iJ

L i k e w i s e , t h e s o l u t i o n o f E q u a t i o n 5 f o r rj u s i n g t h e c o n t a c t d i s t r i b u t i o n s g i v e n by E q u a t i o n s 13 and 14 i s t h e r e a l r o o t o f

-3

6G

1

+

1 -2

60' + 1 -

.

N

f

/ /- \ 1

where G i s e v a l u a t e d w i t h t h e MCSL c o n t a c t d i s t r i b u t i o n f u n c t i o n . To i l l u s t r a t e t h e d i f f e r e n c e s between t h e new m i x i n g r u l e s and the VDW-1 r u l e s , F i g u r e s 3-4 show t h e hard sphere r e s u l t s o b t a i n e d w i t h E q u a t i o n s 6, 13 and 14 r e l a t i v e t o t h a t from t h e VDW-1 m i x i n g r u l e , E q u a t i o n 8 a t two p a c k i n g f r a c t i o n s , n = 0.2 and n = 0.4. We note t h a t t h e e f f e c t i s asymmetric i n t h e c o n c e n t r a t i o n , X£, o f t h e l a r g e r component i n t h a t s m a l l c o n c e n t r a t i o n s o f l a r g e m o l e c u l e s have l a r g e e f f e c t s on t h e e f f e c t i v e mean d e n s i t y . I n F i g u r e 5 t h e same r a t i o o f new o n e - f l u i d dimater t o the van der Waals r e s u l t i s shown as a f u n c t i o n o f d e n s i t y ( p a c k i n g f r a c t i o n ) a t a c o n s t a n t c o m p o s i t i o n o f x-j = 0 . 8 . As one would e x p e c t , t h e d i f f e r e n c e i s most pronounced a t t h e h i g h e s t d e n s i t i e s and i n c r e a s e s as t h e s i z e r a t i o increases. In F i g u r e 6, t h e d i f f e r e n c e between t h e VDW C a r n a h a n - S t a r l i n g c o m p r e s s i b i l i t y f a c t o r i s p l o t t e d v e r s u s t h e " e x a c t " v a l u e g i v e n by the MCSL e q u a t i o n . I n t h i s case t h e r e i s no e r r o r a s s o c i a t e d w i t h the new m i x i n g r u l e s . F i g u r e 7, however, p l o t s t h e e r r o r i n the H e l m h o l t z f r e e energy from t h e van der Waals r u l e and t h a t o b t a i n e d from E q u a t i o n s 6, 13 and 14. The l i n e s above t h e z e r o l i n e r e p r e s e n t t h e VDW-1 case and t h e l i n e s below t h e z e r o l i n e a r e the new r e s u l t s . The VDW-1 r e s u l t s a r e u n i f o r m l y i n g r e a t e r e r r o r than the new r u l e s and a r e a l s o o f o p p o s i t e s i g n . F i n a l l y , F i g u r e s 8 and 9 compare t h e o n e - f l u i d d i a m e t e r s c a l c u l a t e d u s i n g E q u a t i o n s 11, 12 and 15, i . e . , t h e P e r c u s - Y e v i c k s o l u t i o n f o r hard spheres t o t h e MCSL r e s u l t s a t two c o m p o s i t i o n s . We note t h a t t h e r e i s n e g l i g i b l e d i f f e r e n c e between t h e two (maximum b e i n g 0.04$) thus one i s j u s t i f i e d i n u s i n g t h e a l g e b r a i c a l l y simpler Percus-Yevick r e s u l t s f o r p r a c t i c a l c a l c u l a t i o n s . A l t h o u g h the van der Waals diameter i s r e a s o n a b l e , E q u a t i o n s 15 and 16 a r e d e f i n i t e improvements. A l s o , these r e s u l t s f o r hard

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

338

g g I D

10. 8

-

1.07

-

10. 6

-

10. 5

-

Composition, Xi F i g u r e 4. R a t i o o f the h a r d sphere d i a m e t e r r e s u l t i n g from the MMDA t o t h a t o f the VDW1 model as a f u n c t i o n o f s i z e r a t i o and c o m p o s i t i o n a t p a c k i n g f r a c t i o n o f 0.2.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

16.

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One-Fluid Conformal Solution Calculations

339

Packing Fraction

F i g u r e 5 . R a t i o o f the MMDA hard sphere diameter t o t h a t o f t h e VDW1 model as a f u n c t i o n o f p a c k i n g f r a c t i o n and s i z e r a t i o a t a f i x e d c o m p o s i t i o n o f x<| = 0 . 8 .

3.000 i

1

r

Packing Fraction, rj

F i g u r e 6. E r r o r i n t h e m i x t u r e hard sphere c o m p r e s s i b i l i t y f a c t o r as a f u n c t i o n o f s i z e r a t i o and p a c k i n g f r a c t i o n a t x-j = 0 . 8 . The MMDA i s e x a c t f o r t h i s hard sphere p r o p e r t y .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

340

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

10 . 00 .900 .800 .700 .600 .500 .400 < .300 < .200 I 1 . 00 g .000 2 -.100 -.200 -.300 -.400 -.500 .00

i

~i

i

i

i

i

i

I

I

I

I

I

i

r

Xi=0.5

_J

I

.20 .30 Packing Fraction, r\

1 .0

L_

.40

.50

F i g u r e 7. E r r o r i n t h e m i x t u r e r e s i d u a l H e l m h o l t z f r e e energy a s a f u n c t i o n o f s i z e r a t i o and p a c k i n g f r a c t i o n a t x-| = 0.5. The c u r v e s above t h e z e r o l i n e a r e o b t a i n e d w i t h t h e VDW1 diameter w h i l e t h o s e below t h e z e r o l i n e were c a l c u l a t e d w i t h t h e MMDA.

i

1

1

r

Xi=0.9

Packing Fraction F i g u r e 8. Comparison o f t h e MMDA hard sphere d i a m e t e r o b t a i n e d w i t h t h e MCSL c o n t a c t h a r d sphere d i s t r i b u t i o n f u n c t i o n s and t h o s e obtained with the Percus-Yevick v i r i a l equation. Results are shown a s a f u n c t i o n o f s i z e r a t i o ( r ) a t v a r i o u s p a c k i n g f r a c t i o n s a t a f i x e d c o m p o s i t i o n o f x-j = 0 . 9 .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

16.

ELY

One-Fluid Conformai Solution Calculations

.20

.30

341

.40

.50

Packing Fraction, η

F i g u r e 9. Comparison o f t h e MMDA hard sphere d i a m e t e r o b t a i n e d w i t h t h e MCSL c o n t a c t hard sphere d i s t r i b u t i o n f u n c t i o n s and those o b t a i n e d w i t h t h e Perçus-Yevick v i r i a l e q u a t i o n . Results are shown as a f u n c t i o n o f s i z e r a t i o ( r ) a t v a r i o u s p a c k i n g f r a c t i o n s a t a f i x e d c o m p o s i t i o n o f x-j = 0 . 5 .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

342

s p h e r e s c l e a r l y demonstrate t h a t m i x i n g r u l e s must be d e n s i t y dependent t o a c c u r a t e l y r e p r e s e n t c o n t r i b u t i o n s from r e p u l s i v e forces. Lennard-Jones M i x t u r e s

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

In o r d e r t o t e s t t h e new m i x i n g r u l e s f o r m i x t u r e s c o n s i s t i n g o f more r e a l i s t i c m o l e c u l e s , c o n f o r m a i s o l u t i o n c a l c u l a t i o n s u s i n g E q u a t i o n s 6, 7, 11 and 12 have been performed u s i n g a Lennard-Jones 6-12 r e f e r e n c e f l u i d . The 32 term BWR t y p e e q u a t i o n o f s t a t e r e p o r t e d by N i c o l a s , e t a l . (27) was used t o o b t a i n t h e Lennard-Jones 6-12 p r o p e r t i e s . I n g e n e r a l , the c a l c u l a t i o n procedure y i e l d e d t h e d e n s i t y , e x c e s s volume, e x c e s s e n t h a l p y , e x c e s s Gibbs energy and e x c e s s i n t e r n a l energy. M i x i n g was performed a t c o n s t a n t p r e s s u r e , i . e . , Ν X (T,p,[x h

])

-

X

m i X

( T , p , [ x ]) - £ i=1

x.

Χ.(Ρ,Τ)

where X i s any e x c e s s p r o p e r t y and X ^ denotes a pure f l u i d v a l u e o f t h a t p r o p e r t y . I n a l l c a l c u l a t i o n s t h e L o r e n t z - B e r t h e l o t combing r u l e s were used f o r t h e u n l i k e p a i r p o t e n t i a l parameters. Comparisons were made w i t h t h r e e d i f f e r e n t s e t s o f computer simulation results: S i n g e r and S i n g e r * s (28) Monte C a r l o r e s u l t s and t h e more r e c e n t m o l e c u l a r dynamics s i m u l a t i o n s o f Gupta (1_8) and H o h e i s e l , e t a l . (|_6). F i g u r e s 10-11 compare the e x c e s s volumes c a l c u l a t e d by t h e HSE-MDA model ( 2 9 ) , VDW-1 and t h e MMDA p r e s e n t e d i n t h i s work a s a f u n c t i o n o f s i z e r a t i o a t t h r e e d i f f e r e n t energy ratios. I n g e n e r a l , t h e MDA i s s l i g h t l y b e t t e r than t h e MMDA f o r the e x c e s s volume but b o t h t h e MMDA and MDA a r e s u b s t a n t i a l improve ments o v e r t h e VDW-1 model. T a b l e I compares t h e p r e d i c t i o n s f o r V , H and G t o t h o s e o b t a i n e d w i t h t h e VDW-1 model and t h e Monte C a r l o calculations. F i g u r e 12 compares t h e MMDA t o t h e e x c e s s volume and F i g u r e 13 compares t h e e x c e s s Gibbs energy t o t h e r e c e n t i s o t h e r m a l - i s o c h o r i c r e s u l t s o f Gupta and H a i l e f o r 02/= 2. I n c l u d e d on t h e s e f i g u r e s are t h e c o r r e s p o n d i n g r e s u l t s f o r t h e VDW-1 model. F i g u r e 13 a l s o shows t h e c a l c u l a t e d Perçus-Yevick r e s u l t s r e p o r t e d by Gupta. T a b l e I I compares t h e c a l c u l a t e d r e s u l t s t o s i m u l a t i o n s . It is apparent from t h e s e comparisons t h a t t h e MMDA o f f e r s a g r e a t l y improved p r e d i c t i o n o f t h e e x c e s s volume f o r systems where t h e s i z e differences are large. I t i s a p p a r e n t , however, t h a t even though the p r e d i c t i o n s o f t h e e x c e s s Gibbs energy a r e b e t t e r than t h e VDW-1 model, some improvement i s s t i l l i n o r d e r . To e x p l o r e t h e e f f e c t o f l a r g e energy and s i z e d i f f e r e n c e s , T a b l e I I I compares MMDA and VDW-1 p r e d i c t i o n s t o t h e r e c e n t r e s u l t s of H o h e s i e l , e t a l . (1_6). A g a i n , t h e MMDA o f f e r s a s u b s t a n t i a l improvement over t h e VDW-1 r e s u l t s , e s p e c i a l l y f o r t h e e x c e s s volume. e

E

E

E

Summary and C o n c l u s i o n s We have p r e s e n t e d a s i m p l e m o d i f i c a t i o n t o t h e mean d e n s i t y a p p r o x i mation which improves t h e r e s u l t s o b t a i n e d from c o n f o r m a i s o l u t i o n predictions. The model i s e x a c t f o r t h e c o m p r e s s i b i l i t y f a c t o r o f h a r d sphere m i x t u r e s . The r e s u l t s f o r r e a l i s t i c f l u i d s a r e not

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

16.

ELY

One-Fluid Conformai Solution Calculations

343

ει/ε2 = .656

Φ ο

ε «• Ε

-2.0

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

Ο

F i g u r e 10. Comparisons o f c a l c u l a t e d and s i m u l a t e d excess volumes as a f u n c t i o n o f s i z e r a t i o . The s i m u l a t e d and VDW1 v a l u e s were taken from (28) w h i l e the MDA v a l u e s were o b t a i n e d from ( 2 9 ) .

I

I

I

ο ο

1

0.0 -«-—α

ο \

-0.2

\

Ε ϋ

^ M D A (97,117)

N

\

ο Ε to"

^MMDA (97) \ \ \ \ \ \ \ ~^MMDA (117)

-0.4

\ ν \ \ ^-VDW1 (97,

- £2/£l = 1 -0.6

1

-

-0.8

α 97 Κ ο 117 Κ ι

1.0

I

117)

\\

I

1.1

ι

1.2

ι

1.3

02/01

F i g u r e 11. Comparisons o f c a l c u l a t e d and s i m u l a t e d excess volumes as a f u n c t i o n o f u n i t y r a t i o . The s i m u l a t e d and VDW1 v a l u e s were t a k e n from (28) w h i l e t h e MDA v a l u e s were o b t a i n e d from ( 2 9 ) .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

344

Table I .

Comparison o f t h e MMDA, VDW-1 and Monte C a r l o R e s u l t s (28) f o r E q u i m o l a r Lennard-Jones Mixture Excess P r o p e r t i e s .

V

ε

ε

2^ 1

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

0.656

σ

Model

97 Κ

117 Κ

1 .000

MC MMDA VDW1

-0.89 -0.84 -0.92

-1 .66 -1.53 -1.77

1 .0619

MC MMDA VDW1

-1 .08 -0.97 -1.11

1 .1277

MC MMDA VDW1

E

117 Κ

0.15 0.13 0.11

0.12 0.09 0.09

0.176 0.181 0.186

0.159 0.158 0.167

-2.02 -1 .74 -2.08

0.24 0.27 0.23

0.19 0.21 0.19

0.242 0.268 0.263

0.208 0.222 0.224

-1 .29 -1.17 -1 .39

-2.43 -2.07 -2.48

0.34 0.41 0.35

0.26 0.32 0.29

0.305 0.352

0.250 0.282 0.274

MC MMDA VDW1

-1 .50 -1 .39 -1.76

-2.87 -2.42 -3.00

0.45 0.53 0.47

0.34 0.42 0.39

0.368

MC MMDA VDW1

-1.76

-3.35

-2.68 -3.61

0.58 0.64 0.58

0.42 0.49 0.48

0.423 0.489 0.454

0.319

-1 .68 -2.23

MC MMDA VDW1

0.00 0.00 0.00

0.00 0.00 0.00

0.00 0.00 0.00

0.00 0.00 0.00

0.000 0.000 0.000

0.000 0.000 0.000

1 .0619 MC MMDA VDW1

0.01

-0.02 -0.02 -0.05

0.01

0.00 0.00 0.00

-0.001

-0.02 -0.05

-0.003 -0.003

-0.003 -0.003 -0.003

MC MMDA VDW1

0.02 -0.08 -0.18

-0.05 -0.11 -0.20

0.03 -0.02 0.00

0.01 -0.01

MC MMDA VDW1

-0.05 -0.20 -0.41

-0.14 -0.26 -0.49

0.05 -0.05 0.00

0.02 -0.03 0.00

-0.013

MC MMDA VDW1

-0.10

-0.24 -0.46 -0.79

0.09 -0.09 0.00

0.04 -0.05 0.00

-0.023

-0.34 -0.72

-0.054 -0.043

-0.044 -0.052 -0.042

MC MMDA VDW1

-0.96 -0.84 -0.92

-1 .72 -1 .55 -1 .77

0.04 0.13 0.11

0.08 0.18 0.09

0.178 0.181 0.185

0.161 0.159 0.167

1 .1277

1 .1978

1.235

G 97 Κ

1 .000

1 .2727

1 .000

97 Κ

E 117 Κ

σ

1 .2727

1.000

H

2/ 1

1 .1978

1.000

E

0.00 0.00

0.00

0.334 0.402 0.402

-0.005

-0.013 -0.011

-0.030 -0.024

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

0.287 0.331 0.317

0.368 0.354

-0.011 -0.013 -0.011 -0.024 -0.029 -0.024

16.

ELY

345

One-Fluid Conformai Solution Calculations

Table I .

Continued. HE

97 Κ

117 Κ

97 κ

117 Κ

97 Κ

117 Κ

1 .0619 MC MMDA VDW1

-0. 80 -0 76 -0 81

-1 .43 -1 .58

-0. 01 -0. 27 0 00

0.00 -0.04 -0.01

0.122 0.084 0.103

0.018 0.086 0.105

1.1277

MC MMDA VDW1

-0 68 -0 69 -0 81

-1.18 -1 .24 -1 .48

-0 10 -0 20 -0 12

-0.07 -0.17 -0.11

0.040 -0.021 0.014

0.049 0.016 0.037

1.1978

MC MMDA VDW1

-0 55 -0 69 -0 .90

-0.95 -1.18 -1.47

-0 17 -0 38 -0 .25

-0.14 -0.21

-0.033 -0.132 -0.082

-0.016 -0.081 -0.036

MC MMDA VDW1

-0 .45 -0.72 -1 .06

-0.76 -1.16 -1 .57

-0 .24 -0 .58 -0 .36

-0.20 -0.46 -0.31

-0.113 -0.249 -0.178

-0.087 -0.173 -0.115

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

02/0]

1.2727

Model

-1.36

-0.31

1 1 1 θ2/σι = 2, £2/ει = 1

ω ο Ε

m Ε

Ο

-10 .25

.50

.75

1.0

Χι

F i g u r e 12. Comparison o f c a l c u l a t e d and s i m u l a t e d excess volumes f o r a system o f Lennard-Jones m o l e c u l e s as a f u n c t i o n o f c o m p o s i t i o n . The open c i r c l e s a r e the s i m u l a t e d p o i n t s c a l c u l a t e d from (1_8). Note t h e s u b s t a n t i a l improvement shown by t h e MMDA as compared t o the VDW1 model.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

346

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

F i g u r e 13. Comparison o f s i m u l a t e d and c a l c u l a t e d excess Gibbs energy v a l u e s as a f u n c t i o n o f c o m p o s i t i o n . S i m u l a t e d r e s u l t s a r e from ( 1 8 ) .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

16.

ELY

347

One-Fluid Conformai Solution Calculations

Table I I .

Comparison o f MMDA and VDW1 P r e d i c t i o n s t o S i m u l a t i o n R e s u l t s (1_8) (ε22/ 11 a l l cases). £

σ 2/σιι

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

2

X1

Model

0.25

1

i

n

, cm3/mole

G /NkT

MD MMDA VDW1

-0.098 -0.131

-0.373

-0.053 -0.035 -0.036

0.50

MD MMDA VDW1

-0.145 -0.175 -0.500

-0.075 -0.057 -0.050

0.75

MD MMDA VDW1

-0.131 -0.133 -0.377

-0.059 -0.042 -0.040

0.25

MD MMDA VDW1

-0.378 -0.437

-0.070 -0.109 -0.119

MD MMDA VDW1

-0.388 -0.530

MD MMDA VDW1

-0.383

MD MMDA VDW1

-0.369 -0.784 -3.629

-0.133 -0.196 -0.235

0.50

MD MMDA VDW1

-0.813 -1.071 -4.824

-0.229 -0.311 -0.350

0.75

MD MMDA VDW1

-0.865 -0.817 -3.629

-0.236 -0.294 -0.305

0.25

MD MMDA VDW1

-0.228 -1 .172 -6.672

-0.207 -0.287 -0.377

0.50

MD MMDA VDW1

-0.480 -1.669

-0.339

MD MMDA

-0.644 -1 .265

0.50

0.75

0.25

0.75

V

=

E

-1.542

-2.067

-0.405 -1.558

-8.888

American $emical SooëÊi^: Library vl

1155 16th St., N.W. Washington, D.C, 20036

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E

-0.111 -0.166 -0.174 -0.099 -0.147 -0.147

-0.468 -0.566 -0.320

-0.463 -0.504

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

348

Table I I I .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

02/01

ε /ε

Comparisons o f MMDA P r e d i c t i o n s t o S i m u l a t e d R e s u l t s f o r E q u i m o l a r M i x t u r e s W i t h Large S i z e and Energy Differences (16).

, cm3/mole

G /NkT

H /NkT

MD MMDA VDW1

-0,,040 -0.,011 -0.,284

0,.041 -0..029 -0,.088

-0..025 -0..066 -0,.102

3551

MD MMDA VDW1

-0..302 -0..287 -0.,362

-0,.140 -0,.117 -0..112

-0..108 -0,.056 -0,.064

198.4

1629

MD MMDA VDW1

-0..268 -0,.894 - 3 . .306

-0..564 -0,.586 -0,.540

-0,.161 -0..226 -0,.563

199.0

1904

MD MMDA VDW1

.261 -1 . -0,.625 -5,.246

-0,.409 -0,.485 -0,.890

-0,.375 -1 , .299 -1 , .107

Τ, Κ

Ρ, B a r

Model

1 .5 ,

201 .4

3563

4,.5

200.5

1 .5 .

4,.5

2

1

V

E

E

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E

16.

ELY

One-Fluid Conformai Solution Calculations

349

q u i t e as a c c u r a t e as those o b t a i n e d from t h e HSE t h e o r y , b u t a r e s i m p l e r t o i n c o r p o r a t e s i n c e t h e MMDA g i v e s t h e o p t i m a l e f f e c t i v e hard sphere diameter d i r e c t l y . Other s i m i l a r a p p r o x i m a t i o n s c o u l d be made which might f u r t h e r improve t h e r e s u l t s o f t h i s type o f model. To some degree p r o g r e s s i n t h i s a r e a has been hampered by the l a c k o f d e t a i l e d i n f o r m a t i o n c o n c e r n i n g t h e r a d i a l d i s t r i b u t i o n f u n c t i o n s i n mixtures. This s i t u a t i o n i s improving r a p i d l y w i t h the a v a i l a b i l i t y o f new m i x t u r e s i m u l a t i o n r e s u l t s and thus one can s p e c u l a t e t h a t improved r a d i a l d i s t r i b u t i o n f u n c t i o n based c o n f o r m a i s o l u t i o n models w i l l be f o r t h c o m i n g . Acknowledgments The a u t h o r would l i k e t o acknowledge P r o f e s s o r J . M. H a i l e o f Clemson U n i v e r s i t y f o r p r o v i d i n g Dr. G u p t a s s i m u l a t i o n r e s u l t s p r i o r t o p u b l i c a t i o n , Ms. Karen Bowie f o r h e r a s s i s t a n c e i n t h e p r e p a r a t i o n o f t h e m a n u s c r i p t and t h e NASA-Lewis Research C e n t e r , Grant No. C-60875-D f o r i t s f i n a n c i a l s u p p o r t .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

1

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Vidal, J. Chem. Eng. Sci. 1978, 33, 787-91. Huron, M.; Vidal, J. Fluid Phase Equil. 1979, 3, 255-271. Mathias, P. M.; Copeman, T. W. Fluid Phase Equil. 1983, 13, 91-108. Whiting, W. B.; Prausnitz, J. M. Fluid Phase Equil. 1982, 9, 119-147. Dieters, U. Chem. Eng. Sci. 1981, 36, 1139-51; 1982, 37, 855-61; Fluid Phase Equil. 1983, 13, 109-20. Rowlinson, J. S.; Watson, I. D. Chem. Eng. Sci. 1969, 24, 1156-1574. Leland, T. W.; Chappelear, P. S. Ind. Eng. Chem. 1968, 60, 16-43. Massih, A. R.; Mansoori, G. A. Fluid Phase Equil. 1983, 10, 57-72. Mollerup, J. Adv. Cryo. Eng. 1975, 20, 172-94; 1978, 23, 550-60. Ely, J. F. Proc. 63rd Gas Processors Association Annual Convention, 1984, p. 9. Naumann, Κ. H.; Leland, T. W. Fluid Phase Equil. 1984, 18, 1-45. Hang, T.; Ely, J. F.; Leland, T. W. Fluid Phase Equil. 1985, submitted for publication. Soave, G. Chem. Eng. Sci. 1978, 33, 787-91. Peng, D. Y.; Robinson, D. B. Ind. Eng. Fundam. 1976, 15, 59-64. Mentzer, R. Α.; Greenkorn, R. Α.; Chao, K. C. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 240-52. Hoheisel, C.; Dieters, U.; Lucas, K. Mol. Phys. 1983, 49, 159-70. Shing, K. S.; Gubbins, Κ. E. Mol. Phys. 1982, 46, 1109-1128; 1983, 49, 1121-38. Gupta, S. Ph.D. Dissertation, Clemson University, 1984. Mansoori, G. Α.; Leland. T. W. J. Chem. Soc. Faraday Trans. II 1972, 6, 320-44.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch016

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20. Hanley, H. J. M.; Evans, D. J. Int. J. Thermophysics 1981, 2, 1-9. 21. Mo, K. C.; Gubbins, Κ. E.; Jacucci, G.; McDonald, I. R. Mol. Phys. 1974, 27, 1173-83. 22. McQuarrie, D. A. "Statistical Mechanics"; Harper and Row: New York, 1976; Ch. 13. 23. Wertheim, M. S. Phys. Rev. Lett. 1963, 10, 321-4. 24. Thiele, E. J. Chem. Phys. 1963, 39, 474-9. 25. Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1970, 53, 600-3. 26. Mansoori, G. Α.; Carnahan, N. F.; Starling, Κ. E.; Leland, T. W. J. Chem. Phys. 1970, 54, 1523-5. 27. Nicolas, J. J.; Gubbins, Κ. E.; Street, W. B.; Tildesley, D. J. Mol. Phys. 1979, 37, 1429-54. 28. Singer, J. V. L.; Singer, K. Mol. Phys. 1972, 24, 357-90. 29. Chang, I. J. M.S. Thesis, Rice University, 1978. 30. Hoheisel, G.; Lucas, K. Mol. Phys. 1984, 53, 51-7. RECEIVED November

8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

17 Recent Mixing Rules for Equations of State A n Industrial Perspective Thomas W. Copeman and Paul M . Mathias

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

Air Products and Chemicals, Inc., Allentown, PA 18105

The number of applications for equations of state capable of describing phase behavior of mixtures with asymmetric interactions is high - and continuing to grow. Gas-processing applications such as enhanced oil recovery and glycol dehydration involve carbon dioxide, water, hydrogen sulfide, and hydrocarbons at high pressure. Organic-chemical and polymer production involve an increasingly wide variety of conditions, not restricted to low pressures (1). Supercritical extraction is actively being applied to the purification of natural products, which involves mixtures of complex polar compounds near the critical point of a gas like carbon dioxide (2). Engineering models to describe phase equilibrium can be divided into two broad categories: equations of state and activity coefficient models. Equations of state have been successfully applied to mixtures of nonpolar and slightly polar compounds at all conditions of engineering interest. These models have been used most extensively by the gas processing industry for the design of various processes (see for example Refs. 3 and 4). C o n v e r s e l y the mathematical f l e x i b i l i t y o f a c t i v i t y c o e f f i c i e n t models has c o n v e n t i o n a l l y been c o n s i d e r e d n e c e s s a r y t o d e s c r i b e systems w h i c h e x h i b i t h i g h l i q u i d - p h a s e n o n i d e a l i t y . These models have been most e x t e n s i v e l y used by the c h e m i c a l s and polymer i n d u s t r i e s f o r d e s i g n o f v a r i o u s p r o c e s s e s . P r o c e s s d e s i g n f o r the p r o d u c t i o n o f p o l y v i n y l a l c o h o l (5) i s one example. The a c t i v i t y c o e f f i c i e n t approach i s adequate a t low reduced temperatures where the l i q u i d phase i s i n c o m p r e s s i b l e and up t o moderate p r e s s u r e s . The use o f d i f f e r e n t models f o r the v a r i o u s phases p r e c l u d e s c o r r e c t d e s c r i p t i o n o f m i x t u r e c r i t i c a l p o i n t s and a d d i t i o n a l problems a r i s e f o r s u p e r c r i t i c a l components (6). The e q u a t i o n - o f - s t a t e approach does not i n h e r e n t l y s u f f e r f r o m these l i m i t a t i o n s and the e x t e n s i o n o f e q u a t i o n s o f s t a t e t o d e s c r i b e asymmetric i n t e r a c t i o n s i s c u r r e n t l y a h i g h l y a c t i v e a r e a o f r e s e a r c h . T h i s paper r e v i e w s r e c e n t developments o f m i x i n g r u l e s f o r e q u a t i o n s o f s t a t e w i t h an e n g i n e e r i n g - d e s i g n p e r s p e c t i v e . 0097-6156/ 86/ 0300-O352$06.00/ 0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

17. C O P E M A N A N D Μ ΑΤΗ IAS

Mixing Rules: An industrial Perspective

353

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

E v o l u t i o n of Equation of State Mixing Rules C o n s i d e r a b l e e f f o r t has been expended o v e r the p a s t f i f t y y e a r s to d e v e l o p new e q u a t i o n s o f s t a t e f o r pure components. Much l e s s a t t e n t i o n has been g i v e n t o i m p r o v i n g m i x i n g r u l e s . The o n e - f l u i d m i x i n g r u l e s o f van der Waals (7) (known h e r e a f t e r a s the vdW-1 m i x i n g r u l e s ) a r e s t i l l i n w i d e s p r e a d use; the e q u a t i o n s o f s t a t e o f Soave (8) and Peng and R o b i n s o n (9) a r e two such examples. I t s h o u l d be noted t h a t the vdW-1 m i x i n g r u l e s a r e r e a s o n a b l y w e l l based i n b o t h t h e o r y and the t y p i c a l e x c e s s - f u n c t i o n b e h a v i o r of normal f l u i d s . L e l a n d , C h a p p e l e a r , and Gamson (10) have shown t h a t t h e s e m i x i n g r u l e s a r i s e from v e r y r e a s o n a b l e assumptions f o r the j i r a d i a l d i s t r i b u t i o n f u n c t i o n s . ( A l s o see R e f s . _Π and 1 2 ) . Henderson and Leonard ( 1 3 , 14) have shown t h a t t h e s e m i x i n g r u l e s p r o v i d e good agreement w i t h the q u a s i - e x p e r i m e n t a l machines i m u l a t i o n r e s u l t s f o r h a r d - s p h e r e and Lennard-Jones (6:12) m i x t u r e s . F u r t h e r , V i d a l (l_5) has demonstrated t h a t the vdW-1 mixing r u l e s p r e d i c t excess f u n c t i o n s very s i m i l a r t o r e g u l a r s o l u t i o n t h e o r y and thus t h e y s h o u l d p r o v i d e a good d e s c r i p t i o n o f most n o n p o l a r m i x t u r e s . An i m p o r t a n t landmark i n the development o f i n d u s t r i a l l y s i g n i f i c a n t m i x i n g r u l e s was the work o f S t o t l e r and B e n e d i c t (16) who suggested the use o f b i n a r y i n t e r a c t i o n p a r a m e t e r s . They found t h a t the o n e - f l u i d m i x i n g r u l e s suggested f o r the Benedict-WebbR u b i n e q u a t i o n (17, 18) c o u l d be markedly improved by the use o f s m a l l p a i r - d e p e n d e n t c o r r e c t i o n terms. Another i m p o r t a n t e f f o r t was the work o f P r a u s n i t z and Gunn (19) who noted t h a t the i n t e r a c t i o n parameter was not a t o t a l l y e m p i r i c a l c o r r e c t i o n f a c t o r , but was r e l a t e d t o the t h e o r y o f i n t e r m o l e c u l a r f o r c e s . Many e f f o r t s t o improve the vdW-1 m i x i n g r u l e s attempted t o f i n d b e t t e r forms w h i l e r e t a i n i n g the o n e - f l u i d c o n c e p t . Plocker et a l . (20), Radosz e t a l . (21) and Lee e t a l . (22) v a r i e d an exponent i n the m i x i n g r u l e s , which can be r e p r e s e n t e d a s εσ

=

η

m

o

=

Z Z x i X j

E l X i X j

e j i a j i

a j i

where η = m = 3 r e p r e s e n t s the van der Waals o n e - f l u i d m i x i n g

( l )

(2)

rules.

P l o c k e r (20) chose n=0.75 and m=3, w h i l e Radosz (21) chose n=-0.25 and m=3. Lee (22) used n=m=4.5. W h i l e some degree o f s u c c e s s i n i m p r o v i n g m i x t u r e p r e d i c t i o n s i s c l a i m e d f o r each o f the methods, t h e i r a p p l i c a t i o n t o h i g h l y asymmetric p o l a r - n o n p o l a r systems i s l i m i t e d . P e r t u r b a t i o n t h e o r y i s u s e f u l as a g u i d e t o the development o f b o t h pure f l u i d and m i x t u r e e q u a t i o n s o f s t a t e f o r e n g i n e e r i n g calculations ( 2 3 ) . Donohue and P r a u s n i t z (24) d e v e l o p e d an equation o f s t a t e f o r mixtures o f molecules w i t h l a r g e s i z e d i f f e r e n c e s based on a p e r t u r b a t i o n e x p a n s i o n f o r s q u a r e - w e l l f l u i d s at low d e n s i t i e s and i d e a s from polymer s o l u t i o n t h e o r y . The e x p a n s i o n was then t r u n c a t e d a f t e r the f o u r t h a t t r a c t i v e term. W h i l e a s i m p l e q u a d r a t i c c o m p o s i t i o n dependence was o b t a i n e d f o r the f i r s t a t t r a c t i v e term, each o f the h i g h e r - o r d e r terms y i e l d s a d d i t i o n a l ( n o n - q u a d r a t i c ) m i x i n g r u l e s . Good r e s u l t s were o b t a i n e d

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

354

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

f o r m i x t u r e s c o n t a i n i n g a l k a n e s (up t o C 3 0 ) , a r o m a t i c s and l i g h t inorganic gases. W i l s o n ( 2 5 ) has s u g g e s t e d t h a t t h e l i q u i d - p h a s e e x c e s s G i b b s energy d a t a c a n be used t o p r o v i d e i n f o r m a t i o n f o r i m p r o v i n g m i x i n g r u l e s a t h i g h d e n s i t i e s . V i d a l ( 1 5 ) , Huron and V i d a l ( 2 6 ) , Heyen ( 2 7 ) , and Won ( 2 8 ) have f u r t h e r proposed n o n - q u a d r a t i c m i x i n g r u l e s based on t h e b e h a v i o r o f l i q u i d - p h a s e a c t i v i t y c o e f f i c i e n t s . These m i x i n g r u l e s do n o t reproduce t h e r e q u i r e d q u a d r a t i c c o m p o s i t i o n dependence o f t h e second v i r i a l c o e f f i c i e n t , as p o i n t e d o u t by Huron and V i d a l ( 2 6 ) and l a t e r by W h i t i n g and P r a u s n i t z ( 2 9 , 30) and Mollerup (31). These i n i t i a l l o c a l - c o m p o s i t i o n e q u a t i o n s o f s t a t e , however, d i d p r o v i d e an i m p o r t a n t advance by d e m o n s t r a t i n g t h a t complex p o l a r - n o n p o l a r systems c o u l d be c o r r e l a t e d w i t h e q u a t i o n s o f s t a t e i n t h e same ( e m p i r i c a l ) manner as w i t h a c t i v i t y - c o e f f i c i e n t models. F o r example, v a p o r - l i q u i d e q u i l i b r i u m c o r r e l a t i o n s were d e v e l o p e d f o r e t h a n o l - b e n z e n e and a c e t o n e - w a t e r by Heyen ( 2 7 ) and Huron and V i d a l ( 2 6 ) , r e s p e c t i v e l y . To overcome t h e above-mentioned d e f i c i e n c y a t low p r e s s u r e , W h i t i n g and P r a u s n i t z ( 2 9 , 3 0 ) , M o l l e r u p ( 3 1 ) and Won ( 3 2 ) p r o p o s e d d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n m i x i n g r u l e s . A common p r o p o s a l i n a l l t h r e e models i s t h e use o f c o n v e n t i o n a l m i x i n g r u l e s f o r t h e r e p u l s i v e term i n t h e e q u a t i o n o f s t a t e and l o c a l - c o m p o s i t i o n m i x i n g r u l e s w i t h d e n s i t y - d e p e n d e n t Boltzmann f a c t o r s f o r t h e a t t r a c t i v e term. These e q u a t i o n s o f s t a t e p l e a s i n g l y resemble t h e W i l s o n ( 3 3 ) a c t i v i t y - c o e f f i c i e n t model a t h i g h d e n s i t i e s and c o n v e n t i o n a l e q u a t i o n s o f s t a t e a t low d e n s i t i e s . M a t h i a s and Copeman ( 3 4 ) however demonstrated t h a t t h e l o c a l - c o m p o s i t i o n Peng-Robinson e q u a t i o n o f s t a t e d e r i v e d by M o l l e r u p ( 3 1 ) l a c k s adequate p r e d i c t i v e c a p a b i l i t y f o r n o n p o l a r m i x t u r e s w i t h even moderate s i z e differences. The l o c a l - c o m p o s i t i o n e f f e c t i s t o o l a r g e , w h i c h c a n r e s u l t i n g r e a t l y u n d e r p r e d i c t e d f u g a c i t y c o e f f i c i e n t s ( f o r example, 3-4 o r d e r s o f magnitude f o r methane i n d e c a n e ) . S i z a b l e i n t e r a c t i o n parameters along w i t h s i z e parameters (e.g., s u r f a c e area) a r e n e c e s s a r y f o r r e a s o n a b l e p r e d i c t i o n s o f phase e q u i l i b r i u m . The M o l l e r u p ( 3 1 ) Peng-Robinson e q u a t i o n o f s t a t e , a l o n g w i t h t h e Redlich-Kwong and o t h e r a n a l o g s , was c o n c l u d e d t o be t o o u n r e l i a b l e f o r g e n e r a l a p p l i c a t i o n t o i n d u s t r i a l problems. An i m p o r t a n t i d e a has been proposed by D i m i t r e l i s and P r a u s n i t z ( 3 5 ) . They propose t h a t l o c a l - c o m p o s i t i o n e f f e c t s do n o t r e s u l t when j i i n t e r a c t i o n s a r e d i f f e r e n t f r o m i i but r a t h e r when j i a r e d i f f e r e n t f r o m some " i d e a l " c o m b i n a t i o n s o f i i and j j i n t e r a c t i o n s , r e f e r r e d t o as ji°. D i m i t r e l i s and P r a u s n i t z d e f i n e d ji° as an a r i t h m e t i c mean i n t h e i r work. M a t h i a s and Copeman ( 3 4 ) and M o l l e r u p ( 3 6 ) , however, chose t h e g e o m e t r i c mean and d e v e l o p e d a d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n Peng-Robinson e q u a t i o n o f s t a t e t h a t meets an i m p o r t a n t a d d i t i o n a l l i m i t : non-conformal e f f e c t s a r e not n e c e s s a r i l y l a r g e ( o r even n o n - z e r o ) f o r an asymmetric system. T h i s f e a t u r e e n a b l e s t h e model t o r e t a i n t h e good p r e d i c t i o n s o f t h e s t a n d a r d Peng-Robinson e q u a t i o n o f s t a t e f o r n o n p o l a r systems and f a c i l i t a t e s c o r r e l a t i o n o f phase e q u i l i b r i u m o f h i g h l y n o n - i d e a l p o l a r systems. A d d i t i o n a l forms o f t h e e q u a t i o n o f s t a t e were s u g g e s t e d by M a t h i a s and Copeman ( 3 4 ) based on e x p a n s i o n o f t h e Boltzmann f a c t o r s i n the a t t r a c t i v e part w i t h t r u n c a t i o n a f t e r the second term i n t h e s e r i e s . Reduced c o m p u t a t i o n a l time a l o n g w i t h s u r p r i s i n g l y good r e s u l t s f o r b i n a r y h y d r o c a r b o n - w a t e r l i q u i d - l i q u i d e q u i l i b r i a were shown.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

17.

C O P E M A N A N D MATH IAS

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Ludecke and P r a u s n i t z (37) have proposed a s i m i l a r e q u a t i o n o f s t a t e based on the van der Waals form. They chose the M a n s o o r i e t a l . (38) e x p r e s s i o n f o r the h a r d - s p h e r e p a r t and a s i m p l e van d e r Waals e x p r e s s i o n f o r the a t t r a c t i v e p a r t . For m i x t u r e s , the Mansoori e t a l . e x p r e s s i o n i s used f o r the h a r d - s p h e r e c o n t r i b u t i o n , w h i l e f o r the a t t r a c t i v e c o n t r i b u t i o n the c o n v e n t i o n a l d e n s i t y - i n d e p e n d e n t q u a d r a t i c m i x i n g r u l e i s used as a l e a d i n g term and a d e n s i t y - d e p e n d e n t c o r r e c t i o n ( c u b i c i n mole f r a c t i o n ) i s used t o d e s c r i b e " n o n - c e n t r a l " f o r c e s . Good r e s u l t s were o b t a i n e d f o r v a p o r - l i q u i d and l i q u i d - l i q u i d e q u i l i b r i a i n b i n a r y m i x t u r e s c o n t a i n i n g w a t e r , p h e n o l , p y r i d i n e , methanol and h y d r o c a r b o n s . The e q u a t i o n o f s t a t e , however, o v e r - p r e d i c t s the t w o - l i q u i d r e g i o n i n the t e r n a r y systems s t u d i e d . The above approach i s a s i m p l e , y e t p o t e n t i a l l y v e r y e f f e c t i v e , way t o d e s c r i b e p o l a r , asymmetric systems and s h o u l d be d e v e l o p e d f u r t h e r f o r i n d u s t r i a l a p p l i c a t i o n s . A p o s s i b l e way t o a v o i d v i o l a t i n g the q u a d r a t i c dependence o f the second v i r i a l c o e f f i c i e n t and y e t have s e p a r a t e m i x i n g r u l e s f o r h i g h and low d e n s i t i e s has been suggested by L a r s e n and P r a u s n i t z ( 3 9 ) . These r e s e a r c h e r s have d i v i d e d the a t t r a c t i v e c o n t r i b u t i o n t o the H e l m h o l t z energy i n t o a s e c o n d - v i r i a l p o r t i o n and a d e n s e - f l u i d p o r t i o n , t h e r e b y a l l o w i n g an a r b i t r a r y m i x i n g r u l e f o r the d e n s e - f l u i d p a r t . Good r e s u l t s have been o b t a i n e d f o r the methane-water b i n a r y . T h i s work i s i m p o r t a n t s i n c e i t can be extended t o p r o p e r i d e n t i f i c a t i o n o f v a r i o u s t y p e s o f c o n t r i b u t i o n s at the p u r e - f l u i d l e v e l , l e a d i n g t o the use o f t h e o r e t i c a l l y suggested m i x i n g r u l e s f o r each c o n t r i b u t i o n . L i e t a l . (40) have combined the p u r e - f l u i d e q u a t i o n o f s t a t e d e v e l o p e d by Chung e t a l . (41) w i t h l o c a l - c o m p o s i t i o n m i x i n g r u l e s . The l o c a l - c o m p o s i t i o n m i x i n g r u l e s a l o n g w i t h van der Waals o n e - f l u i d m i x i n g r u l e s were e v a l u a t e d a g a i n s t b o t h v a p o r - l i q u i d e q u i l i b r i u m and d e n s i t y d a t a f o r b i n a r y systems. The p r e d i c t i v e c a p a b i l i t y o f the L i e t a l . model s h o u l d be e v a l u a t e d . T a b l e I p r e s e n t s a l i s t o f systems s t u d i e d by v a r i o u s i n v e s t i g a t o r s u s i n g l o c a l - c o m p o s i t i o n m i x i n g r u l e s . The m a j o r i t y o f s t u d i e s have not i n c l u d e d e v a l u a t i o n o f l i q u i d - l i q u i d e q u i l i b r i u m and d e n s i t y p r e d i c t i o n s . I t appears t h a t improved m i x t u r e models w i l l come f r o m the concept o f d e n s i t y - d e p e n d e n t m i x i n g r u l e s . However, i t i s not c l e a r at the p r e s e n t time w h i c h o f the proposed forms i s the " b e s t . " Most appear t o p r o v i d e s i g n i f i c a n t improvement o v e r vdW-1 for correlation o f the phase e q u i l i b r i u m o f b i n a r y systems, i n c l u d i n g complex b e h a v i o r l i k e l i q u i d - p h a s e i m m i s c i b i l i t y . However, v e r y few s t u d i e s have a d d r e s s e d the more d i f f i c u l t problem o f multicomponent predictions. P a r a l l e l work i n computer s i m u l a t i o n and t h e o r e t i c a l s t a t i s t i c a l mechanics i s e x t r e m e l y i m p o r t a n t as i t p r o v i d e s a g u i d e l i n e f o r the development o f e m p i r i c a l models. Approaches t o D e r i v e

Local Composition Mixing

Rules

The e a r l y d e r i v a t i o n s o f l o c a l - c o m p o s i t i o n m i x i n g r u l e s were based on l a t t i c e t h e o r i e s and/or e m p i r i c a l f o r m u l a t i o n s f o r i n t e r n a l , H e l m h o l t z , o r G i b b s energy (33, 42 - 4 4 ) . The d e r i v a t i o n by Kemeny and Rasmussen ( 4 4 ) , based on l a t t i c e p a r t i t i o n f u n c t i o n s

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Table I.

B i n a r y Systems S t u d i e s w i t h E q u a t i o n o f S t a t e L o c a l Composition Mixing Rules

Investigators

Year

Systems

Type

Huron and V i d a l (25) Heyen (26 ) Won (27) W h i t i n g and P r a u s n i t z (29) Won (31) M a t h i a s and Copeman (33) M o l l e r u p (35) Ludecke and P r a u s n i t z (36) L i e t a l . (39)

1979 1981 1981 1982 1983 1983 1983 1985 1984

1-8 9-11 12, 13 14 5, 15- 18 1, 10, 11, 119-24 1, 5, 20, 25-34 20, 22, 23, 25, 35-41 1, 5, 6, 14, 16, 19, 20, 25 , 29, 31, 42-46

VLE VLE VLE VLE VLE VLE, LLE VLE VLE, LLE VLE, density

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

acetone-water carbon dioxide-ethane ethane-acetone ethane-methyl a c e t a t e methanol-carbon d i o x i d e propane-ethanol methanol-1,2 d i c h l o r o e t h a n e ac e tone-eye1ohexane e t h a n o l benzene butanol-water carbon dioxide-methane carbon dioxide-napthalene ethylene-napthalene water-methane carbon dioxide-n-octane carbon dioxide-n-decane carbon dioxide-butanol water-carbon d i o x i d e methane-n-decane methanol-benzene isobutylene-methanol benzene-water hexane-water

24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

1-methylnapthalene-water methanol-water ethanol-water 2-propanol-water acetone-carbon d i o x i d e methano1-n-hexane me t hano1-eye1ohexane e thano1-n-hexane e t hano1-n-hep t ane e thano1-eye1ohexane e thano1-me thy1eye1ohexane pyridine-water pyridine-benzene wa t e r-cyc1ohexane water-heptane water-octane phenol-water propane-water ethane-butane carbon dioxide-benzene ammonia-water e t hano1-n-dec ane ethane-water

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

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and t w o - l i q u i d t h e o r y , p r o v i d e d a thorough e x p l a n a t i o n o f the assumptions r e q u i r e d t o o b t a i n the W i l s o n e q u a t i o n . F o l l o w i n g the q u a s i c h e m i c a l approach, the c o m b i n a t o r i a l f a c t o r i n the configurâtional p a r t i t i o n f u n c t i o n i s e x p r e s s e d i n terms o f the count o f c o n t a c t s and a c o r r e c t i o n f a c t o r , denoted by h. To meet the p r o p e r random-mixture l i m i t , an e x p r e s s i o n was d e r i v e d f o r h u s i n g a T a y l o r s e r i e s e x p a n s i o n about complete randomness. The l a t t i c e t h e o r y d e r i v a t i o n c l e a r l y shows t h a t the W i l s o n e q u a t i o n i s most v a l i d f o r near-random m i x t u r e s o f s i m i l a r s i z e d m o l e c u l e s . Kemeny and Rasmussen a l s o p r e s e n t e d an e q u a t i o n f o r the combinatorial f a c t o r which leads to expressions f o r l o c a l area f r a c t i o n s , as used i n the s u c c e s s f u l UNIQUAC a c t i v i t y c o e f f i c i e n t model ( 4 5 ) . L a t t i c e t h e o r i e s have p r o v i d e d l i t t l e f u r t h e r p r o g r e s s i n d e v e l o p i n g improved m i x i n g r u l e s s i n c e the mid 1970*s. More r e c e n t l y , Lee et a l . (46) d e r i v e d g e n e r a l r e l a t i o n s f o r l o c a l c o m p o s i t i o n s i n terms o f r a d i a l d i s t r i b u t i o n f u n c t i o n s and r e l a t e d p o t e n t i a l s o f mean f o r c e t o the W i l s o n f o r m u l a t i o n . T h i s work r e p r e s e n t s a s i g n i f i c a n t c o n t r i b u t i o n , b r i d g i n g the gap between c o n v e n t i o n a l l o c a l - c o m p o s i t i o n m i x i n g r u l e s and r i g o r o u s f l u i d - p h a s e s t a t i s t i c a l mechanics. Mansoori and E l y (47) have f u r t h e r d e r i v e d a u n i f i e d treatment o f the l o c a l - c o m p o s i t i o n concept f o r f l u i d - p h a s e m i x t u r e s . The energy, p r e s s u r e , and c o m p r e s s i b i l i t y e q u a t i o n s were e x p r e s s e d i n terms o f l o c a l p a r t i c l e numbers. For example, the i n t e r n a l energy i s g i v e n by

Ε

= -NkT/2

Σ Σ χ ι

/ " n j i i r ) (dUj ι ( r ) / d r ) d r

(3)

V a r i o u s a p p r o x i m a t i o n s f o r n j i were a n a l y z e d by Mansoori and E l y . T h e i r g e n e r a l procedure i s based on e q u a t i n g the m i x t u r e p o t e n t i a l energy f u n c t i o n and l o c a l p a r t i c l e number t o t h a t o f a h y p o t h e t i c a l pure f l u i d . An advantage o f the Mansoori and E l y approach i s t h a t m i c r o s c o p i c a p p r o x i m a t i o n s t o the r a d i a l d i s t r i b u t i o n f u n c t i o n s can be t r a n s l a t e d i n t o e x p r e s s i o n s f o r l o c a l p a r t i c l e numbers and m i x i n g r u l e s . The r e l a t i o n s , based on the c o m p r e s s i b i l i t y e q u a t i o n , a r e n o v e l and m e r i t f u r t h e r e v a l u a t i o n . One p r o b a b l e d i s a d v a n t a g e f o r i n d u s t r i a l a p p l i c a t i o n i s t h a t the m i x i n g - r u l e e x p r e s s i o n s a r e g e n e r a l l y not e x p l i c i t w i t h r e s p e c t t o the m i x t u r e parameters, r e q u i r i n g s i g n i f i c a n t a d d i t i o n a l c o m p u t a t i o n a l expense. Seaton and G l a n d t (48) have e x p l o r e d l o c a l c o m p o s i t i o n s f o r m i x t u r e s d e s c r i b e d by the a d h e s i v e i n t e r m o l e c u l a r p o t e n t i a l . The unique f e a t u r e o f t h e i r work i s the unambiguous d e f i n i t i o n o f nearest neighbors s i n c e a t t r a c t i v e f o r c e s are extremely short-ranged and n e i g h b o r i n g m o l e c u l e s a r e i n c o n t a c t . The l o c a l p a r t i c l e numbers were shown t o be n j i = i r p i X j io] i / 3

(4)

where the X j ι a r e o b t a i n e d from the s o l u t i o n o f t h r e e q u a d r a t i c a l g e b r a i c e q u a t i o n s f o r a b i n a r y m i x t u r e . The W i l s o n e q u a t i o n was compared t o the a d h e s i v e - p o t e n t i a l model and c o n c l u d e d t o be inaccurate i n d e s c r i b i n g l o c a l compositions f o r mixtures of

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unequal-sized molecules. T h i s c o n c l u s i o n i s s i m i l a r t o t h a t reached by G i e r y c z and N a k a n i s h i (49) and s u g g e s t s t h a t more a t t e n t i o n s h o u l d be g i v e n t o s i z e d i f f e r e n c e s i n l o c a l - c o m p o s i t i o n formulations. Comparison o f L o c a l C o m p o s i t i o n F o r m u l a t i o n s

t o Computer S i m u l a t i o n

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N a k a n i s h i and co-workers have used m o l e c u l a r dynamics and Monte Carlo simulations t o provide u s e f u l information f o r the l o c a l s t r u c t u r e o f v a r i o u s k i n d s o f m i x t u r e s (49 - 52). Insight into l o c a l f l u i d s t r u c t u r e h a s been o b t a i n e d by comparing l o c a l c o m p o s i t i o n s c a l c u l a t e d from computer s i m u l a t i o n s t o l o c a l c o m p o s i t i o n s p r e d i c t e d from s e m i - e m p i r i c a l models ( l i k e W i l s o n , 2 5 ) . The number o f p a r t i c l e s j around a p a r t i c l e i i s d e t e r m i n e d f r o m the s i m u l a t i o n r e s u l t s as R

nji

The

= 4irNj/V J r g j i ( r ) d r 2

l o c a l composition xji

= n i / (E n d

k

(5)

o f j around i i s k i

)

(6)

N a k a n i s h i and co-workers have s e t t h e upper l i m i t o f i n t e g r a t i o n i n E q u a t i o n ( 5 ) t o t h e d i s t a n c e o f t h e f i r s t peak i n t h e r a d i a l d i s t r i b u t i o n f u n c t i o n . Lennard-Jones p o t e n t i a l and L o r e n t z B e r t h e l o t combining r u l e s were chosen f o r c a l c u l a t i o n s on m i x t u r e s w i t h v a r i e d s i z e and energy p a r a m e t e r s . The Lennard-Jones energy parameters were s u b s t i t u t e d f o r t h e p o t e n t i a l s o f mean f o r c e i n t h e local-composition formulations. I n t h e i r most e x t e n s i v e works (49, 5 2 ) , m o l e c u l a r dynamic c a l c u l a t i o n s were made o v e r a range o f o v e r a l l mole f r a c t i o n s and the l o c a l c o m p o s i t i o n s were compared t o t h e W i l s o n (25) and Renon and P r a u s n i t z (42) f o r m u l a t i o n s . The W i l s o n e q u a t i o n was found t o p r e d i c t l o c a l c o m p o s i t i o n s p o o r l y f o r t h e m i x t u r e s d e s c r i b e d above, w h i l e t h e Renon and P r a u s n i t z e q u a t i o n w i t h <* = 0.4 showed b e t t e r agreement ( i s t h e nonrandomness p a r a m e t e r ) . S i n c e t h e Boltzmann f a c t o r s o f t h e s e two models d i f f e r o n l y by a f a c t o r o f 0.4, t h e Wilson formulation over-predicts the local-composition e f f e c t . For m i x t u r e s o f u n e q u a l - s i z e d m o l e c u l e s t h e R e n o n - P r a u s n i t z model predictions are l e s s accurate. F i r m c o n c l u s i o n s cannot be drawn on the a b i l i t y o f t h e s e models t o c o r r e l a t e t h e a c t i v i t y - c o e f f i c i e n t b e h a v i o r o f r e a l f l u i d s , but t h e s e s t u d i e s do p r o v i d e a g u i d e l i n e f o r model development. G i e r y c z and N a k a n i s h i (49) proposed improvements t o t h e R e n o n - P r a u s n i t z model based on t h e i r comparisons. E v a l u a t i o n o f t h e s e improvements u s i n g r e a l m i x t u r e s of i n d u s t r i a l i n t e r e s t i s a valuable next-step. W h i l e a number o f assumptions were used i n N a k a n i s h i s comparisons ( s e t t i n g R t o t h e f i r s t peak o f t h e r a d i a l d i s t r i b u t i o n f u n c t i o n and s u b s t i t u t i n g Lennard-Jones energy parameters f o r p o t e n t i a l s o f mean f o r c e ) , t h i s work c o n t r i b u t e s t o t h e u n d e r s t a n d i n g o f l o c a l l i q u i d s t r u c t u r e and l o c a l - c o m p o s i t i o n formulations. œ

1

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A d d i t i o n a l work t o a s s e s s the impact o f d e f i c i e n c i e s i n m i x i n g r u l e s on p r o p e r t i e s used i n p r o c e s s d e s i g n , such as a c t i v i t y c o e f f i c i e n t s , i s needed. R e c e n t l y , Hu e t a l . (53) have compared t h e i r e q u a t i o n - o f - s t a t e p r e d i c t i o n s t o the computer-simulation r e s u l t s o f S h i n g and Gubbins ( 5 4 ) . Of p a r t i c u l a r i n t e r e s t i s the d i r e c t comparison o f r e s i d u a l c h e m i c a l p o t e n t i a l s at low d i l u t i o n , a s t r i n g e n t t e s t o f the e q u a t i o n o f s t a t e .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

van der Waals P a r t i t i o n F u n c t i o n and Square-Well F l u i d s I n an on-going s e r i e s o f p a p e r s , S a n d l e r and co-workers (55-57) have attempted t o use a c o m b i n a t i o n o f t h e o r y and computer s i m u l a t i o n s t o d e v e l o p f u n d a m e n t a l l y - b a s e d e q u a t i o n s o f s t a t e . They have used the g e n e r a l i z e d van der Waals p a r t i t i o n f u n c t i o n t o s i m p l i f y the development o f new models and t o examine the assumptions i n e x i s t i n g models. F u r t h e r , they have performed Monte C a r l o s i m u l a t i o n s on s q u a r e - w e l l f l u i d s t o d e v e l o p new models o f engineering u t i l i t y . The a n a l y s i s by S a n d l e r and co-workers o f " l o c a l - c o m p o s i t i o n s " and t h e i r e f f e c t on e q u a t i o n - o f - s t a t e m i x i n g r u l e s i s o f p a r t i c u l a r importance t o the p r e s e n t r e v i e w . A s e r i e s of i n t e r e s t i n g o b s e r v a t i o n s and c o n c l u s i o n s were p r e s e n t e d i n t h e i r s t u d i e s . •

•

•

•

•

The s q u a r e - w e l l p o t e n t i a l i s v a l u a b l e f o r i n v e s t i g a t i v e s t u d i e s because i t i s r e a l i s t i c , y e t has a w e l l - d e f i n e d , f i n i t e a t t r a c t i v e range. The Boltzmann f a c t o r t h a t i s commonly used t o model l o c a l c o m p o s i t i o n s i s o n l y c o r r e c t at low d e n s i t i e s . At h i g h d e n s i t i e s the Boltzmann f a c t o r o v e r p r e d i c t s the l o c a l c o m p o s i t i o n s s i n c e the f l u i d s t r u c t u r e i s p r e d o m i n a n t l y d e t e r m i n e d by r e p u l s i v e f o r c e s . L o c a l c o m p o s i t i o n s a r e best d e f i n e d i n terms o f l o c a l p a r t i c l e numbers, w h i c h i s equal t o an i n t e g r a l o v e r the r a d i a l d i s t r i b u t i o n f u n c t i o n . I n the case o f i n f i n i t e range p o t e n t i a l s ( e . g . , Lennard-Jones) i t i s p r o b a b l y b e t t e r t o model the configurâtional energy d i r e c t l y . The vdW-1 and l o c a l - c o m p o s i t i o n models are p o o r l y named s i n c e b o t h can be viewed t o a r i s e from l o c a l - c o m p o s i t i o n a s s u m p t i o n s . The key d i f f e r e n c e between the two i s t h a t the f i r s t c l a s s o f models a r i s e s from the a s s u m p t i o n t h a t the c o o r d i n a t i o n number v a r i e s w i t h c o m p o s i t i o n , whereas i n the second c l a s s the c o o r d i n a t i o n number i s f i x e d . The d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n models o f W h i t i n g and P r a u s n i t z (29, 3 0 ) , M o l l e r u p (31) and Won (31) a r e q u a l i t a t i v e l y i n e r r o r s i n c e t h e y assume t h a t the e f f e c t o f the a t t r a c t i v e f o r c e s i s more important a t h i g h d e n s i t i e s than at low d e n s i t i e s .

We p r e s e n t section:

our v i e w p o i n t

on the s t a t e m e n t s i n the f o l l o w i n g

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

360

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Analysis There seems t o be on-going c o n f u s i o n about v a r i o u s c o n c e p t s and c o n f l i c t r e g a r d i n g r e s u l t s and assumptions among t h e s e v e r a l approaches t o model development. vdW-1 V s . Random M i x i n g . Many r e s e a r c h e r s assume o r imply t h a t t h e vdW-1 m i x i n g r u l e s r e s u l t from t h e assumption t h a t the m i x t u r e i s random. I n terms o f d i s t r i b u t i o n f u n c t i o n t h e o r y , a random m i x t u r e i s one i n w h i c h a l l t h e j i p a i r d i s t r i b u t i o n f u n c t i o n s a r e i d e n t i c a l . L e l a n d and Chappelear (12) have shown t h a t t h i s assumption l e a d s t o a m i x i n g r u l e t h a t depends o n t h e p a i r potential. I f a l l p a i r s i n t h e m i x t u r e obey t h e Lennard-Jones p o t e n t i a l , the m i x i n g r u l e t h a t f o l l o w s i s

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

ε

=

(Σ Σ

Χι

x j εjι a j

i

(Σ Σ

Xi

x j εjι a j

i

6

2

)

(7) 1 2

)

Xi

x j εjι o

(Σ Σ

Xi

x j εj

12

(Σ Σ

1

)

3i

(8) i

aj

i

6

)

1

/

The above m i x i n g r u l e s l e a d t o e x t r e m e l y poor p r e d i c t i o n s even when t h e r e a r e s m a l l d i f f e r e n c e s i n m o l e c u l a r s i z e s and e n e r g i e s o f attraction. The vdW-1 m i x i n g r u l e s r e s u l t s from a c o n s i d e r a b l y more a p p e a l i n g assumption. I t i s assumed t h a t t h e i j r a d i a l d i s t r i b u t i o n f u n c t i o n c a n be approximated as

(9)

E q u a t i o n (9) assumes t h a t t h e j i d i s t r i b u t i o n f u n c t i o n a t d i m e n s i o n l e s s d i s t a n c e ( r / a j i ) i s t h e same as t h a t o f a pure h a r d - s p h e r e f l u i d a t t h e same reduced d e n s i t y as t h a t o f t h e m i x t u r e w i t h p e r t u r b a t i o n terms t o account f o r the a t t r a c t i v e f o r c e s . The power o f t h e model i s t h a t t h e f u n c t i o n s g and Ϋ need not be known. Now i t c a n be shown t h a t the vdW-1 m i x i n g r u l e s ( E q u a t i o n s (1) and ( 2 ) ) a r e exact up t o o r d e r 1/T ( 1 2 ) . The vdW-1 m i x i n g r u l e s a r e c l e a r l y a s i m p l i f i c a t i o n . Terms o r o r d e r ( l / T ) and g r e a t e r a r e n e g l e c t e d , w h i c h s h o u l d i n t r o d u c e e r r o r s a t low t e m p e r a t u r e s . A d d i t i o n a l l y , t h e model assumes t h a t g a t t h e p o i n t o f c o n t a c t i s t h e same f o r a l l j i p a i r s , w h i c h ( 0 >

2

< 0 >

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

17.

COPEMAN AND

MATH

IAS

Mixing Rules: An Industrial Perspective

361

i s known t o be wrong at h i g h d e n s i t i e s . However, the r e a s o n a b l e n e s s and the c l a r i t y o f the a s s u m p t i o n s must a l s o be r e c o g n i z e d as we attempt t o d e v e l o p improved m i x i n g r u l e s ( 5 8 ) . The P r e d i c t i v e and C o r r e l a t i v e Power o f the vdW-1 M i x i n g R u l e s . In a d d i t i o n t o t h e i r f o u n d a t i o n i n t h e o r y , the vdW-1 m i x i n g r u l e s have p r o v i d e d good p r e d i c t i v e and c o r r e l a t i v e c a p a b i l i t y . We have a l r e a d y mentioned the work o f Henderson and Leonard ( 1 3 , 14) who found t h a t the vdW-1 m i x i n g r u l e s p r o v i d e d good p r e d i c t i o n s o f m a c h i n e - s i m u l a t i o n r e s u l t s f o r h a r d - s p h e r e and Lennard-Jones m i x t u r e s . We r e i t e r a t e t h a t c o m p a r i s o n w i t h computer s i m u l a t i o n s i s a s t r o n g t e s t o f t h e o r e t i c a l f o r m u l a t i o n s s i n c e the parameters a r e known and cannot be a d j u s t e d . However, t h e s e c o m p a r i s o n s s h o u l d be made i n the d i l u t e r e g i o n not (as i s u s u a l ) f o r e q u i m o l a r m i x t u r e s . S a n d l e r and co-workers have o b t a i n e d m a c h i n e - s i m u l a t i o n r e s u l t s f o r b o t h pure f l u i d s and m i x t u r e s i n t e r a c t i n g w i t h the s q u a r e - w e l l p o t e n t i a l s (56, 5 7 ) , thus e n a b l i n g a t e s t o f the s i m p l e vdW-1 mixing r u l e s . These s i m u l a t i o n s a r e f o r e q u i s i z e d m o l e c u l e s . Therefore the vdW-1 m i x i n g r u l e s reduce t o the s i m p l e f o r m ε

=

Σ

Σ

XiXj

εjι

(10)

The r e s u l t s f o r the vdW-1 p r e d i c t i o n s a r e p r e s e n t e d i n F i g u r e s 1 and 2. F i g u r e 1 shows t h a t i n the case where the temperature i s r e l a t i v e l y h i g h ( n o t e t h a t ε/kTc - 0.8), the vdW-1 mixing r u l e s p r o v i d e e x c e l l e n t p r e d i c t i o n s o f the configurâtional e n e r g y a t a l l c o m p o s i t i o n s and d e n s i t i e s . I n the case where the temperature i s lower ( F i g u r e 2 ) , the vdW-1 m i x i n g r u l e s p r o v i d e good p r e d i c t i o n s at h i g h d e n s i t i e s , and u n d e r p r e d i c t the configurâtional energy a t low d e n s i t i e s . A b e t t e r mixing rule f o r low-density mixtures i s a l o c a l - c o m p o s i t i o n model based on the second v i r i a l c o e f f i c i e n t : Ε

=

Σ

Σ

XiXj

Ε

(ε^;

Τ,ρ)

(il)

E q u a t i o n ( i l ) i s e x a c t a t z e r o d e n s i t y ( 5 5 ) . The dashed l i n e i n F i g u r e 2 shows t h a t the l o w - d e n s i t y m i x i n g r u l e p r o v i d e s improved p r e d i c t i o n s o f the m i x t u r e d a t a . But the model s t i l l u n d e r p r e d i c t s the d a t a . T h i s i s perhaps caused by m i x t u r e c o n d i t i o n s w i t h i n the u n s t a b l e r e g i o n o f the f l u i d , w h i c h i s s u p p o r t e d by the s c a t t e r i n the s i m u l a t i o n d a t a . S e v e r a l i n v e s t i g a t o r s have p o i n t e d out t h a t the s i m p l e vdW-1 m i x i n g r u l e s p r o v i d e good p r e d i c t i o n s f o r systems c o n t a i n i n g e q u i - s i z e d molecules w i t h d i f f e r i n g energies of i n t e r a c t i o n ( f o r example, see F i s c h e r , 5 8 ) . However, the vdW-1 f o r m i s i n a d e q u a t e f o r the more d i f f i c u l t m i x t u r e s c o n t a i n i n g m o l e c u l e s w i t h l a r g e s i z e d i f f e r e n c e s (54, 4 8 ) . I t s h o u l d be noted t h a t i t i s not n e c e s s a r y t o use the vdW-1 a p p r o x i m a t i o n f o r the r e p u l s i v e p a r t o f an e q u a t i o n o f s t a t e s i n c e t h e o r e t i c a l l y - b a s e d , t r a c t a b l e models a r e a v a i l a b l e f o r u n e q u a l - s i z e d h a r d - s p h e r e m i x t u r e s (60, 3 8 ) . Hu e t a l . (53) have used the Mansoori-Carnahan S t a r l i n g - L e i a n d e q u a t i o n (38) f o r the r e p u l s i v e term and o b t a i n e d good agreement w i t h c o m p u t e r - s i m u l a t i o n data.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

362

A p p l i c a t i o n s t o Important I n d u s t r i a l E q u a t i o n s o f S t a t e . It i s u s e f u l t o a p p l y t h e r a d i a l - d i s t r i b u t i o n - f u n c t i o n framework f o r l o c a l c o m p o s i t i o n s ( 4 6 , 55) t o t h e i m p o r t a n t i n d u s t r i a l e q u a t i o n s o f s t a t e . T h i s a l l o w s e x a m i n a t i o n o f t h e l o c a l c o m p o s i t i o n s i m p l i e d by the p r a c t i c a l l y - s u c c e s s f u l vdW-1 m i x i n g r u l e s and t h o s e o f t h e a t t e m p t s t o improve them (29 - 3 1 ) . Perhaps more i m p o r t a n t , t h e r e are i n d i c a t i o n s that the s i m p l i f i c a t i o n s inherent i n the p u r e - f l u i d e q u a t i o n s o f s t a t e must be c o n s i d e r e d when a p p l y i n g t h e m i x i n g r u l e s s u g g e s t e d by a r i g o r o u s t h e o r e t i c a l framework. C o n s i d e r t h e i n t e r n a l e n e r g y o b t a i n e d by a p p l y i n g t h e vdW-1 m i x i n g r u l e s t o t h e common c u b i c e q u a t i o n s o f s t a t e 3aji/T Ε

=

-

F

Σ

v

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

1

(12)

Σ XiXj J

3 1/T

where t h e f u n c t i o n F i s dependent on t h e f o r m chosen f o r t h e a t t r a c t i v e term o f t h e e q u a t i o n o f s t a t e . F o r example: v

van d e r Waals:

F

v

R e d 1 i ch-Kwong:

F

v

Peng-Robinson:

F

v

(13) (14)

1 l n ( l + pb) b

1 4- (1+72) bp 1 + ( W 2 ) bp

In

= 2/2b

(15)

The Redlich-Kwong and Peng-Robinson forms f o r F reduce t o t h e s i m p l e l i n e a r dependence on d e n s i t y ( E q u a t i o n ( 1 3 ) ) i n t h e l i m i t o f low d e n s i t y . (See F i g u r e 3.) Now c o n s i d e r t h e s t a t i s t i c a l m e c h a n i c a l e x p r e s s i o n f o r t h e i n t e r n a l energy o f a system where t h e t o t a l i n t e r m o l e c u l a r p o t e n t i a l i s pairwise additive. v

Ε = ρ/2 Σ Σ X i X j J u j i ( r ) g j i ( r ) 4 t r r d r 2

(16)

As p-»0, t h e r a d i a l d i s t r i b u t i o n f u n c t i o n g j i ( r ) approaches the Boltzmann f a c t o r -Uji(r)/RT

lim p-»0

(17)

gji(r) = e

I f E q u a t i o n ( 1 7 ) i s s u b s t i t u t e d i n t o E q u a t i o n ( 1 6 ) , i t c a n be shown ( 4 6 ) t h a t t h e i n t e r n a l energy a t low d e n s i t i e s i s r e l a t e d t o the t e m p e r a t u r e d e r i v a t i v e o f t h e second v i r i a l c o e f f i c i e n t : 3B2 j ι

(18)

Ε = Rp Σ Σ X i X j 1

J

3 1/T

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Mixing Rules: An Industrial Perspective

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

COPEMAN A N D MATHIAS

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

F i g u r e 3. D e n s i t y dependence o f a t t r a c t i v e term - Simple equations o f s t a t e

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

17.

COPEMAN AND

Mixing Rules: An Industrial Perspective

MATHIAS

365

Comparing E q u a t i o n (18) w i t h the l o w - d e n s i t y l i m i t o f E q u a t i o n (12) ( F - p ) , we o b t a i n an e x p r e s s i o n f o r a j i i n terms o f the second v i r i a l c o e f f i c i e n t . v

3B2 j

a i d

= -RT

J

1/T

-RT

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

-RT

>2ji

ι d(l/T)

1 / T

=0

r\

-

- B j ι (T = °°) 2

- bj

(19)

i

Thus, as i s w e l l known, a j ι a r i s e s from the a t t r a c t i v e p o r t i o n o f the j i p a i r p o t e n t i a l . E q u a t i o n s (16) and (17) show t h a t , a t low d e n s i t i e s , the s i m p l e e q u a t i o n s q u a l i t a t i v e l y d e s c r i b e the l o c a l c o m p o s i t i o n s r e s u l t i n g from the Boltzmann f a c t o r . We can now get some i d e a o f the h i g h - d e n s i t y r a d i a l d i s t r i b u t i o n f u n c t i o n assumed by the s i m p l e e q u a t i o n s o f s t a t e . The van der Waals e q u a t i o n assumes t h a t t h e r e i s no e f f e c t o f d e n s i t y . The Redlich-Kwong and Peng-Robinson models assume t h a t the l o w - d e n s i t y Boltzmann f a c t o r r e s u l t s i n an o v e r p r e d i c t i o n o f the h i g h - d e n s i t y configurâtional e n e r g y , but t h i s can be c o r r e c t e d by a s i m p l e f u n c t i o n of d e n s i t y , F /p. We emphasize t h a t t h i s i s a nonunique i n t e r p r e t a t i o n . More important t o the d i s c u s s i o n at hand, the vdW-1 m i x i n g r u l e s assume t h a t the l o c a l c o m p o s i t i o n s are independent o f d e n s i t y s i n c e the c o r r e c t i o n f a c t o r i s the same f o r a l l j i p a i r s . T h i s p o i n t has been noted by S a n d l e r (55) who a l s o s t a t e s t h a t t h i s b e h a v i o r i s i n c o n s i s t e n t w i t h the i n v e s t i g a t i o n s o f C h a n d l e r and Weeks (61) and Weeks et a l . (62) who have shown t h a t the s t r u c t u r e o f a h i g h - d e n s i t y f l u i d i s l a r g e l y d e t e r m i n e d by the r e p u l s i v e f o r c e s . However, these s i m p l e e q u a t i o n s o f s t a t e , t o g e t h e r w i t h the vdW-1 m i x i n g r u l e s , have r e s u l t e d i n r e l i a b l e and f r e q u e n t l y a c c u r a t e p r e d i c t i o n s f o r a wide v a r i e t y o f n o n p o l a r m i x t u r e s i n c l u d i n g those whose components have l a r g e d i f f e r e n c e s i n t h e i r intermolecular forces (63). I t i s i m p o r t a n t t o pursue the n a t u r e and impact o f t h e s e d i f f e r e n c e s t o g u i d e improvements t o the vdW-1 m i x i n g r u l e s f o r m i x t u r e s where they a r e o b v i o u s l y i n a d e q u a t e ( e . g . , p o l a r - n o n p o l a r b i n a r i e s l i k e methanol-benzene). We t h e r e f o r e r e v i e w the assumptions l e a d i n g t o E q u a t i o n ( 1 9 ) : a j i i s not s t r i c t l y a l o w - d e n s i t y parameter. I n f a c t , i t i s more c o r r e c t l y a h i g h - d e n s i t y parameter s i n c e i t s v a l u e i s d e t e r m i n e d by f i t t i n g vapor p r e s s u r e o r phase e q u i l i b r i u m d a t a , f o r w h i c h the v a r i a t i o n o f the l i q u i d - p h a s e f u g a c i t i e s u s u a l l y dominate. Further, we a l s o note t h a t the second v i r i a l c o e f f i c i e n t s p r e d i c t e d by the c u b i c e q u a t i o n s are u s u a l l y l e s s n e g a t i v e t h a n the e x p e r i m e n t a l v a l u e s , i n d i c a t i n g t h a t the e f f e c t o f the a t t r a c t i v e f o r c e s i s u n d e r p r e d i c t e d at low d e n s i t i e s . v

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

366

STATE: T H E O R I E S A N D

APPLICATIONS

Perhaps a b e t t e r i n t e r p r e t a t i o n o f E q u a t i o n s (16) and (17) i s t h a t the a j i a r i s e s f r o m an e f f e c t i v e p a i r p o t e n t i a l w h i c h p e r m i t s a r e a s o n a b l e d e s c r i p t i o n o f the i n t e r n a l energy at h i g h d e n s i t i e s . However, t h e s e s i m p l e models do not r e v e a l the form they assume f o r m i c r o s c o p i c q u a n t i t i e s , s i n c e we cannot deduce the form o f a f u n c t i o n from a s i n g l e value of i t s i n t e g r a l . T h i s u n c e r t a i n t y i n the j i d i s t r i b u t i o n f u n c t i o n s f o r the vdW-1 case makes an a n a l y s i s o f the d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n models v e r y tenuous i n d e e d . One argument, o f f e r e d by S a n d l e r and c o - w o r k e r s , s u g g e s t s t h a t the l o c a l c o m p o s i t i o n s caused by the a t t r a c t i v e f o r c e s a r e h i g h e s t at low d e n s i t i e s and d e c r e a s e w i t h d e n s i t y s i n c e h i g h - d e n s i t y s t r u c t u r e i s d e t e r m i n e d by the r e p u l s i v e f o r c e s . A c c o r d i n g t o t h i s argument the d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n models (29 - 31) a r e q u a l i t a t i v e l y wrong s i n c e t h e y assume t h a t the l o c a l - c o m p o s i t i o n e f f e c t i s z e r o at low d e n s i t i e s and i n c r e a s e s w i t h d e n s i t y . F u r t h e r comments a r e needed t o put t h i s statement i n t o p e r s p e c t i v e . I t s h o u l d f i r s t be n o t e d t h a t the d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n models do not p r e d i c t random b e h a v i o r at low d e n s i t i e s s i n c e they d e s c r i b e second v i r i a l c o e f f i c i e n t b e h a v i o r c o r r e c t l y ( E q u a t i o n 19). Now the q u e s t i o n i s : I s i t q u a l i t a t i v e l y c o r r e c t t o assume t h a t the l o c a l - c o m p o s i t i o n e f f e c t i n c r e a s e s w i t h density? S a n d l e r ' s s t u d y , c o n s i s t e n t w i t h o t h e r s t u d i e s , shows us t h a t the W i l s o n Boltzmann f a c t o r s do o v e r e s t i m a t e the l o c a l c o m p o s i t i o n e f f e c t at h i g h d e n s i t i e s . The computer s i m u l a t i o n s o f N a k a n i s h i and co-workers have shown t h a t t h e r e i s some l o c a l - c o m p o s i t i o n e f f e c t a t h i g h d e n s i t i e s due t o the a t t r a c t i v e f o r c e s and t h i s c l u s t e r i n g i s a p p r o x i m a t e l y d e s c r i b e d by the NRTL model. Thus t h e r e i s some j u s t i f i c a t i o n f o r the model at h i g h d e n s i t i e s and the d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n models at l e a s t meets m a c r o s c o p i c boundary c o n d i t i o n s at low and h i g h d e n s i t i e s . We s t r e s s t h a t " l o c a l - c o m p o s i t i o n " i s an ambiguous and i n t e r m e d i a t e q u a n t i t y . I t i s perhaps more a p p r o p r i a t e t o e v a l u a t e models a c c o r d i n g t o t h e i r d e s c r i p t i o n o f more m e a n i n g f u l p r o p e r t i e s l i k e the configurâtional energy. The

Future

vdW-1 m i x i n g r u l e s a r e r e a s o n a b l y based i n t h e o r y and have enabled r e l i a b l e p r e d i c t i o n s i n many n o n p o l a r systems. F u r t h e r work u s i n g vdW-1 m i x i n g r u l e s as a b a s i s f o r e x t e n s i o n appears t o be warranted. P e r t u r b a t i o n theory suggests that separate mixing r u l e s are needed f o r d i f f e r e n t i n t e r m o l e c u l a r e f f e c t s . Approaches based on p e r t u r b a t i o n t h e o r y s h o u l d c o n t r i b u t e t o the c o n t i n u e d development o f new m i x i n g r u l e s . Most l o c a l c o m p o s i t i o n m i x i n g r u l e f o r m u l a t i o n s now resemble W i l s o n ' s e q u a t i o n , w h i c h b o t h o v e r e s t i m a t e s the l o c a l - c o m p o s i t i o n e f f e c t and becomes l e s s a c c u r a t e f o r m i x t u r e s w i t h i n c r e a s i n g s i z e d i f f e r e n c e s . New i d e a s a r e needed t o improve t h e s e d e f i c i e n c i e s . For example, E l y (64) has r e c e n t l y shown f o r s q u a r e - w e l l m i x t u r e s

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

17.

COPEMAN AND

-βε

j

nji - e

Mixing Rules: An Industrial Perspective

MATHIAS

367

3pj

ι

δ j

i

-

+

Bpj

8pi

Hard

4TT

R

3

(20)

Sphere

which can be s u b s t i t u t e d i n t o the energy e q u a t i o n ( E q u a t i o n 3) t o d e r i v e the m i x t u r e e q u a t i o n o f s t a t e . E l y ' s development i n c o r p o r a t e s hard-sphere s i z e e f f e c t s i n t o the l o c a l - c o m p o s i t i o n f o r m u l a t i o n . F u r t h e r work i n t h i s d i r e c t i o n i s s t r o n g l y encouraged.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

Conclusions The c a p a b i l i t y o f e q u a t i o n s o f s t a t e t o d e s c r i b e asymmetric m i x t u r e s has improved s u b s t a n t i a l l y o v e r the p a s t f i v e y e a r s . The o l d r u l e o f thumb t h a t one must use a c t i v i t y c o e f f i c i e n t models f o r m i x t u r e s w i t h p o l a r components i s b e g i n n i n g t o f a d e away. T h i s t r e n d w i l l b e n e f i t c h e m i c a l p r o c e s s d e s i g n and o p t i m i z a t i o n , as more than one approach has been needed, i n some c a s e s , t o s i m u l a t e one e n t i r e process. Equations of s t a t e w i t h l o c a l - c o m p o s i t i o n mixing r u l e s o f f e r s h o r t - t e r m p o t e n t i a l t o d e s c r i b e h i g h l y p o l a r - a s y m m e t r i c systems i n the same ( e m p i r i c a l ) manner as a c t i v i t y c o e f f i c i e n t models. These models now l a r g e l y resemble W i l s o n ' s e q u a t i o n and new i d e a s a r e needed t o d e v e l o p improved f o r m u l a t i o n s . Computer s i m u l a t i o n s t u d i e s a r e b e i n g used t o p r o v i d e i n f o r m a t i o n on l o c a l f l u i d s t r u c t u r e . Some improvements t o m i x i n g r u l e s have been s u g g e s t e d , however, have not y e t been e v a l u a t e d on r e a l systems. F u r t h e r work t o improve m i x i n g r u l e s f o r a p p l i c a t i o n to r e a l systems based on computer s i m u l a t i o n i s b a d l y needed. While comparisons based on l o c a l c o m p o s i t i o n s can p r o v i d e some guidance i n d e v e l o p i n g b e t t e r models, a n a l y s i s o f m a c r o s c o p i c p r o p e r t i e s , such as a c t i v i t y c o e f f i c i e n t s and i n t e r n a l energy, i s recommended t o a v o i d p o t e n t i a l a m b i g u i t i e s and p i t f a l l s . A unified formulation of the energy, p r e s s u r e and c o m p r e s s i b i l i t y e q u a t i o n s i n terms o f l o c a l p a r t i c l e numbers now e x i s t s and can s t r a i g h t f o r w a r d l y be used t o d e v e l o p m i x t u r e e q u a t i o n s o f s t a t e f o r new f o r m u l a t i o n s . Acknowledgments We w i s h t o thank P r o f e s s o r s S. S a n d l e r , J . M. P r a u s n i t z and L. L. Lee and Dr. J . F. E l y f o r s e n d i n g us p r e p r i n t s o f t h e i r u n p u b l i s h e d p a p e r s , and P r o f e s s o r P r a u s n i t z f o r h i s h e l p f u l comments. L i s t o f Symbols a b Β Ε g k η Ν R Τ u

a t t r a c t i v e c o n s t a n t i n van der Waals-based e q u a t i o n o f s t a t e s i z e c o n s t a n t i n van d e r Waals-based e q u a t i o n o f s t a t e second v i r i a l c o e f f i c i e n t i n t e r n a l energy radial distribution function Boltzmann's c o n s t a n t l o c a l p a r t i c l e number number o f m o l e c u l e s l i m i t of integration a b s o l u t e temperature intermolecular pair potential

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

368 V X β δ ε μ ρ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

σ

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

volume f l u i d - p h a s e mole f r a c t i o n 1/KT Kronecker d e l t a energy parameter i n p o t e n t i a l chemical p o t e n t i a l density hard-sphere diameter

function

Literature Cited 1. Sandler, S. I. Proceedings of the National Science Foundation Workshop, "Thermodynamic Needs for the Decade Ahead - Theory and Experiment," Washington, D. C., October 28-29, 1983. 2. Ely, J. F.; Baker J. K. "A Review of Supercritical Extraction;" NBS Technical Note 1070, 1983. 3. Chappelear, P. S.; Chen, R. J.; Elliot, D. E. Hydrocarbon Processing, 1977, 215-217. 4. George, B. A. Proceedings of the Sixty-first Annual Convention, Gas Processors Association, Dallas, March 15-17, 1982. 5. Miller, E. J.; Geist, J. M. Presented at the Joint Meeting of the Chemical Industry Engineering Society of China and the AIChE, Beijing, China, 1982. 6. Abrams, D. S.; Seneci, F.; Chueh, P. L.; and Prausnitz, J. M. Ind. Eng. Chem. Fundam., 1975, 14:52-54.31 7. van der Waals, J. D. Z. Phys. Chem., 1890, 5:133-173. 8. Soave, G. Chem. Eng. Sci., 1972, 27:1197. 9. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam., 1976, 15:59-64. 10. Leland, T. W.; Chappelear, P. S.; Gamson, B. W. AIChEJ, 1962, 8:482-489. 11. Reid, R. C.; Leland, T. W. AIChEJ, 1965, 11:228-237. 12. Leland, T. W.; Chappelear, P. S. Ind. Eng. Chem., 1968, 60(7):15-43. 13. Henderson, D.; Leonard, P. J. Proc. Natl. Acad. Sciences, 1970, 67:1818-1823 14. Henderson, D.; Leonard, P. J. Proc. Natl. Acad. Sciences, 1971, 68:632-635 15. Vidal, J. Chem. Eng. Sci., 1978, 33:787-791. 16. Stotler, H. H.; Benedict, M. Chem. Eng. Prog. Symp. Ser., No. 6, 1953, 49:25-36. 17. Benedict, M.; Webb, G. B.; and Rubin, L. C. J. Chem. Phys., 1940, 8:334. 18. Benedict, M.; Webb, G. B.; and Rubin, L. C. Chem-Eng. Prog., 1951, 47:419. 19. Prausnitz, J. M.; Gunn, R. D. AIChEJ, 1958, 4:430-435. 20. Plocker, U.; Knapp, H.; Prausnitz, J. M. Ind. Eng. Chem. Proc. Des. Dev., 1978, 17:324. 21. Radosz, M.; Lin, H. M.; and Chao, K. C. Ind. Eng. Chem. Proc. Des. Dev., 1982, 21:653-658. 22. Lee, T. J.; Lee, L. L.; Starling, Κ. E. "Equations of State in Engineering and Research," 1979; Meeting of the American Chemical Society, September 11-14, 1978, 125-141. 23. Henderson, D. "Equations of State in Engineering and Reasearch," 1979, Meeting of the American Chemical Society, September 11-14, 1978, 1-30.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

17. COPEMAN AND MATHIAS

Mixing Rules: An industrial Perspective

24. Donohue, M. D.; Prausnitz, J. M. AIChEJ, 1978, 24:849-860. 25. Wilson, G. M. Contribution No. 29, 1972, Center for Thermochemical Studies, Brigham Young University, March. 26. Huron, M. J.; Vidal, J. Fluid Phase Equilibria, 1979, 3:255-271. 27. Heyen, G. Proceedings of the 2nd World Congress of Chemical Engineering, October 4-9, 1981, 41-46. 28. Won, K. W. Presented at the Annual AIChE Meeting, New Orleans, November 8-12, 1981. 29. Whiting, W. B.; Prausnitz, J. M. Paper presented at Spring National Meeting, American Institute Chemical Engineers, Houston, Texas, April 5-9, 1981. 30. Whiting, W. B.; Prausnitz, H. M. Fluid Phase Equilibria, 1982, 9:119-147. 31. Mollerup, J. Fluid Phase Equilibria, 1981, 7:121-138. 32. Won, K. W. Fluid Phase Equilibria, 1983, 10:191-210. 33. Wilson, G. M. J. Am. Chem. Soc., 1964, 86:127-130. 34. Mathias, P. M.; Copeman, T. W. Presented at the Third International Conference on Fluid Properties and Phase Equilibria for Chemical Process Design, Callaway Gardens, Georgia, April 10-15, 1983. Also published in Fluid Phase Equilibria, 13:91-108. 35. Dimitrelis, D.; Prausnitz, J. M. Presented at Fall Annual Meeting, American Institute of Chemical Engineering, Los Angeles, CA, Nov. 14-19, 1982. 36. Mollerup, J. Fluid Phase Equilibria, 1983, 15:189-207 37. Ludecke, D.; Prausnitz, J. M. Fluid Phase Equilibria, 1985, 22:1-19. 38. Mansoori, G. Α.; Carnahan, N. F.; Starling, Κ. E.; Leland, T. W. J. Chem. Phys., 1971, 54:1523-1525. 39. Larsen, E. R.; Prausnitz, J. M. AIChEJ, 1984, 30:732-738. 40. Li, M. H.; Chung, F. T.; Lee, L. L.; Starling, Κ. E. Presented at the Fall Annual Meeting, American Institute of Chemical Engineers, San Francisco, CA, November 25-30, 1984. 41. Chung, T. H.; Khan, M. M.; Lee, L. L.; Starling, Κ. E. Fluid Phase Equilibria, 1984, 17:351. 42. Renon, H.; Prausnitz, J. M. AIChEJ, 1968, 14:135-144. 43. McDermott, C.; Ashton, N. Fluid Phase Equilibria, 1977, 1:33-35. 44. Kemeny, S.; Rasmussen, P. Fluid Phase Equilibria, 1981, 7:197-203. 45. Abrams, D. S.; Prausnitz, J. M. AIChEJ 1975, 21:116-128. 46. Lee, L. L.; Chung, R. H.; and Starling, Κ. E. Fluid Phase Equilibria, 1983, 12:105-124. 47. Mansoori, G. Α.; Leland, Jr., T. W. J. Chem. Soc., Faraday Trans. II, 1972, 68:320-344. 48. Seaton, Ν. Α.; Glandt, E. D. Presented at the Fall Annual Meeting, American Institute of Chemical Engineers, San Francisco, CA, November 25-30, 1984. 49. Gierycz, P.; Nakanishi, K. Fluid Phase Equilibria, 1984, 16:255-273. 50. Nakanishi, K.; Tanaka, H. Fluid Phase Equilibria, 1983, 13:371-380. 51. Nakanishi, K.; Taukabo, K. J. Chem. Phys., 1979, 70:5848-5850.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

369

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

370

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

52. Gierycz, P.; Tanaka, H.; Nakanishi, K. Fluid Phase Equilibria, 1984, 16:241-253. 53. Hu, Y.; Ludecke, D.; Prausnitz, J. M. Fluid Phase Equilibria, 1984, 17:217-241. 54. Shing, K. S.; Gubbins K. E. Mol. Phys., 1983, 49:1121-1138. 55. Sandler, S. I. Fluid Phase Equilibria, 1985, 19:233-257. 56. Lee, K. H.; Lombardo, M.; Sandler, S. I. "The Generalized van der Waal's Partition Function II. Application to Square-Well Fluid," Fluid Phase Equilibria, 1985, in press. 57. Lee, Κ. H.; Sandler, S. I.; Patel, N. C. "The Generalized van der Waal's Partition Function III. Local-Composition Models for a Mixture of Equal Size Square-Well Molecules," Fluid Phase Equilibria, 1985, in press. 58. Mansoori, G. Α.; Ely, J. F. "Statistical Mechanical Theory of Local Compositions," submitted to Fluid Phase Equilibria, 1984, 59. Fischer, J. Fluid Phase Equilibria, 1983, 10:1-7. 60. Lebowitz, J. L.; Rowlinson, J. S. J. Chem. Phys., 1964, 41:133-138. 61. Chandler, D.; Weeks, J. D. Phys. Rev. Lettrs., 1970, 24:849 62. Weeks, J. D.; Chandler, D.; Anderson, H. C. J. Chem. Phys., 1971, 54:5237. 63. Oellrich, L.; Plocker, V.; Prausnitz, J. M.; Knapp, H. International Chemical Engineering, 1981, 21:1-16. 64. Ely, J. F., 1985, to be submitted for publication. RECEIVED November

5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18 Calculation of Fluid-Fluid and Solid-Fluid Equilibria in Cryogenic Mixtures at High Pressures Ulrich K. Deiters

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

Lehrstuhl für Physikalische Chemie II, Ruhr-Universität, D-4630 Bochum 1, Federal Republic of Germany

A non-cubic equation of state, which includes quantum corrections and special corrections for molecular size differences, is used to correlate fluid-fluid phase equilibria (vapour-liquid, liquid-liquid, and gas-gas) in binary cryogenic mixtures (containing H , CO, noble gases, etc.) up to 2000 bar. Volumetric properties of these mixtures are predicted successfully. The cubic Redlich-Kwong equation of state is shown to perform less satisfactorily. The thermodynamic functions of the mixtures are calculated from modified versions of either one-fluid theory or mean density approximation. An expression for the Gibbs energy of a pure solid is derived; from this it is possible to predict solid-fluid equilibria, including solid-liquid-gas three-phase lines, from experimental fluid-fluid equilibrium data, and vice versa. 2

The

calculation

pressure low to

of

requires

pressure regard

phase

much

e q u i l i b r i a .

one o r more

retical

approaches

because

appropriate

equations culation -

of

standard

states

for several

proper

to liquid,

Because

of

their

b u i l t - i n

tions

these

between

Calculations is

state also

with

aim of

of

permissible

ideal,

and

theo-

inefficient,

are

On

other

lacking.

basis

i n

f o r phase

the

hand,

equilibrium

and s u p e r c r i t i c a l states

the v i c i n i t y

with

a c t i v i t y

predicting

(solid-supercritical

consistency

equations but also

this

are not only of

longer become

work

capable

from

of

of

c r i t i c a l

coefficient

state

input

cal-

a l i k e , points,

methods,

are

that

of

transi-

the equilibrium equations

equilibria,

e q u i l i b r i a

melting)

calculate

vaporization.

modern

f l u i d - f l u i d

solid-fluid

to

and a l l

information.

yield

and heats

demonstrate

equilibria,

possible

not only

correlating and

is

equilibria,

t h e same

densities to

of

PVT data fluid

i t

l i q u i d - l i q u i d

two k i n d s

compositions, the

no

elevated

coefficients

vapour,

encountered

equilibria,

It

i s

at

calculations

avoided.

vapour-liquid

phase

i t

than

reasons:

d i f f i c u l t i e s

are so often

mixtures

effort

i n e q u i l i b r i u m as

on a c t i v i t y

can form a

conceptual

completely

-

the phases

so

that

binary

pressure

They a r e a p p l i c a b l e which

-

of

i n

computational

At high

based

state

methods

e q u i l i b r i a

more

of

successfully.

0097-6156/ 86/ 0300-0371 $06.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

of but

mixtures

372

E Q U A T I O N S O F STATE: T H E O R I E S A N D

Equations of

APPLICATIONS

State

Pure S u b s t a n c e s . For the c a l c u l a t i o n s d e s c r i b e d below we have used two van der Waals type e q u a t i o n s of s t a t e , i . e . e q u a t i o n s c o n s i s t i n g o f d i s t i n c t r e p u l s i o n and a t t r a c t i o n terms, namely: - the Redlich-Kwong (RK) e q u a t i o n (1) as a r e p r e s e n t a t i v e o f the w i d e l y used c l a s s o f c u b i c e q u a t i o n s o f s t a t e : ρ _ ρ

_ ρ rep

(Here a and b a r e and r e p u l s i o n . )

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

- and

our Ρ

own

the

_ att

R

T

Ά

_

V -b m

VTV

(1 )

(V + b) mm

(substance-specific)

equation of state

parameters o f a t t r a c t i o n

(2):

_ Η ν m

with ξ =

Τ

c

ρ

•^£-y e f f " Y ( p ,« cc)) 0

= 0.6887

ρ = ^—

(reduced

densities)

Τ = —

(reduced

temperatures)

a

λ = -0.07c

The o t h e r e x p r e s s i o n s i n E q u a t i o n 2 are f u n c t i o n s of ρ and c, which have been e x p l a i n e d elsewhere ( 2 ) . T h i s e q u a t i o n o f s t a t e i s c l e a r l y n o n - c u b i c ; i t s t h r e e parameters a, b, and c may be u n d e r s t o o d as c h a r a c t e r i s t i c t e m p e r a t u r e , covolume, and a n i s o t r o p y . At low t e m p e r a t u r e s , some l i g h t gases such as h e l i u m , neon, h y d r o gen e t c . show d e v i a t i o n s from c l a s s i c a l mechanics, which are u s u a l l y r e f e r r e d to as quantum e f f e c t s . The l a r g e r p a r t of t h e s e d e v i a t i o n s can be e x p l a i n e d by r e s t r i c t i o n s of the p a r t i c l e m o t i o n s : a t h i g h d e n s i t i e s each m o l e c u l e i s more or l e s s c o n f i n e d t o a c e l l by the r e p u l s i o n of i t s neighbour m o l e c u l e s . Then the t r a n s l a t i o n a l e i g e n s t a t e s are no longer c l o s e l y s p a c e d , so t h a t i t i s not p e r m i s s i b l e t o c a l c u l a t e the p a r t i t i o n f u n c t i o n by i n t e g r a t i o n . I t i s p o s s i b l e , however, t o r e p r e sent the quantum c o r r e c t e d p a r t i t i o n f u n c t i o n as a p r o d u c t of the c l a s s i c a l p a r t i t i o n f u n c t i o n and a c o r r e c t i o n f u n c t i o n (3):

Q

3 N

quant

= Q , q class corr

For s m a l l d e v i a t i o n s from c l a s s i c a l b e h a v i o u r can be approximated by

q corr with y = Λ / L Λ =

π

^γ~

(3)

the c o r r e c t i o n

= 1 - \ 2

(reduced w a v e l e n g t h )

(thermal

de

B r o g l i e wavelength)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

function

(4)

18.

Fluid-Fluid and Solid-Fluid

DEITERS

(cell

Equilibria

373

size)

The c e l l s i z e depends on t h e f r e e volume o f t h e f l u i d . F o r l a r g e r e d u ced wavelengths t h e c o r r e c t i o n f u n c t i o n can be o b t a i n e d from a s e r i e s e x p a n s i o n ; the c o e f f i c i e n t s a r e g i v e n i n T a b l e I : 13 In q

- I

(5)

r .y J

j=1 T a b l e I . Expansion C o e f f i c i e n t s o f the Quantum C o r r e c t i o n

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

j 0 1 2 3 4 5 6

r. J

r. J

j

0 -0.499 951 -0.128672 +0.026 360 -0.552 534 +2.240 993 -5.570201

7 8 9 10 11 12 13

749 27 732 48 453 309 071 60 820 8 592 1

Function

+8 .663 291 -8 .727837 +5 .742 346 -2 .444507 +0 .648 968 -0 .097 758 +0 .006 387

244 9 8272 562 0 6274 1 16 33 376489 718839

The f r e e volume depends on the e q u a t i o n o f s t a t e . F o r van d e r Waals type e q u a t i o n s , which c o n s i s t o f a r e p u l s i o n and an a t t r a c t i o n term, the f r e e volume i s d e f i n e d by

rep nRT

In

The

f r e e volume a s s o c i a t e d

V

f

(6)

dV

w i t h the equation o f s t a t e

2 i s g i v e n by

= V exp|

From E q u a t i o n s 3, 5, and 6 i t i s e v i d e n t t h a t the quantum c o r r e c t e d v e r s i o n o f any van der Waals type e q u a t i o n o f s t a t e i s g i v e n by

Ρ = Ρ

rep

(1

I

r

J

j

jr.y ) - Ρ 1 att

(8)

S i n c e the d o m i n a t i n g e x p a n s i o n c o e f f i c i e n t s a r e n e g a t i v e , t h e quantum c o r r e c t i o n amounts t o an i n c r e a s e o f the r e p u l s i o n p r e s s u r e . T h i s i s i n agreement w i t h the p e r t u r b a t i o n t h e o r y r e s u l t s o f S i n g h and S i n h a ( 4 ) . The i n f l u e n c e o f the quantum c o r r e c t i o n on the r e p r e s e n t a t i o n o f PVT d a t a and c r i t i c a l d a t a o f pure s u b s t a n c e s has been d i s c u s s e d elsewhere ( 3 , 5 ) . I t s h o u l d be n o t e d t h a t the i n t r o d u c t i o n o f quantum c o r r e c t i o n s always l e a d s t o n o n - c u b i c e q u a t i o n s o f s t a t e . Mixtures. One o f the most s u c c e s s f u l methods o f g e n e r a l i z i n g a pure substance e q u a t i o n o f s t a t e t o m i x t u r e s i s the o n e - f l u i d t h e o r y . I t amounts t o a p p l y i n g the p u r e s u b s t a n c e e q u a t i o n o f s t a t e t o a m i x t u r e , u s i n g c o n c e n t r a t i o n dependent p a r a m e t e r s . The c o n c e n t r a t i o n dependence i s u s u a l l y c o n t a i n e d i n a s e t of i n t e r p o l a t i o n f o r m u l a s , r e f e r r e d t o as

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

374

m i x i n g r u l e s . F o r the Redlich-Kwong m i x i n g r u l e s have been u s e d :

equation

x

the f o l l o w i n g q u a d r a t i c

a

a = x^a^ ^ + 2x^x^a^2 + £ 2 2 2 b = x^b^

2 + 2 x ^ 2 ^ 2 + 2 22 x

These m i x i n g r u l e s do n o t i m p l y random m i x i n g ( 6 ) , n o r a r e they compa t i b l e w i t h the van der Waals m i x i n g t h e o r y ; the Redlich-Kwong a t t r a c t i o n parameter i s r e l a t e d t o the m o l e c u l a r p o t e n t i a l w e l l depth and d i a m e t e r by a ^ ε ' σ , whereas the a t t r a c t i o n parameter o f the v a n d e r Waals equation o f s t a t e i s p r o p o r t i o n a l t o ε o . A d e t a i l e d d i s c u s s i o n o f the p h y s i c a l meaning o f a t t r a c t i o n parameters i n s e v e r a l e q u a t i o n s o f s t a t e and i m p l i c a t i o n s f o r improved m i x i n g r u l e s have been g i v e n by Mansoori in t h i s symposium ( 7 ) . Here we o n l y want t o make the p o i n t t h a t o t h e r than van d e r Waals type m i x i n g r u l e s have s u c c e s s f u l l y been used f o r a long t i m e , and t h a t i t i s n o t n e c e s s a r y f o r a s u c c e s s f u l m i x i n g r u l e t o be o f the v a n d e r Waals t y p e . For E q u a t i o n 2 t h e f o l l o w i n g m i x i n g r u l e s have been used (8): 3

2

3

1

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

b

X

3

C

c = x-jC^ + 2 22 2 b = i i i x

a = x.a., 1

1

a

w i t h Aa =

a

^ 12 S

S

12 X

+ x,a„,

1

11 11

b

2

2

2

+

x

x

b

^ i 2 12

+

, 1

+

+

2 x

/1 • 4

x

2 2 22 b

1 11 2 s

q i

q

A a

• q [exp(^)-1]

„ (13; M

2

a

22 S

(exchange energy)

22

S

i i i q. = (surface fraction) ι 22 The c a l c u l a t i o n o f t h e e f f e c t i v e s u r f a c e r a t i o s s has been d e s c r i b e d elsewhere (9) (compare E q u a t i o n 2 4 ) . The square r o o t term i n E q u a t i o n 13 accounts f o r non-randomness a c c o r d i n g t o a q u a s i c h e m i c a l model ( 8 0 . I t c o n t r i b u t e s t o t h e e q u a t i o n o f s t a t e a t low temperatures o n l y . I n the h i g h temperature l i m i t E q u a t i o n 13 becomes a p p r o x i m a t e l y e q u i v a l e n t to a v e r a g i n g ε σ ; the a c t u a l v a l u e o f the σ-exponent depends somewhat on t h e s i z e r a t i o . The c o n t r i b u t i o n s o f the quantum e f f e c t s t o t h e e q u a t i o n o f s t a t e are averaged a c c o r d i n g t o t h e f o l l o w i n g f o r m u l a : and

X

S

1

+

1

X

1

S

2

2

Ρ = Ρ (1 - χ, Y j r . y j - x £ j r . y h rep 1 . 1 1 2 r ι 2 J J y. = A./L ι ι 0

with

- Ρ _ att

(14)

J

The reduced wavelengths a r e c a l c u l a t e d s e p a r a t e l y f o r each component of t h e m i x t u r e . The e f f e c t i v e c e l l s i z e L i s o b t a i n e d from t h e mean reduced d e n s i t y u s i n g E q u a t i o n s 12, 2, and 7.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18.

Fluid-Fluid

DEITERS

and Solid-Fluid

Equilibria

375

I f t h e m o l e c u l e s o f a m i x t u r e d i f f e r v e r y much i n s i z e ( d i a m e t e r r a t i o above 1 . 6 ) , t h e m i x i n g r u l e s g i v e n above a r e no l o n g e r s u f f i c i e n t . I t i s p o s s i b l e , however, t o extend E q u a t i o n 2 t o m i x t u r e s w i t h even l a r g e r s i z e r a t i o s by s u b s t i t u t i n g the r i g i d sphere c o m p r e s s i b i l i t y f a c t o r by an a p p r o p r i a t e l y m o d i f i e d r i g i d sphere m i x t u r e e x p r e s s i o n ( 1 0 ). Here t h e e x p r e s s i o n o f M a n s o o r i , Carnahan, S t a r l i n g , and L e l a n d ( 1 1 ) has been u s e d .

^r e p•• V~ ^ I[ 'i ^° . ' ^ΓΪ j r ^ ] r

+

(is)

t c

m 3

with

Ε =

(

ν

Χ

2

Κ

2 1

)

(

Χ

1

+

Χ

^

2

Κ

2 1

}

+ 1 Λ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

χ .j •+• χ 2 ^ 2 Ί (χ,| + X 2 R 2 1 )

and

F =

R 1

^—^ (χ ^ + X 2 R 2

R2 1 0 1

)

• 1 NT

= \\

(diameter

ratio)

The a v e r a g i n g p r o c e d u r e s i m p l i e d by E q u a t i o n 1 4 and the use o f E q u a t i o n 1 5 c o n s t i t u t e a d e p a r t u r e from o n e - f l u i d t h e o r y . C o n v e r s e l y , i f f o r m a l v a l i d i t y o f o n e - f l u i d t h e o r y i s assumed, t h e s e two e q u a t i o n s would l e a d to d e n s i t y and temperature dependent p a r a m e t e r s . The parameters and b a r e e s t i m a t e d from t h e u s u a l g e o m e t r i c , r e s p . a r i t h m e t h i c , mean r u l e s . T h e i r e x a c t v a l u e s a r e always d e t e r m i n e d by f i t t i n g the e q u a t i o n o f s t a t e t o b i n a r y phase e q u i l i b r i u m d a t a . u

Phase

Equilibria

Fluid-Fluid Equilibria. The thermodynamic c o n d i t i o n s o f phase brium i n a b i n a r y f l u i d m i x t u r e a r e ( 1 2 ) :

equili

P' = P" (16)

T' = T" i = 1

1

μ . = μ'.' 1

1

The c h e m i c a l p o t e n t i a l s μ a r e o b t a i n e d as d e r i v a t i v e s energy a t c o n s t a n t p r e s s u r e and temperature:

J

i

of the Gibbs

(17)

lôn.L i P,T,n. ι . m

L

In the case of a f l u i d mixture, from t h e f o l l o w i n g f o r m u l a :

J

the Gibbs

energy

G (Ρ,Τ)

i s obtained

V J P(V,T) dV

f

G (P,T) = η ^ ^ Ρ ' , Τ ) + n G£(P°,T) 2

+ PV - nRT

V° + nRT(x^ln χ

Ί

+ x l n x^) 2

(18)

The G° denote Gibbs e n e r g i e s o f the pure components i n t h e p e r f e c t gas s t a t e ( v e r y s m a l l p r e s s u r e P° , v e r y l a r g e volume V ° ) . These terms c a n -

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

376

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

e e l i n the c o u r s e o f the phase e q u i l i b r i u m c a l c u l a t i o n , i f c h e m i c a l r e a c t i o n s a r e e x c l u d e d . The i n t e g r a n d i s the e q u a t i o n of s t a t e of the m i x t u r e . E q u a t i o n 18 i s a p p l i e d t o v a p o u r s , l i q u i d s , and s u p e r c r i t i c a l s t a t e s a l i k e . Because of the c o n t i n u i t y p r i n c i p l e i t i s p o s s i b l e to i n t e g r a t e the e q u a t i o n o f s t a t e " t h r o u g h " the vapour p r e s s u r e l i n e (see F i g u r e 1 ) . Together w i t h E q u a t i o n 18, 16 r e p r e s e n t s a system o f non l i n e a r e q u a t i o n s , which can be s o l v e d n u m e r i c a l l y f o r the e q u i l i b r i u m phase c o m p o s i t i o n s ( 1 3 ) . The c o n d i t i o n s o f phase e q u i l i b r i u m can a l s o be f o r m u l a t e d u s i n g f u g a c i t i e s ; the r e s u l t i n g e q u a t i o n s are e q u i v a l e n t to those g i v e n above, but can be made to appear l e s s c o m p l i c a t e d , i f the f u n c t i o n s o f the s t a n d a r d s t a t e a r e s u p p r e s s e d and r e s i d u a l p r o p e r t i e s a r e u s e d . But i n t h i s work the use o f c h e m i c a l p o t e n t i a l s and Gibbs e n e r g i e s i s p r e f e r r e d f o r two r e a s o n s : E q u a t i o n 18 i s b e t t e r adapted to our phase e q u i l i b r i u m a l g o r i t h m ( 1 3 ) , and the g e n e r a l i z a t i o n o f our c o m p u t a t i o n a l method t o m i x t u r e s w i t h c h e m i c a l r e a c t i o n s i s more s t r a i g h t f o r w a r d . Solid-Fluid Equilibria. S i n c e the c o n t i n u i t y p r i n c i p l e cannot be ex tended to s o l i d phases, i t i s n e i t h e r p o s s i b l e nor d e s i r a b l e to use the same e q u a t i o n o f s t a t e f o r f l u i d and s o l i d phases w i t h o u t c r e a t i n g con c e p t u a l d i f f i c u l t i e s . In some c a s e s a s u c c e s s f u l c o r r e l a t i o n o f s o l i d gas e q u i l i b r i a has been a c c o m p l i s h e d by f i t t i n g a f l u i d e q u a t i o n o f s t a t e to s o l i d s t a t e d a t a , thus t r e a t i n g the s o l i d - g a s e q u i l i b r i u m f o r m a l l y as a l i q u i d - v a p o u r e q u i l i b r i u m . But as t h i s approach r e n d e r s the d e s c r i p t i o n of s o l i d - l i q u i d - g a s three-phase l i n e s i m p o s s i b l e and i s n o t c a p a b l e o f e x p l a i n i n g the complete phase diagram o f a m i x t u r e , i t i s n o t used i n t h i s work. I n s t e a d , a s e p a r a t e e q u a t i o n o f s t a t e i s employed f o r the s o l i d s t a t e . U s u a l l y the f o l l o w i n g r e l a t i o n p r o v i d e s a s u f f i c i e n t d e s c r i p t i o n o f s o l i d PVT b e h a v i o u r : V (P,T) = v (p S

s

s f

,T)(i

- *(p - p

s f

))

(19)

sf Ρ i n E q u a t i o n 19 denotes the s u b l i m a t i o n o r m e l t i n g p r e s s u r e (which ever i s more a p p r o p r i a t e a t the temperature T ) ; κ i s the i s o t h e r m a l s o l i d c o m p r e s s i b i l i t y . In most c a s e s t h e r e w i l l not be s u f f i c i e n t ex p e r i m e n t a l d a t a a v a i l a b l e to c o n s t r u c t a more s o p h i s t i c a t e d s o l i d equa t i o n o f s t a t e . In o r d e r t o o b t a i n an e x p r e s s i o n f o r the Gibbs energy o f a pure s o l i d a t a r b i t r a r y p r e s s u r e , which can be used f o r phase e q u i l i b r i u m c a l c u l a t i o n s , we have to l i n k E q u a t i o n 19 to 18 i n a c o n s i s t e n t manner ( 1 0 ) : Ρ S

f

G (P,T) = G ( P

S f

,T) +

f J

S

V ( P , T ) dP

(20)

psf

T h i s e q u a t i o n i s o b t a i n e d by (compare F i g u r e 1) - d e t e r m i n i n g the Gibbs energy o f the f l u i d phase a t the s u b l i m a t i o n / melting point, - r e a l i z i n g t h a t at t h i s p o i n t s o l i d and f l u i d phase have the same molar Gibbs energy, ^ - i n t e g r a t i n g the s o l i d e q u a t i o n o f s t a t e from Ρ up t o the p r e s s u r e desired. E q u a t i o n 20 i n c o n j u n c t i o n w i t h 18 enables us to c a l c u l a t e phase e q u i l i b r i a between a f l u i d m i x t u r e and a pure c r y s t a l l i n e phase. g

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18.

DEITERS

Fluid-Fluid

and Solid-Fluid

Equilibria

317

Other a u t h o r s have used a s i m i l a r c o n c e p t f o r t r e a t i n g s o l i d - f l u i d phase e q u i l i b r i a (14, 1 5 ) . The f o r m a l i s m proposed h e r e i s thermodynamic a l l y e q u i v a l e n t , but a v o i d s the use o f a c t i v i t y c o e f f i c i e n t s and metas t a b l e or h y p o t h e t i c a l r e f e r e n c e s t a t e s . Furthermore our formalism accounts f o r the c o m p r e s s i b i l i t y o f the s o l i d phase, the i n f l u e n c e o f which on phase e q u i l i b r i a i s n o t always n e g l i g i b l e .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

M o d i f i e d Mean D e n s i t y

Approximation

An a l t e r n a t i v e m i x i n g t h e o r y , which i n i t s o r i g i n a l form i s o n l y s l i g h t l y i n f e r i o r t o the o n e - f l u i d t h e o r y ( 1 6 ) , i s the mean d e n s i t y a p p r o x i m a t i o n . I t has the advantage, however, t h a t i t can be m o d i f i e d w i t h o u t too much m a t h e m a t i c a l e f f o r t . For example, the i n t r o d u c t i o n o f d e n s i t y dependent a t t r a c t i o n parameters i n t o E q u a t i o n 2 would y i e l d an unwieldy f o r m a l i s m w i t h i n o n e - f l u i d t h e o r y , but can be made e a s i l y w i t h i n the mean d e n s i t y a p p r o x i m a t i o n . Our p r o p o s i t i o n i s to r e p l a c e the i n t e g r a l i n E q u a t i o n 18, which r e p r e s e n t s a H e l m h o l t z energy o f c o m p r e s s i o n , by I I x.x.g.. Γ P(V ,T;a..,b,c..,...) dV m J J ι J ij m ij ij π A = - -±-J 1 comp y y x.x.g.. r Jr J J

J

1

(21)

1

where the weight f a c t o r s g a r e the c o n t a c t v a l u e s o f the ( i , j ) - r a d i a l d i s t r i b u t i o n f u n c t i o n o f a r i g i d sphere m i x t u r e . The average covolume parameter b i s a g a i n o b t a i n e d by E q u a t i o n 12. A c c o r d i n g t o s c a l e d p a r t i c l e theory (17), the v a l u e s o f the g.. a r e f u n c t i o n s o f c o m p o s i t i o n and d e n s i t y a c c o r d i n g t o ~' 1

g..

= ^ ( ( C . .+1 ) -C . J 3

(22)

3

2 R

ξ with

i1

X

C.. = -r-—'" 1+R.. 1 J

1

+ X

2 21 R

(23)

r— χ,+XoRoi

Ξ

1

ij

2 21

For a more d e t a i l e d e x p l a n a t i o n o f the t h e o r e t i c a l background of the m o d i f i e d mean d e n s i t y a p p r o x i m a t i o n see ( 1 8 ) . F u r t h e r m o r e , two o t h e r c o n t r i b u t i o n s to t h i s symposium a r e a l s o c o n c e r n e d w i t h m o d i f i e d mean d e n s i t y a p p r o x i m a t i o n (19, 20). A t h i g h d e n s i t i e s t h i s t h e o r y y i e l d s the same s t a t i s t i c a l w e i g h t f a c t o r s f o r the a t t r a c t i v e i n t e r a c t i o n as E q u a t i o n 13, but - i n con t r a s t t o t h i s e q u a t i o n - i t a l s o l e a d s t o a c o r r e c t low p r e s s u r e l i m i t (18). T h e r e f o r e t h i s m o d i f i e d mean d e n s i t y a p p r o x i m a t i o n ( E q u a t i o n 21) i s i n t e r e s t i n g f o r m i x t u r e s of m o l e c u l e s o f v e r y d i f f e r e n t s i z e s , i n which a l s o a l a r g e r a n g e o f d e n s i t i e s must be a c c o u n t e d f o r . 8

lim

ζ-0 ,. lim ξ^Ο

g

8

22

g

= 1

11

12 11

8

ξ-1 8 Λ

= 1 g

lim

22

= R

11

11

s

? 9

L L

f =

8

=

?1

(24)

12

lim

2

Z l

2 h

υ

+

R

Ί — 1 J

2

=

s

1 0

1 2

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

378

E Q U A T I O N S O F STATE: T H E O R I E S A N D

APPLICATIONS

In t h i s case the s u r f a c e r a t i o s s have been c a l c u l a t e d a c c o r d i n g t o E d u l j e e and S a n d l e r ( 9 , 21_, 22). I t might be argued t h a t the a v e r a g i n g p r o c e d u r e E q u a t i o n 21 s h o u l d be w r i t t e n i n terms of p r e s s u r e s r a t h e r than i n terms of Helmholtz e n e r g i e s . I t would be d i f f i c u l t , however, t o p e r f o r m a f o r m a l i n t e g r a t i o n on the p r e s s u r e e q u a t i o n t o o b t a i n A, G, o r μ, whereas the d i f f e r e n t i a t i o n o f the H e l m h o l t z energy e q u a t i o n t o o b t a i n the p r e s s u r e i s s i m p l e . F i n a l l y we want to p o i n t out t h a t the m o d i f i e d mean d e n s i t y a p p r o x i m a t i o n cannot be a p p l i e d to c u b i c e q u a t i o n s of s t a t e w i t h o u t l o o s i n g the " c u b i c c h a r a c t e r " of the m i x t u r e e q u a t i o n o f s t a t e . Results In a l l the f o l l o w i n g c a l c u l a t i o n s two a d j u s t a b l e b i n a r y p a r a m e t e r s , a and b^ , have been u s e d . Because o f the g r e a t s e n s i t i v i t y o f phase e q u i l i b r i u m c a l c u l a t i o n s f o r dense m i x t u r e s , one a d j u s t a b l e parameter i s u s u a l l y not s u f f i c i e n t . I t must be n o t e d , however, t h a t the ad j u s t a b l e parameters a r e d e t e r m i n e d o n l y once f o r a b i n a r y m i x t u r e ; they are c o n s i d e r e d independent o f temperature, d e n s i t y , and c o m p o s i t i o n . Furthermore, the same parameters have always been used even f o r the c a l c u l a t i o n o f d i f f e r e n t thermodynamic p r o p e r t i e s . I f n o t s t a t e d o t h e r w i s e , o n e - f l u i d t h e o r y has been u s e d .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

n

The hydrogen/methane system i s an e s p e c i a l l y f a s c i n a t i n g system f o r t e s t i n g c o m p u t a t i o n a l methods: The e x p e r i m e n t a l d a t a (23) cover a wide range o f r e d u c e d t e m p e r a t u r e s , because o f the h i g h p r e s s u r e s b o t h c o e x i s t i n g phases have h i g h d e n s i t i e s and t h e r e f o r e l a r g e d e v i a t i o n s from i d e a l i t y , and a t low temperature quantum e f f e c t s become i m p o r t a n t . F i g u r e 2 shows t h r e e i s o t h e r m s o f the / C l ^ system. The b i n a r y p a r a meters o f the e q u a t i o n s o f s t a t e have been d e t e r m i n e d from the m i d d l e (130 K) i s o t h e r m o n l y . I t i s e v i d e n t t h a t E q u a t i o n 2 l e a d s t o a r a t h e r good r e p r e s e n t a t i o n o f the e x p e r i m e n t a l m a t e r i a l , whereas the RK equa t i o n i s good a t low p r e s s u r e s , but has d i f f i c u l t i e s a t low t e m p e r a t u r e s and h i g h p r e s s u r e s . The 100 Κ i s o t h e r m has a l s o been c a l c u l a t e d w i t h the m o d i f i e d mean d e n s i t y a p p r o x i m a t i o n ( u s i n g E q u a t i o n 2 ) . I t s r e s u l t s agree w e l l w i t h the e x p e r i m e n t a l d a t a i n the c r i t i c a l r e g i o n , where the o n e - f l u i d t h e o r y shows some d e v i a t i o n s ; at low p r e s s u r e s , t h e r e s u l t s of the mean d e n s i t y a p p r o x i m a t i o n and o f the o n e - f l u i d t h e o r y c o i n c i d e . We have used E q u a t i o n 2 to c a l c u l a t e the c r i t i c a l l i n e of the H /CH system. The r e s u l t s a r e shown i n F i g u r e 3. E v i d e n t l y t h i s equa t i o n o f s t a t e , t o g e t h e r w i t h the e x t e n s i o n s and m i x i n g r u l e s d e s c r i b e d above, g i v e s r e l i a b l e e x t r a p o l a t i o n s o f phase envelopes t o d i f f e r e n t temperatures. As a s e v e r e t e s t o f the c o n s i s t e n c y o f our approach we have used E q u a t i o n 2 i n c o n n e c t i o n w i t h 19 to p r e d i c t the s o l i d - l i q u i d - g a s t h r e e phase e q u i l i b r i u m , which has been o b s e r v e d i n t h i s system a t low tempe r a t u r e s ( 2 3 ) . I t t u r n s out t h a t the s l g t h r e e - p h a s e l i n e i s indeed r e p r e s e n t e d s u c c e s s f u l l y ; the d e v i a t i o n s a t h i g h p r e s s u r e a r e m o s t l y due to the i n a c c u r a c y of the e x p e r i m e n t a l v o l u m e t r i c d a t a of s o l i d methane ( 2 4 ) . Even so the temperature minimum o f the s l g l i n e , which has been o b s e r v e d e x p e r i m e n t a l l y , i s r e p r o d u c e d by the c a l c u l a t i o n . A g a i n we must p o i n t o u t t h a t the parameters o f the e q u a t i o n o f s t a t e have n o t been f i t t e d to the s o l i d - f l u i d e q u i l i b r i u m , but a r e the same as f o r the VLE c a l c u l a t i o n mentioned above. Furthermore we have t r i e d t o c a l c u l a t e v o l u m e t r i c d a t a o f homoge neous H 2 / C H m i x t u r e s from the e q u a t i o n s o f s t a t e 1 and 2, a g a i n u s i n g 2

4

4

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

18.

DEITERS

Fluid-Fluid and Solid-Fluid

Equilibria

379

Figure 1. C a l c u l a t i o n of the Gibbs energy of a pure s u b s t a n c e . Key: o, c r i t i c a l p o i n t ; Δ, t r i p l e p o i n t ; , sublimation, melting, or vapour p r e s s u r e c u r v e ; — i n t e g r a t i o n of the f l u i d e q u a t i o n of s t a t e ; — » , i n t e g r a t i o n o f the s o l i d e q u a t i o n o f s t a t e .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

380

E Q U A T I O N S O F STATE: T H E O R I E S A N D

APPLICATIONS

the same s e t o f parameters as b e f o r e . I t t u r n s out t h a t b o t h e q u a t i o n s r e p r e s e n t the e x p e r i m e n t a l d a t a (25) q u i t e w e l l f o r m i x t u r e s r i c h i n hydrogen, but t h a t the RK e q u a t i o n shows l a r g e d e v i a t i o n s a t o t h e r comp o s i t i o n s . A p p a r e n t l y the c u b i c e q u a t i o n of s t a t e cannot f i t phase e q u i l i b r i u m d a t a and v o l u m e t r i c d a t a s i m u l t a n e o u s l y , whereas the o t h e r e q u a t i o n of s t a t e does not s u f f e r from t h i s l i m i t a t i o n ( F i g u r e 4 ) . E v i d e n t l y the RK e q u a t i o n does not p r o p e r l y r e p r e s e n t the molar volumes of l i q u i d methane. Of c o u r s e i t i s p o s s i b l e t o f i n d a parameter s e t which would l e a d t o a b e t t e r r e p r e s e n t a t i o n of the methane PVT d a t a , but then the agreement o f c a l c u l a t e d and e x p e r i m e n t a l phase e q u i l i b r i u m d a t a would d e t e r i o r a t e . We c o n s i d e r t h i s an i m p o r t a n t d i s a d v a n t a g e of the RK e q u a t i o n , because the aim of t h i s work i s n o t the c o r r e l a t i o n o f s i n g l e thermodynamic p r o p e r t i e s , but the s i m u l t a n e o u s d e s c r i p t i o n of s e v e r a l p r o p e r t i e s w i t h the same c o m p u t a t i o n a l method. As another example o f a phase e q u i l i b r i u m c a l c u l a t i o n based on the m o d i f i e d mean d e n s i t y a p p r o x i m a t i o n , we p r e s e n t the P-x diagram o f the hydrogen/carbon monoxide system a t 70K ( F i g 5) In view o f the h i g h p r e s s u r e s i n v o l v e d , the o n e - f l u i d t h e o r y as w e l l as the mean d e n s i t y approx i m a t i o n can be s a i d t o r e p r e s e n t the e x p e r i m e n t a l d a t a s u c c e s s f u l l y , but i t i s e v i d e n t t h a t i n t h i s case the mean d e n s i t y a p p r o x i m a t i o n i s s u p e r i o r . A t "low" p r e s s u r e , however, both methods agree v e r y w e l l . Because o f i t s h i g h e r CPU time r e q u i r e m e n t s the m o d i f i e d mean d e n s i t y a p p r o x i m a t i o n s h o u l d be a p p l i e d t o h i g h d e n s i t y phase e q u i l i b r i a o n l y . In p r i n c i p a l , t h e r e a r e no d i f f e r e n c e s between l i q u i d s and g a s e s , and our f o r m u l a t i o n s h o u l d a p p l y to s u b l i m a t i o n and m e l t i n g diagrams a l i k e . T h i s i s demonstrated i n F i g u r e 6, which c o n t a i n s an i s o t h e r m a l m e l t i n g diagram o f the t e t r a f l u o r o m e t h a n e / k r y p t o n system. The agreement between the e x p e r i m e n t a l d a t a (27) and the c a l c u l a t i o n ( u s i n g E q u a t i o n 2) i s q u i t e good. The s t r o n g c u r v a t u r e o f the e q u i l i b r i u m c u r v e i n d i c a t e s t h a t t h i s system i s f a r from i d e a l i t y . The s y s t e m a t i c d e v i a t i o n s of the c a l c u l a t e d l i n e a t low p r e s s u r e s a r e p r o b a b l y due t o the format i o n o f a s o l i d s o l u t i o n i n s t e a d o f a pure c r y s t a l l i n e k r y p t o n phase (21). The e q u i l i b r i u m l i n e o r i g i n a t i n g a t the m e l t i n g p o i n t o f C F has been c a l c u l a t e d from the same parameter s e t as the o t h e r e q u i l i b r i u m l i n e . U n f o r t u n a t e l y no v o l u m e t r i c d a t a o f s o l i d CF f o r the r e q u i r e d temperature and p r e s s u r e range a r e r e p o r t e d i n l i t e r a t u r e , so t h a t i t s s o l i d molar volume had t o be f i t t e d t o some b i n a r y d a t a . The b i n a r y system ethene/naphthalene shows an i n t e r e s t i n g i n t e r a c t i o n between s o l i d - f l u i d and f l u i d - f l u i d e q u i l i b r i a . A t moderate temperatures t h e r e i s a s o l i d - s u p e r c r i t i c a l f l u i d e q u i l i b r i u m ( F i g u r e 7 ) , t o which we have f i t t e d the b i n a r y parameters o f E q u a t i o n 2/15. The agreement between the e x p e r i m e n t a l d a t a and the c a l c u l a t e d e q u i l i b r i u m s t a t e s i s q u i t e good. R e v e r s i n g the p r o c e d u r e d e s c r i b e d f o r the h y d r o gen/methane system, we have now used the parameters e s t a b l i s h e d from the s o l i d - f l u i d e q u i l i b r i u m t o p r e d i c t the v a p o u r - l i q u i d e q u i l i b r i u m . The r e s u l t i n g phase d i a g r a m i s shown i n F i g u r e 8: The c r i t i c a l l i n e o f the m i x t u r e , which would n o r m a l l y j o i n the c r i t i c a l p o i n t s o f ethene and n a p h t h a l e n e , i s i n t e r r u p t e d by the s l g t h r e e - p h a s e l i n e . The s o l i d s u p e r c r i t i c a l f l u i d phase e q u i l i b r i a c o n t a i n e d i n F i g u r e 7 b e l o n g to the P-T r e g i o n marked " s g " . The s e c t i o n o f the c r i t i c a l l i n e , which i s o u t s i d e the s l g a r e a , i s p r e d i c t e d a c c u r a t e l y by E q u a t i o n 2. F u r t h e r more, the s l g l i n e o r i g i n a t i n g a t the t r i p l e p o i n t o f naphthalene i s r e p r e s e n t e d q u i t e w e l l ; the d e v i a t i o n s from the e x p e r i m e n t a l d a t a (28, 29) , which a r e p r o b a b l y due to the l a c k o f r e l i a b l e c o m p r e s s i b i l i t y d a t a o f n a p h t h a l e n e near i t s m e l t i n g p o i n t , are never l a r g e r than 4 K. 4

4

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18.

DEITERS

Fluid-Fluid

and Solid-Fluid

Equilibria

381

P/MPa 140 120 100 80 60 40 20

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

60

80 ~ 100

120

140

160

180 T/K

F i g u r e 3. P-T phase diagram of the hydrogen/methane system. Key: ·, A , c r i t i c a l p o i n t and t r i p l e p o i n t of methane; , methane m e l t i n g o r vapour p r e s s u r e l i n e ; o, e x p e r i m e n t a l b i n a r y c r i t i c a l p o i n t s ( 2 3 ) ; +, exp. s l g t h r e e - p h a s e s t a t e s ; , calculated c r i t i c a l l i n e ( w i t h E q u a t i o n 2/14); , c a l c u l a t e d three-phase line. Reproduced w i t h p e r m i s s i o n from ( 3 2 ) . C o p y r i g h t 1985, E l s e v i e r . V m 3

cm mol

-1

55 50 45 40 I 35| 30

0.0

0.2

0.4 0.6 x(H )

0.8

2

F i g u r e 4. V o l u m e t r i c p r o p e r t i e s of the hydrogen/methane system. Key: •, e x p e r i m e n t a l d a t a ( 2 5 ) ; , c a l c u l a t e d w i t h E q u a t i o n 2/14; , c a l c u l a t e d w i t h the Redlich-Kwong e q u a t i o n .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

382

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

V

0.0

0.2

0.4

0.6

0.8

x(H ) 2

F i g u r e 4. V o l u m e t r i c p r o p e r t i e s o f t h e hydrogen/methane system. K e y : B , e x p e r i m e n t a l d a t a ( 2 5 ) ; — , c a l c u l a t e d w i t h E q u a t i o n 2/14; , c a l c u l a t e d w i t h t h e Redlich-Kwong e q u a t i o n .

Ρ ΜΡα 60

0.0

0.2

0.4

0.6

0.8

1.0

x(H ) 2

F i g u r e 5. P-x diagram o f the h y d r o g e n / c a r b o n monoxide system. Key: Δ, exp. data ( 2 6 ) ; , c a l c u l a t e d w i t h E q u a t i o n 2/14 u s i n g onef l u i d theory; , u s i n g m o d i f i e d mean d e n s i t y a p p r o x i m a t i o n .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

18.

DEITERS

Fluid-Fluid

and Solid-Fluid

0.0

0.2

0.4

Equilibria

0.6

0.8

383

1

x(CF ) 4

F i g u r e 6. I s o t h e r m a l m e l t i n g diagram o f the CF / k r y p t o n system. Key: Α, Δ, e x p e r i m e n t a l d a t a ( 2 7 ) ; , c a l c u l a t e d w i t h E q u a t i o n 2. U

0.00

0.02

0.04

0.06

0.08 *(C

1

0

H ) 8

F i g u r e 7. I s o t h e r m a l s o l i d - s u p e r c r i t i c a l f l u i d phase e q u i l i b r i a of the ethene/naphthalene system. Key: Δ, V, exp. d a t a (28, 29) ; , c a l c u l a t e d w i t h E q u a t i o n 2/15.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

384

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Oh^ 250

· 300

* 350

J

/

K

F i g u r e 8. Phase diagram of the ethene/naphthalene system. Key: •, t r i p l e p o i n t o f n a p h t h a l e n e ; ·, c r i t i c a l p o i n t o f ethene; , m e l t i n g o r vapour p r e s s u r e l i n e s ; o, exp. b i n a r y c r i t i c a l p o i n t ; + , exp. s l g t h r e e - p h a s e s t a t e ; , calc. c r i t i c a l line (Equation 2/15); , c a l c . s l g three-phase l i n e .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18.

DEITERS

Fluid-Fluid

and Solid-Fluid

Equilibria

385

F i n a l l y we want to demonstrate t h a t i t i s p o s s i b l e to p r e d i c t v a p o u r - l i q u i d e q u i l i b r i a from 1 i q u i d - l i q u i d phase s e p a r a t i o n s even i n d i f f i c u l t c a s e s . In many t e t r a f l u o r o m e t h a n e / a l k a n e m i x t u r e s a l i q u i d l i q u i d phase s e p a r a t i o n o c c u r s , where the c r i t i c a l l i n e has a s h a l l o w temperature minimum. F i g u r e 9 shows the c r i t i c a l lines (calculated w i t h E q u a t i o n 2) of the systems CF / propane, /butane, and C F / p e n tane, t o g e t h e r w i t h some e x p e r i m e n t a l d a t a (30, 3 1 ) . The agreement i s q u i t e good. The CF / a l k a n e systems a r e i n t e r e s t i n g from a p h a s e - t h e o r e t i c a l p o i n t of view: In the CF /propane system the v a p o u r - l i q u i d and l i q u i d - l i q u i d c r i t i c a l c u r v e s a r e s e p a r a t e ; i n the CY /pentane system a c o n t i n u o u s t r a n s i t i o n between v a p o u r - l i q u i d and l i q u i d - l i q u i d e q u i l i b r i a o c c u r s . The b e h a v i o u r o f the C F / b u t a n e system i s s i m i l a r to t h a t of C F / p r o p a n e , but h e r e tern the c r i t i c a l e n d p o i n t o c c u r s a t a tempera t u r e above the c r i t i c a l temperature o f CF . T h i s i s perhaps u n u s u a l , but i n agreement w i t h phase t h e o r y . An i s o t h e r m a l P-x diagram o f the C F / b u t a n e system taken a t a temperature s l i g h t l y above the temperature minimum ( F i g u r e 10) c o n t a i n s t h r e e c r i t i c a l p o i n t s : one f o r the v a p o u r l i q u i d e q u i l i b r i u m , two f o r the l i q u i d - l i q u i d e q u i l i b r i a . Having f i t t e d the b i n a r y parameters to the h i g h p r e s s u r e l i q u i d - l i q u i d e q u i l i b r i u m , we f i n d t h a t E q u a t i o n 2 g e n e r a t e s the lower l i q u i d - l i q u i d and the v a p o u r l i q u i d e q u i l i b r i a i n good agreement w i t h the e x p e r i m e n t a l d a t a ( 3 0 ) , whereas the r e s u l t s o f the RK e q u a t i o n a r e o n l y q u a l i t a t i v e l y c o r r e c t . 4

4

4

4

k

4

4

4

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

4

We c o n c l u d e t h a t the non-cubic e q u a t i o n of s t a t e 2, e v e n t u a l l y enhanced by E q u a t i o n s 14 and 15, does not o n l y p e r m i t s u c c e s s f u l c o r r e l a t i o n s o f f l u i d - f l u i d e q u i l i b r i a , but a l s o p r e d i c t i o n s o f f l u i d - f l u i d e q u i l i b r i a o f a d i f f e r e n t t y p e , v o l u m e t r i c d a t a , and s o l i d - f l u i d e q u i l i b r i a i n a wide temperature and p r e s s u r e range. The Redlich-Kwong e q u a t i o n performs w e l l a t low d e n s i t i e s , but f a i l s a t h i g h p r e s s u r e s and a t p r e d i c t i o n s o f d e n s i t i e s . In some c r y o g e n i c phase e q u i l i b r i a a t h i g h p r e s s u r e s the r e s u l t s o f the c a l c u l a t i o n can be s i g n i f i c a n t l y im proved, i f the o n e - f l u i d t h e o r y i s r e p l a c e d by the m o d i f i e d mean den s i t y approximation. Symbols a A b c g G h k L m η Ν Ρ q

partition function expansion c o e f f i c i e n t u n i v e r s a l gas c o n s t a n t diameter r a t i o effective surface temperature volume molar f r a c t i o n reduced w a v e l e n g t h isothermal c o m p r e s s i b i l i t y thermal de B r o g l i e w a v e l e n g t h chemical p o t e n t i a l

a t t r a c t i o n parameter H e l m h o l t z energy covolume parameter a n i s o t r o p y parameter c o n t a c t v a l u e o f RDF Gibbs energy Planck constant Boltzmann c o n s t a n t effective c e l l size m o l e c u l a r mass amount o f s u b s t a n c e number o f m o l e c u l e s pressure surface f r a c t i o n

Subscripts

Superscripts

i component i n d e x m molar p r o p e r t y rep r e p u l s i o n att attraction f f r e e volume

° s f 1

,

11

p e r f e c t gas s t a t e solid fluid phase i n d i c a t o r

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

386

200

250

300

350

400

450

F i g u r e 9. Phase diagram of some CF / a l k a n e systems. The c u r v e s are marked w i t h t h e carbon number n. Key: ·, pure s u b s t a n c e c r i t . point; , vapour p r e s s u r e l i n e ; o, + , exp. b i n a r y c r i t . p o i n t , (30, 31, r e s p . ) ; , c a l c u l a t e d c r i t i c a l l i n e s ( E q u a t i o n 2 ) . For the CF^ /pentane system, the c r i t i c a l l i n e o r i g i n a t i n g a t the CF^ c r i t i c a l p o i n t has been o m i t t e d .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

18.

DEITERS

Fluid-Fluid

0

and Solid-Fluid

0.2

Equilibria

0.6

0Â

387

0.8

1

xlCFJ F i g u r e 10. P-x diagram o f t h e CF^ /butane system ( l o g a r i t h m i c s c a l e ) . o, exp. d a t a ( 3 0 ) ; , c a l c u l a t e d w i t h E q u a t i o n 2; c a l c . with the Redlich-Kwong e q u a t i o n ; t h r e e - p h a s e s t a t e . Reproduced w i t h p e r m i s s i o n from ( 1 0 ) . C o p y r i g h t 1984, Ver l a g Chemie, Weinheim.

Acknowledgment The a u t h o r wishes t o thank P r o f . Dr. G.M. S c h n e i d e r f o r f r i e n d l y s u p p o r t of t h i s work. F i n a n c i a l s u p p o r t by the Deutsche F o r s c h u n g s g e m e i n s c h a f t i s g r a t e f u l l y acknowledged.

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Redlich, O.; Kwong, J. N. S. Chem. Reviews 1949, 44, 233. Deiters, U. K. Chem. Eng. Sci. 1981, 36, 1139, 1147. Deiters, U. K. Fluid Phase Equil. 1983, 10, 173. Singh, Y.; Sinha, S. K. Phys. Rep. 1981, 79, 213. Deiters, U. K. Fluid Phase Equil. 1983, 13, 109. Rowlinson, J. S.; Swinton, F. L. "Liquids and Liquid Mixtures"; Butterworths: London, 1982; p. 279 ff. Mansoori, G. Α., this symposium. Deiters, U. K. Chem. Eng. Sci. 1982, 37, 855. Deiters, U. K. Fluid Phase Equil. 1983, 12, 193. Deiters, U. K.; Swaid, I. Ber. Bunsenges. Phys. Chemie 1984, 88, 791.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch018

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EQUATIONS OF STATE: THEORIES AND APPLICATIONS

11. Mansoori, G. Α.; Carnahan, Ν. F.; Starling, Κ. Ε.; Leland, T. W. J. Chem. Phys. 1971, 54, 1523. 12. Haase, R. "Thermodynamik der Mischphasen"; Springer-Verlag: Berlin, 1956. 13. Deiters, U. K. Fluid Phase Equil. 1985, 19, 287. 14. Preston, G. T.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Develop. 1970, 9, 264. 15. Teller, M.; Knapp, H. Ber. Bunsenges. Phys. Chemie 1983, 87, 532. 16. Hoheisel, C.; Deiters, U. K.; Lucas, K. Mol. Phys. 1983, 49, 159. 17. Lebowitz, J. L.; Helfand, E.; Praestgaard, E. J. Chem. Phys. 1965, 43, 774. 18. Deiters, U. K. Habilitation Thesis, Ruhr-Universität, Bochum, FRG, 1985. 19. Ely, J. F., this symposium. 20. Li, M. H.; Chung, F. T. H.; So, C.-K.; Lee, L. L.; Starling, Κ. Ε., this symposium. 21. Eduljee, G. H. Fluid Phase Equil. 1983, 12, 190. 22. Sandler, S. I. Fluid Phase Equil. 1983, 12, 189. 23. Tsang, C. Y.; Clancy, P.; Calado, J. C. G.; Streett, W. B. Chem. Eng. Commun. 1980, 6, 365. 24. Breitling, S. M.; Jones, A. D.; Boyd, R.H. J. Chem. Phys. 1971, 54, 3959. 25. Machado, J. R. S.; Streett, W. B., unpublished data. 26. Tsang, C. Y.; Streett, W. B. Fluid Phase Equil. 1981, 6, 261. 27. Jeschke, P.; Schneider, G. M. J. Chem. Thermodynamics 1982, 14, 743. 28. Diepen, G. A. M.; Scheffer, F. E .C. J. Phys. Chem. 1953, 57, 575. 29. van Gunst, C. Α.; Scheffer, F. E .C.; Diepen, G. A .M. J. Phys. Chem. 1953, 57, 578. 30. Jeschke, P.; Schneider, G. M. J. Chem. Thermodynamics 1982, 14, 547. 31. Mukherjee, A. Ph.D. Thesis, University of London, 1978. 32. Deiters, U. K. Fluid Phase Equil. 1985, 20, 275. RECEIVED November

12, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

19 Optimal Temperature-Dependent Parameters for the Redlich-Kwong Equation of State Randall W. Morris and Edward A. Turek

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

Amoco Production Company, Tulsa, OK 74102

Redlich-Kwong "a" and "b" parameters for several light hydrocarbons, carbon dioxide, nitrogen, and hydrogen sulfide have been optimized at many temperatures using accurate correlations of PVT data available in the l i t erature. The regressed parameter values have been fit as functions of temperature for convenient use in cal culations. The accuracy of volumetric calculations using the new parameters is compared to that obtained using the original Redlich-Kwong equation and the Soave and Yarborough modifications. The new parameters sig nificantly improve the accuracy of volumetric calcula tions using the Redlich-Kwong equation. One a p p l i c a t i o n o f e q u a t i o n s o f s t a t e i n the p e t r o l e u m i n d u s t r y i s i n the n u m e r i c a l modeling o f p e t r o l e u m r e s e r v o i r s . S i n c e t h e s e mathe m a t i c a l models a r e v e r y complex, a r e l a t i v e l y s i m p l e e q u a t i o n o f s t a t e i s needed f o r computer s o l u t i o n o f the model t o be p r a c t i c a l . The Redlich-Kwong e q u a t i o n (1.) i s a common form o f e q u a t i o n o f s t a t e used i n such s i t u a t i o n s . The e q u a t i o n , RT

a

(1)

Ρ = V-b

5

T°* V(V+b)

r e l a t e s t h e p r e s s u r e , molar volume, and temperature o f a f l u i d u s i n g the two p a r a m e t e r s , " a " and "b". F o r a m i x t u r e , t h e v a l u e s o f t h e s e parameters a r e determined by combining t h e parameters o f t h e i n d i v i d u a l components. Thus, i n g e n e r a l , m i x t u r e c a l c u l a t i o n s w i l l y i e l d the b e s t r e s u l t s when t h e i n d i v i d u a l component parameters p r o v i d e t h e best f i t s t o t h e p r o p e r t i e s o f t h o s e components. A c c u r a t e c o r r e l a t i o n s o f PVT d a t a a r e a v a i l a b l e from IUPAC ( 2 - 4 ) f o r carbon d i o x i d e , n i t r o g e n , and methane, t h e N a t i o n a l Bureau o f Standards (5-8) f o r ethane, propane, i s o b u t a n e , and normal butane, and t h e Texas A&M U n i v e r s i t y Thermodynamics R e s e a r c h C e n t e r (9) f o r hydrogen s u l f i d e . We have used d a t a from t h e s e c o r r e l a t i o n s t o e v a l uate t h e a c c u r a c y o f the Redlich-Kwong e q u a t i o n o f s t a t e and t o

0097-6156/86/0300-0389$06.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

390

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

d e v e l o p new parameters. S i n c e the q u a l i t a t i v e r e s u l t s were the same f o r a l l s u b s t a n c e s s t u d i e d , a d e t a i l e d a n a l y s i s of the r e s u l t s w i l l be p r e s e n t e d o n l y f o r carbon d i o x i d e .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

Background In the o r i g i n a l development of the Redlich-Kwong e q u a t i o n of s t a t e , the parameters " a " and "b" were s p e c i f i e d to be c o n s t a n t f o r a g i v e n fluid. For a pure s u b s t a n c e , they were d e f i n e d i n terms of the c r i t i c a l temperature and c r i t i c a l p r e s s u r e of the s u b s t a n c e . These v a l u e s o f the parameters f o r carbon d i o x i d e are r e p r e s e n t e d by the h o r i z o n t a l l i n e s i n F i g u r e s 1 and 2, and w i l l be r e f e r r e d to as the " s t a n d a r d " Redlich-Kwong parameters i n the f o l l o w i n g d i s c u s s i o n . E r r o r s i n v o l u m e t r i c c a l c u l a t i o n s f o r C O 2 u s i n g these parameters a r e shown i n F i g u r e s 3 and 4. Note the l a r g e e r r o r s i n the c a l c u l a t i o n of l i q u i d volumes. In the Soave m o d i f i c a t i o n (10,11 ) of the Redlich-Kwong e q u a t i o n , the " a " parameter i s r e p l a c e d w i t h a f u n c t i o n of temperature which i s a d j u s t e d to f i t vapor p r e s s u r e s . The "b" parameter r e t a i n s i t s s t a n dard v a l u e . T h i s m o d i f i c a t i o n i s i n t e n d e d to improve phase e q u i l i brium c a l c u l a t i o n s and r e s u l t s i n l i t t l e improvement i n the v o l u metric c a l c u l a t i o n s . F i g u r e 1 shows the temperature dependence of the Soave " a " parameter c o n s i s t e n t w i t h E q u a t i o n 1. F i g u r e s 5 and 6 show the e r r o r s i n v o l u m e t r i c c a l c u l a t i o n s f o r C O 2 u s i n g the Soave parameters. Z u d k e v i t c h and J o f f e (_12) a l l o w e d both parameters to v a r y w i t h temperature below the c r i t i c a l temperature, d e t e r m i n i n g t h e i r v a l u e s from the p r e s s u r e , l i q u i d d e n s i t y , and f u g a c i t y c o e f f i c i e n t at s a t u r a t i o n f o r a substance at a g i v e n temperature. They used the v a l u e s of the parameters determined at the c r i t i c a l temperature f o r temperat u r e s above the c r i t i c a l . Yarborough ( O ) a p p l i e d t h i s method u s i n g g e n e r a l i z e d c o r r e l a t i o n s to c a l c u l a t e vapor p r e s s u r e , s a t u r a t e d l i q u i d d e n s i t y , and f u g a c i t y c o e f f i c i e n t . The r e s u l t i n g c o r r e l a t i o n e x p r e s s e s the e q u a t i o n of s t a t e parameters i n d i m e n s i o n l e s s form as f u n c t i o n s o n l y of reduced temperature and a c e n t r i c f a c t o r . V a l u e s of t h e s e parameters f o r C O 2 a r e i n c l u d e d i n F i g u r e s 1 and 2 f o r compari s o n to the s t a n d a r d and Soave parameters. F i g u r e s 7 and 8 show the e r r o r s i n v o l u m e t r i c c a l c u l a t i o n s f o r C O 2 u s i n g these parameters. Note t h a t the a c c u r a c y of the l i q u i d volume c a l c u l a t i o n s i s improved considerably. New

Parameters

While the Yarborough parameters o f f e r s u b s t a n t i a l improvement o v e r the s t a n d a r d and Soave parameters at temperatures below the c r i t i c a l t e m p e r a t u r e , a f u r t h e r improvement i s p o s s i b l e by a l l o w i n g the parame t e r s t o v a r y w i t h temperature above the c r i t i c a l a l s o . We have done t h i s f o r s e v e r a l s u b s t a n c e s , l i s t e d above, which a r e commonly encount e r e d at s u p e r c r i t i c a l temperatures. V a l u e s of the parameters were d e t e r m i n e d by o p t i m i z i n g the f i t of the Redlich-Kwong e q u a t i o n to single-phase volumetric data. We a l s o determined new v a l u e s of the parameters at s u b c r i t i c a l temperatures u s i n g both s i n g l e - p h a s e v o l u m e t r i c d a t a and s a t u r a t i o n d a t a . O p t i m a l v a l u e s of the parameters were determined a t many temperatures. For each temperature, 197 pressure-volume d a t a p o i n t s were

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

MORRIS A N D TUREK

Parameters for the Redlich-Kwong

T = T

360000·

Equation

Legend:

C

S t a n d a r d Redlich-Kwong _ Soave Redlich-Kwong

_

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

Yarborough

-100

0

100

200

300

400

500

600

700

800

Temperature, *F Figure

1.

C O 2 " a " Parameter f o r t h e Redlich-Kwong

Λ

Equation

= Τ

Legend: Standard Redlich-Kwong Yarborough New Parameters . -100

0

1

100

200

1 300

1

1

400

500

600

700

800

e

Temperature, F F i g u r e 2.

C O 2 "b

,f

Parameter f o r t h e Redlich-Kwong

Equation

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

10-H - 1 0 0

1 0

h 100

1

1

1

1

1

1

|

J

1

200

300

400

500

600

700

800

900

1 1000

1 1100

1 1200

Temperature, °F F i g u r e 3. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h S t a n d a r d Parameters

3

Volume, ft /lbmol F i g u r e 4. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h S t a n d a r d Parameters

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

MORRIS A N D TUREK

10-Η -100

1 0

h 100

Parameters for the Redlich-Kwong

1

1

1

1

1

200

300

400

500

600

1 700

r

800

Equation

1 900

1 1000

1 1100

1 1200

e

Temperature, F F i g u r e 5. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h Soave Parameters

F i g u r e 6. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h Soave Parameters

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

394

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

100

200

300

400

500

600

700

600

800

1000

1100

1200

Temperature, °F Figure 7 . E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h Yarborough Parameters

Γ

T=T

C

3

Volume, ft /lbmol F i g u r e 8 . E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h Yarborough Parameters

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

19.

Parameters for the Redlich-Kwong

MORRIS A N D TUREK

Equation

395

o b t a i n e d from t h e c o r r e l a t i o n s . The d i s t r i b u t i o n o f these p o i n t s was l i n e a r w i t h r e s p e c t t o l o g - l o g o f volume i n o r d e r t o g i v e t h e d e s i r e d d i s t r i b u t i o n o f l i q u i d and vapor volumes. Any o f t h e 197 p o i n t s t h a t f e l l i n t h e two-phase r e g i o n f o r t h e g i v e n temperature were e l i m i nated. F o r temperatures below t h e c r i t i c a l temperature, the s a t u r a t i o n p r e s s u r e and s a t u r a t e d l i q u i d and vapor volumes were a l s o used r e s u l t i n g i n a maximum o f 200 d a t a p o i n t s a t a g i v e n temperature. V o l u m e t r i c d a t a were weighted a c c o r d i n g t o t h e d e v i a t i o n o f t h e Ζ f a c t o r from 1, s i n c e t h e f i t t o d a t a e x h i b i t i n g i d e a l gas b e h a v i o r was found t o be i n s e n s i t i v e t o t h e v a l u e s o f t h e e q u a t i o n o f s t a t e parameters. W e i g h t i n g s o f t h e s a t u r a t i o n p r o p e r t y d e v i a t i o n s were a d j u s t e d by t r i a L t o g i v e a r e a s o n a b l e b a l a n c e w i t h t h e v o l u m e t r i c data. The f i n a l o b j e c t i v e f u n c t i o n minimized a t each temperature was 2

F = Σ[(Ζ-1) Δν]

2

2

+ (16ΔΡ

) + (25Δν

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

sat where the Δ q u a n t i t i e s a r e f r a c t i o n a l

ΔΧ = (Χ

Λ

calc

Λ

)

2

+ (2Δν ^

sat,l

- X

)

2

(2)

sat,ν

deviations, i . e . , )/X exp exp

f o r a g i v e n q u a n t i t y X. Parameters were c o n s t r a i n e d d u r i n g t h e r e g r e s s i o n t o be c o n s i s t e n t w i t h t h e e x p e r i m e n t a l c r i t i c a l tempera t u r e , i . e . , parameter v a l u e s y i e l d i n g t h r e e r o o t s i n t h e e q u a t i o n o f s t a t e above t h e e x p e r i m e n t a l c r i t i c a l temperature o r o n l y one r o o t below t h e e x p e r i m e n t a l c r i t i c a l temperature were r e j e c t e d . The o p t i m i z e d parameter v a l u e s were f i t as f u n c t i o n s o f tempera t u r e f o r each s u b s t a n c e . Since petroleum r e s e r v o i r a p p l i c a t i o n s of e q u a t i o n s o f s t a t e a r e g e n e r a l l y a t a c o n s t a n t temperature, t h e p a r ameters must be determined o n l y once f o r a l a r g e number o f c a l c u l a tions. F o r t h i s r e a s o n , t h e f u n c t i o n a l forms used t o f i t t h e o p t i mized parameter v a l u e s c o u l d be made as complex as n e c e s s a r y f o r the d e s i r e d a c c u r a c y w i t h o u t a d v e r s e l y a f f e c t i n g computing t i m e s . I t s h o u l d a l s o be noted t h a t t h e method used t o d e t e r m i n e these parame t e r s r e s u l t s i n a d i s c o n t i n u i t y i n t h e temperature d e r i v a t i v e s o f t h e parameters a t t h e c r i t i c a l t e m p e r a t u r e . T h i s i s not a problem f o r c o n s t a n t temperature a p p l i c a t i o n s but may need t o be c o n s i d e r e d f o r c a l c u l a t i o n s r e q u i r i n g t h e temperature d e r i v a t i v e s o f t h e e q u a t i o n o f s t a t e p a r a m e t e r s , e.g., e n t h a l p y c a l c u l a t i o n s . The f i t s t o t h e o p t i mized parameters a r e d e s c r i b e d i n t h e Appendix. The new parameters f o r C O 2 a r e compared t o t h e s t a n d a r d v a l u e s , and t h e Soave and Yarborough v a l u e s i n F i g u r e s 1 and 2. E r r o r s i n v o l u m e t r i c c a l c u l a t i o n s u s i n g t h e new parameters a r e shown i n F i g u r e s 9 and 10. Note t h a t t h e l a r g e e r r o r s i n c a l c u l a t e d volumes a r e now c o n f i n e d t o t h e near p r o x i m i t y o f t h e c r i t i c a l p o i n t where they a r e u n a v o i d a b l e w i t h t h e Redlich-Kwong e q u a t i o n o f s t a t e . The i n c l u s i o n o f v o l u m e t r i c d a t a i n t h e parameter r e g r e s s i o n s a t s u b c r i t i c a l temperatures s t i l l a l l o w s a r e a s o n a b l e f i t t o t h e s a t u r a t i o n p r e s s u r e s as shown i n F i g u r e 11. S a t u r a t i o n p r e s s u r e s f o r C O 2 c a l c u l a t e d u s i n g t h e new parameters show an average e r r o r o f about 0.3%. Improvements i n t h e a c c u r a c y o f v o l u m e t r i c c a l c u l a t i o n s f o r t h e o t h e r s u b s t a n c e s s t u d i e d a r e s i m i l a r t o those o b t a i n e d f o r C 0 2 E r r o r s i n v o l u m e t r i c c a l c u l a t i o n s a t s e l e c t e d c o n d i t i o n s f o r a l l sub s t a n c e s s t u d i e d a r e shown i n T a b l e s I through V I I I a l o n g w i t h average e r r o r s over t h e e n t i r e range o f p r e s s u r e s and volumes c o n s i d e r e d .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

396

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

r

T=T

C

1

H ι m

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

\ in J 1 V \}f/ Ύ

Y - 1 0 0

0

1

j-

Î

1 • 100

„

=

! 200

1 300

I 400

1 500

1

1

600

700

1• BOO

E Z 3 0.2 to 0.5% WM\ 0.5 to 2.0% i H 2.0 to 5.0% H i 5.0 to 10.0% • ϋ Over 10.0%

! 000

1

1

1000

1100

1 1200

e

Temperature, F F i g u r e 9. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h New Parameters

3

Volume, ft /lbmol F i g u r e 10. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h New Parameters

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

19.

MORRIS A N D TUREK

Parameters for the Redlich-Kwong

Equation

Legend: S t a n d a r d Redlich-Kwong _ Suave Redlich-Kwong

\^

_

V

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

Yarborough

\

New j ) a r a meters

\^

\

-4H

0

Û

20

40

60

80

100

Temperature, *F F i g u r e 11. E r r o r i n Vapor P r e s s u r e C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

397

398

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Table I .

Error

i n C a l c u l a t e d Volume f o r Carbon

Dioxide

r

r

Standard Parameters

Soave Parameters

Yarborough Parameters

New Parameters

0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15 1.50 1.50 1.50 2.00 2.00 2.00

0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

16.68 11.66 5.91 0.10 12.92 3.89 0.59 0.16 2.79 0.89 3.38 1.28 0.75 2.67 4.88

11.49 8.65 4.90 0.42 16.46 4.39 0.49 11.47 4.50 0.70 4.14 5.34 0.71 2.91 5.03

0.17 1.20 3.33 0.61 14.70 6.09 0.93 0.39 4.84 1.00 3.64 0.22 0.78 2.70 3.95

0.27 1.72 3.96 1.07 2.13 4.52 0.27 0.38 3.84 0.41 1.07 0.37 0.34 0.91 0.03

3.15

4.10

2.19

1.02

Τ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

Percent

ρ

Average

Table I I .

Percent

Error in

C a l c u l a t e d Volume f o r N i t r o g e n

r

r

Standard Parameters

Soave Parameters

Yarborough Parameters

New Parameters

2.00 2.00 2.00 3.00 3.00 3.00

0.5 2.0 10.0 0.5 2.0 10.0

0.32 1.06 2.78 0.30 1.10 3.83

0.26 1.20 1.16 0.20 0.71 0.79

0.32 1.08 2.95 0.30 1.12 3.92

0.29 0.72 0.56 0.04 0.09 0.14

1.85

0.88

1.91

0.20

Τ

ρ

Average

Table I I I .

Percent

Error in

C a l c u l a t e d Volume f o r Methane

r

r

Standard Parameters

Soave Parameters

Yarborough Parameters

New Parameters

1.15 1.15 1.15 1.50 1.50 1.50 2.00 2.00 2.00 3.00 3.00

0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0

0.35 7.56 0.38 0.23 0.31 0.63 0.21 0.36 0.62 0.21 0.70

0.27 8.55 0.50 0.08 1.88 1.87 0.22 1.35 2.39 0.20 0.80

0.29 7.08 0.37 0.21 0.30 0.04 0.20 0.38 1.01 0.21 0.73

1.01 0.63 0.22 0.65 1.62 0.81 0.28 0.56 0.21 0.02 0.01

1.12

1.72

1.11

0.75

Τ

ρ

Average

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

19.

MORRIS A N D TUREK

T a b l e IV. τI

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

r 0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15 1.50 1.50 1.50

ρ r 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

Average

T a b l e V. τ 1

r 0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15 1.50 1.50 1.50

ρ r 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

Average

Table VI. τ 1

r 0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15

ρ r r 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

Average

Parameters for the Redlich-Kwong

Percent

Equation

399

E r r o r i n C a l c u l a t e d Volume f o r Ethane

Standard Parameters 11.11 6.83 1.96 0.17 11.24 1.44 0.46 2.90 1.10 0.57 1.67 0.71

Soave Parameters 9.17 5.74 1.60 0.06 12.73 1.65 0.05 8.58 1.87 0.29 2.58 2.81

Yarborough Parameters 1.53 0.76 3.82 0.33 11.66 2.04 0.56 2.73 1.66 0.61 1.77 0.30

New Parameters 0.31 2.34 5.13 0.59 2.48 4.87 0.07 0.40 3.95 0.49 1.13 0.61

2.45

3.14

1.53

1.03

Percent

E r r o r i n C a l c u l a t e d Volume f o r Propane

Standard Parameters 13.72 8.88 3.32 0.29 12.01 1.98 0.54 1.12 1.24 0.49 2.85 1.56

Soave Parameters 10.37 6.97 2.69 0.06 14.37 2.31 0.22 9.40 2.40 0.70 2.83 3.27

Yarborough Parameters 1.33 0.78 3.77 0.59 13.09 3.28 0.74 0.98 2.46 0.55 3.01 0.66

New Parameters 0.41 2.31 5.08 0.68 2.93 4.98 0.17 0.60 2.62 0.31 1.72 1.03

3.65

4.24

2.00

1.33

Percent

E r r o r i n C a l c u l a t e d Volume f o r Isobutane

Standard Parameters 14.88 9.65 3.55 0.60 11.81 1.85 0.91 0.11 0.91

Soave Parameters 10.68 7.25 2.76 0.17 14.69 2.26 0.00 9.67 2.31

Yarborough Parameters 0.77 1.29 4.34 0.99 13.21 3.53 1.17 0.27 2.48

New Parameters 0.37 2.32 5.26 0.24 4.11 4.10 0.76 0.67 0.55

4.42

4.82

2.20

1.64

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

400

Table

τ 1

VII.

Percent

r

0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15

r

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

VIII.

r

0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15 1.50 1.50 1.50

Parameters

Butane

Yarborough

New

Parameters

Parameters

0.37 2.19 5.00 0.66 3.75 4.35 0.37 0.69 1.22

5.07

5.58

2.44

1.59

Error

Volume

in Calculated

Standard

0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

Normal

1.31 0.68 3.69 0.78 14.60 4.69 0.96 0.32 3.58

Soave

Parameters

r

for

11.68 8.28 3.77 0.12 16.21 3.28 0.29 11.04 3.36

Percent

ρ

XT

Volume

16.28 10.92 4.65 0.34 13.06 2.84 0.68 0.50 1.85

Average

τ 1

Calculated Soave

Parameters

0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

Table

in

Standard

ρ

Jr

Error

Parameters

for

Hydrogen

Sulfide

Yarborough

New

Parameters

Parameters

9.75 6.28 2.34 0.28 11.22 2.20 0.08 6.42 2.14 0.24 0.19 1.46

8.05 5.32 2.02 0.49 12.58 2.40 0.40 11.85 2.86 0.58 3.89 4.85

0.68 0.98 3.29 0.14 11.53 2.69 0.16 6.23 2.59 0.27 0.27 1.80

0.39 2.06 4.37 0.63 3.23 2.79 0.07 0.68 3.64 0.11 0.20 2.84

2.87

3.79

1.79

0.93

Average

Conclusion The

accuracy

tion

of

state

parameters

to

temperature. in

the

made

volumetric

was

improved

vary By

without for

with of

the

s a c r i f i c i n g

developed

convenient

in use

calculations

the this

by

both

saturation new

in

using

significantly

temperature

u t i l i z i n g

development

parameters ture

of

above

data

parameters, f i t work

to

vapor

were

the

f i t

Redlich-Kwong

allowing and

as

well

these

the

"a"

below

the

as

The

functions

of

tempera

Nomenclature =

parameter

for

equation

1,

psia

f t

b

=

parameter

for

equation

1,

ft /lbmol

6

°R

0

,

5

/lbmol

data

were

optimized

calculations.

a

"b"

c r i t i c a l

volumetric

improvements

pressures. as

equa

and

2

3

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

19.

b*

= limiting

Parameters for the Redlich-Kwong

v a l u e f o r parameter

F

= objective function

ΔΡ

= fractional

ΔΤ

= (T/T ) - 1 c

Δν

= fractional

optimization

deviation

i n molar volume

= pressure, psia

P^

= reduced p r e s s u r e , P/P

calculation

pressure, psia

R

= gas c o n s t a n t , p s i a

Τ

= t e m p e r a t u r e , °R = critical

3

i n pressure c a l c u l a t i o n

= critical

401

b, f t / l b m o l

f o r parameter

P^

c

Equation

deviation

Ρ

T

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

MORRIS A N D TUREK

c

3

f t / l b m o l °R

t e m p e r a t u r e , °R

= r e d u c e d t e m p e r a t u r e , T/T^ V

= molar volume,

Ζ

= compressibility

3

ft /lbmol factor,

PV/RT

Subscripts c = c r i t i c a l point calc = calculated exp = experimental 1 = liquid r = reduced sat = saturation ν = vapor

Appendix The o p t i m i z e d Redlich-Kwong " a " and "b" parameters d e t e r m i n e d i n t h i s work were f i t as f u n c t i o n s o f t e m p e r a t u r e . The f u n c t i o n s used i n t h e s e f i t s a r e e m p i r i c a l and no p h y s i c a l s i g n i f i c a n c e s h o u l d be a t t a c h e d to t h e i r forms. S i n c e the f i t s were d e v e l o p e d s e p a r a t e l y over a p e r i o d o f time, some v a r i a t i o n i n f u n c t i o n a l form may be seen. For c o n v e n i e n c e , a l l the f u n c t i o n s a r e e x p r e s s e d i n terms o f t h e v a r i a b l e Δ Τ , d e f i n e d as

ΔΤ = (T/T ) - 1 c

i . e . , the d i m e n s i o n l e s s d i s p l a c e m e n t from the c r i t i c a l t e m p e r a t u r e . C r i t i c a l temperatures and ranges o f f i t s o f the parameters a r e shown i n T a b l e IX.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

402

T a b l e IX.

Critical

Substance Carbon D i o x i d e Nitrogen Methane Ethane Propane Isobutane Normal Butane Hydrogen S u l f i d e

Temperatures and Ranges o f Parameter

Critical

temperature, 547.578 227.16 343.00 549.594 665.730 734.130 765.288 672.354

Fits

Range of f i t , °F -63 t o 800 -75 to 650 -75 to 650 -200 t o 600 -200 to 600 -200 to 600 -200 to 600 -90 t o 600

R

The e q u a t i o n o f s t a t e parameters were f i t as d i m e n s i o n a l q u a n t i ties. Parameter " a i s i n p s i a f t °R° · / l b m o l . Parameter "b" i s in f t / l b m o l . S e p a r a t e f i t s were r e q u i r e d f o r s u b c r i t i c a l and s u p e r c r i t i c a l parameters i n o r d e r t o f i t the " s p i k e " i n the parameter v a l u e s a t the c r i t i c a l temperature. In o r d e r t o i n s u r e the p r o p e r b e h a v i o r at temperatures slightly above the c r i t i c a l , a check must be a p p l i e d t o the c a l c u l a t e d v a l u e s of the parameters. The q u a n t i t y "b*", the minimum a l l o w a b l e v a l u e o f the s u p e r c r i t i c a l "b" parameter i s c a l c u l a t e d from the " a " parameter at t h e same temperature w i t h the f o l l o w i n g e q u a t i o n : ff

6

5

2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

3

b* = b i a / [ T

1.5

(3)

2

1.8886163E-02 0.99974

bi b

where:

(l+b AT)]

c

2

I f ΔΤ i s l e s s than 0.004 o r i f the c a l c u l a t e d v a l u e o f the "b" param e t e r i s l e s s than "b*", "b*" must be used as the v a l u e o f the "b" p a r a m e t e r . T h i s check may be o m i t t e d f o r n i t r o g e n and methane s i n c e t h e i r f i t s do not extend down to t h e i r c r i t i c a l t e m p e r a t u r e s . The parameter f i t s f o r a l l s u b s t a n c e s s t u d i e d a r e g i v e n below.

Carbon D i o x i d e Parameters, a

= aoAT

b

= b AT

ao ai a a a as a6 a as

= = = = = = = = =

0

2

3

4

7

3

3

+ aiAT

2

+ biAT

2

AT = 0

+ a AT • a

+ a (a -AT)

a e

3

+ b AT + b

+ b (b -AT)

b 6

3

2

1.1323818E+05 -1.9341561E+05 -2.0363902E+05 -5.2494786E+07 5.2775697E+07 1.6289565E-04 -7.9739155E-05 4.5206914E-05 8.4787834E-09

2

4

4

5

5

+ a /(a -AT) 7

8

+ b /(bg-AT) 7

bo = -2.3973118E+00 bi

b b b b be by bg 2

3

4

5

= = = = = = = =

-1.8705039E+00 -7.0399570E-01 4.6945137E-01 -1.9580259E-01 3.3775759E-06 3.1643771E-01 1.3103149E-11 1.2868684E-09

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

MORRIS A N D T U R E K

19.

Parameters for the Redlich-Kwong

Equation

403

Carbon D i o x i d e Parameters, AT ^ 0 a

= aoAT + a i A T

b

= b AT

ao ai a2 a3 a as ae

= = = = = = =

3

3

+ biAT

0

4

2

+

2

a2AT

+

a

3

+ a (AT-a )

+ b AT + b

3

+ b (AT-b )

2

-1.2327970E+04 -3.9510605E+03 1.5753912E+04 2.9883762E+05 1.2638522E-02 -1.2669980E-01 -6.9947616E+00

b bi b b b b b 0

2

3

4

5

6

4

5

4

5

= 4.9137761E-02 = -1.5082045E-01 = 1.9277928E-01 = 4.0844576E-01 = 1.9737045E-04 = -3.8637696E-02 = -1.7936190E+00

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

N i t r o g e n and Methane Parameters = aoAT

3

b

= b AT

3

ao ai a2 a a as

= = = = = =

Nitrogen -1.6347508E+01 1.2161843E+03 -2.2892243E+04 1.2959794E+05 -4.8072688E+04 -7.5945730E-01

Methane 5.7694890E+03 -3.2063240E+04 3.0940267E+04 1.6229084E+05 -4.6636347E+00 1.1820010E-01

bo bi b b b b

= = = = = =

-2.8577341E-04 3.8658153E-03 -1.7061211E-02 5.2704092E-01 -1.2361065E-01 -1.2508721E+00

3.0568302E-03 -1.5876295E-02 2.2593112E-02 4.8294387E-01 -4.0107835E-04 8.0512599E-02

0

3

4

2

3

4

5

+ aiAT

2

+ biAT

2

& ΔΊ

a

+

+ a

3

+ a /(AT-as)

+ b AT + b

3

+ b /(AT-b )

2

2

4

4

5

Ethane, Propane, I s o b u t a n e , and Normal Butane Parameters, ΔΤ = 0 a

= aiAT

b

= biAT

ai a a a as a6 a7 as a9

= =

2

3

4

= = = = = =

5

5

+

a2AT

+ b AT 2

4

4

+

a AT

3

+ b AT

3

Ethane 0.0 0.0 6.0408944E+04 -3.3524012E+05 -2.4440624E+05 -5.3770245E+06 5.8143088E+06 4.5467943E-05 -9.6079783E-04

3

3

+ a AT

2

+ b AT

2

4

3 9

+ a AT + a

6

+ a ( a g- Δ Τ )

+ b AT + b

6

+ b (b -AT)

5

4

Propane 1.3807587E+05 0.0 0.0 -6.8075273E+05 -5.1847288E+05 -4.1533054E+08 4.1613493E+08 3.2664789E-05 -2.5461817E-05

5

7

7

Isobutane 3.6060539E+05 0.0 0.0 -8.9042232E+05 -7.4744128E+05 1.1845554E+06 3.6937924E+04 1.7804777E-04 -1.8159933E-01

b 9

8

Normal Butane 4.0124834E+05 0.0 0.0 -9.9038190E+05 -8.2677046E+05 1.2222223E+06 6.7757452E+04 1.1836024E-04 -1.3348188E-01

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

404

bi

b b b b b b? b bg 2

3

4

5

6

8

E Q U A T I O N S O F STATE: T H E O R I E S A N D

= = = = = = = =

0.0 0.0 -3.8152843E-•01 -7.6440305E-•01 -4.9300521E-•01 8.3401715E-•01 -2.5418456E--01 2.8828651E-•05 7.9453607E-•02

0.0 2.8826091E-01 0.0 -7.2744096E-01 -5.4966388E-01 -7.0736939E+02 7.0819594E+02 3.7042340E-05 -2.4660292E-05

APPLICATIONS

0.0 0.0 -5.0388863E-01 -1.2016256E+00 -7.5953251E-01 -4.9144994E+02 4.9249286E+02 5.0910144E-05 -5.1232547E-05

0.0 0.0 -5.0788946E-01 -1.2234509E+00 -7.8285224E-01 -4.8981625E+02 4.9086550E+02 6.0889312E-05 -5.1127477E-05

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

Ethane, Propane, I s o b u t a n e , and Normal Butane P a r a m e t e r s , AT = 0 a

= aiAT

b

= biAT

ai a a3 a as ae a as ag

= = = = = = = = =

Ethane 0.0 0.0 0.0 -2.9168739E+04 6.6705082E+04 4.6578248E+05 1.2357826E+00 4.3979460E-02 -3.2057354E+00

Propane 0.0 1.2332546E+06 0.0 -1.7413558E+06 1.2077341E+06 -5.4364187E+08 5.4425460E+08 1.5085713E-02 -1.3187677E-04

Isobutane 0.0 0.0 3.7513525E+06 -4.1274194E+06 1.4272780E+06 1.2027989E+06 2.5675094E+02 4.7010838E-02 -2.1024705E+00

Normal Butane 0.0 5.2547626E+06 0.0 -4.3472555E+06 2.4222201E+06 -3.2439274E+07 3.3361438E+07 1.3513673E-02 -3.6386297E-03

= = = = = = = =

2.3099616E-01 -2.9478506E-01 0.0 0.0 1.9013417E-01 6.4692946E-01 1.1226280E-03 1.5821016E-02 -1.0175663E+00

0.0 1.3021642E+00 0.0 -1.8915504E+00 1.4616927E+00 -1.1210554E+01 1.1849368E+01 9.8746681E-03 -6.5869382E-03

0.0 0.0 3.3538827E+00 -3.6241228E+00 1.4281118E+00 1.1243492E+00 1.3058917E-05 6.0297320E-02 -3.3703921E+00

0.0 1.1231303E+01 -7.6112989E+00 0.0 1.0382612E+00 1.1002003E+00 5.5989194E-04 3.5280578E-02 -1.7457633E+00

2

4

7

bi

b b b b be by b b 2

3

4

5

8

9

5

+ a AT

4

+ b AT

4

2

5

+ a AT

3

+ b AT

3

3

2

+ a AT

2

+ b AT

2

+ a AT + a6 + a ( a i + A T )

4

3

5

7

+ b AT + b

4

5

6

+ b (b +AT) 7

a 9

3

b 9

8

Hydrogen S u l f i d e P a r a m e t e r s , AT ύ 0 a

= aiAT

3

b

= biAT

3

ai a a a as ae a 2

3

4

7

= = = = = = =

+ a AT

2

+ b AT

2

2

+ a AT + a

+ a (a -AT)

a 7

4

+ b AT + b

+ b (b -AT)

b 7

4

3

2

2.7553548E+05 -1.5271335E+05 -2.2153932E+05 -7.8818355E+09 7.8822161E+09 3.7436399E-04 -9.8433542E-07

3

bi b b b b b b 2

3

4

5

6

7

= = = = = = =

5

5

6

6

7.5061764E-02 -2.6513067E-01 -2.7365804E-01 -7.1403337E+03 7.1407259E+03 2.4591169E-04 -1.4554375E-06

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

19.

Parameters for the Redlich-Kwong

MORRIS AND TUREK

Equation

405

Hydrogen S u l f i d e Parameters, AT ^ 0 a

=

aiAT

3

b

=

biAT

3

ai

=

+ a AT

2

+ b AT

2

2

+ a AT + a

4

+ a (a +AT)

+ b AT + b

4

+ b (b +AT)

3

2

3

8.9992497E+04

= = = a7 = 4

as ae

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

1. 2.

3.

4.

5.

6.

7.

8.

9.

10. 11. 12. 13.

6

2

3

4.1713206E+05 5.4872735E+02 3.4032689E-02 -1.1257065E+00

Literature

5

6

1.6468196E-01 bi = b = •-2.7842604E-01 1.7715814E-01 b = b = •-1.8220125E+02 b = 1.8259959E+02 b = 8.3692118E-03 b = •-9.1864978E-05

*2 = -2.1018269E+05 *3 = 9.3022318E+04 a

5

4

5

6

7

Cited

R e d l i c h , O.; Kwong, J. N. S. Chemical Reviews 1949, 44, 233-244 Angus, S.; Armstrong, B.; de Reuck, Κ. M. " I n t e r n a t i o n a l T h e r modynamic T a b l e s of the F l u i d S t a t e - 3 , Carbon D i o x i d e " ; P e r gamon: O x f o r d , 1976 Angus, S.; de Reuck, Κ. M.; Armstrong, B. " I n t e r n a t i o n a l Ther modynamic T a b l e s of the F l u i d S t a t e - 6 , N i t r o g e n " ; Pergamon: O x f o r d , 1979 Angus, S.; Armstrong, B.; de Reuck, Κ. M. " I n t e r n a t i o n a l Ther modynamic T a b l e s of the F l u i d S t a t e - 5 , Methane"; Pergamon: O x f o r d , 1978 Goodwin, R. D.; Roder, H. M.; S t r a t y , G. C. "Thermophysical P r o p e r t i e s of Ethane from 90 to 600 Κ a t P r e s s u r e s to 700 Bar"; N a t i o n a l Bureau of Standards T e c h n i c a l Note 684, N a t i o n a l Bureau of S t a n d a r d s : Washington, D.C., 1976 Goodwin, R. D; Haynes, W. M. " T h e r m o p h y s i c a l P r o p e r t i e s of P r o pane from 85 t o 700 Κ a t P r e s s u r e s to 70 MPa"; N a t i o n a l Bureau of Standards Monograph 170, N a t i o n a l Bureau of S t a n d a r d s : Wash i n g t o n , D.C., 1982 Goodwin, R. D; Haynes, W. M. " T h e r m o p h y s i c a l P r o p e r t i e s of I s o butane from 114 to 700 Κ at P r e s s u r e s to 70 MPa"; National Bureau of Standards T e c h n i c a l Note 1051, N a t i o n a l Bureau of Standards: Washington, D.C., 1982 Haynes, W. M.; Goodwin, R. D. " T h e r m o p h y s i c a l P r o p e r t i e s of Normal Butane from 135 to 700 Κ at P r e s s u r e s to 70 MPa"; N a t i o n a l Bureau of Standards Monograph 169, N a t i o n a l Bureau of Standards: Washington, D.C., 1982 Chen, S. S.; K r e g l e w s k i , A. " S e l e c t e d V a l u e s of P r o p e r t i e s of Chemical Compounds"; Thermodynamics Research C e n t e r Data P r o ject: Texas A&M U n i v e r s i t y , 1976; T a b l e s 14-2-(1.01)-j, 1 4 - 2 - i , 14-2-k, and 14-2-kb Soave, G. Chem. Eng. Sci. 1972, 27, 1197-1203 G r a b o s k i , M. S; Daubert, T. E. Ind. Eng. Chem. P r o c e s s D e s . Dev. 1978, 17, 448-454 Z u d k e v i t c h , D.; J o f f e , J . AIChE J . 1970, 16, 112-119 Yarborough, L. In " E q u a t i o n s of S t a t e i n E n g i n e e r i n g and R e s e a r c h " ; Chao, K. C.; Robinson, R. L., Eds.; ADVANCES IN CHEMISTRY SERIES No. 182, American Chemical S o c i e t y : Wash i n g t o n , D.C., 1979; pp. 385-439

RECEIVED November 8,

1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

20 Phase Behavior of Mixtures of San Andres Formation Oils with A c i d Gases Application of a Modified Redlich-Kwong Equation of State Joseph J . Chaback and Edward A. Turek

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

Amoco Production Company, Tulsa, OK 74102

A modified Redlich-Kwong equation of state, with appropriate parameter adjustments, was successful in reproducing the volumetric properties and complex phase equilibria exhibited by San Andres Formation oil-acid gas systems. These parameter adjustments are carried out by trial and error to match a portion of experimental PVT and phase equilibria data available for these oils and their mixtures with acid gases. The remaining data are used to test the selected parameters. The similarity of the C + fraction throughout the geologic formation allowed application of a common C + compositional distribution for all the San Andres oils. Thus, for any oil from the subject formation, only the overall fluid composition and reservoir temperature are required to calculate volumetric properties and phase behavior. 7

7

Development and e v a l u a t i o n programs f o r m i s c i b l e enhanced o i l r e c o v e r y (EOR) p r o c e s s e s r e q u i r e f l u i d p r o p e r t i e s f o r b o t h t h e r e s e r v o i r o i l and m i x t u r e s o f t h e d i s p l a c i n g gas and t h e r e s e r v o i r oil. Western Texas c o n t a i n s numerous c a n d i d a t e f i e l d s f o r c a r b o n d i o x i d e f l o o d i n g . I n most cases t h e s e f i e l d s have been p r e s s u r e d e p l e t e d and w a t e r f l o o d e d . Y e t t h e s e f i e l d s s t i l l r e p r e s e n t a subs t a n t i a l r e s o u r c e base. The net r e s e r v e s p o t e n t i a l l y r e c o v e r a b l e w i t h c a r b o n d i o x i d e approaches 700 m i l l i o n b a r r e l s . S i x o f t h e s e f i e l d s have been t h e s u b j e c t o f e x t e n s i v e e x p e r i m e n t a l PVT and phase e q u i l i b r i u m s t u d i e s . These f i e l d s produce from t h e San Andres F o r m a t i o n . These e x p e r i m e n t a l i n v e s t i g a t i o n s i n c l u d e f i r s t - c o n t a c t and m u l t i p l e - c o n t a c t s t u d i e s . F i r s t - c o n t a c t s t u d i e s a r e t h o s e i n which o n l y m i x t u r e s o f r e s e r v o i r o i l and c a r b o n d i o x i d e a r e s t u d i e d . I n m u l t i p l e - c o n t a c t s t u d i e s , two phase e q u i l i b r i u m i s a c h i e v e d f o r an o i l - c a r b o n dioxide mixture. Then e i t h e r the o i l - r i c h o r C 0 2 ~ r i c h phase i s r e c o n t a c t e d w i t h f r e s h C O 2 o r f r e s h r e s e r v o i r o i l , r e s p e c 0097-6156/86/0300-0406$08.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

20.

CHABACK AND TUREK

Phase Behavior of San Andres Formation Oils

407

t i v e l y . O n l y f i r s t - c o n t a c t d a t a are d i s c u s s e d h e r e ; a subsequent paper w i l l d i s c u s s t h e e x t e n s i o n t o m u l t i p l e c o n t a c t r e s u l t s . The o v e r a l l o b j e c t i v e o f t h e p r e s e n t i n v e s t i g a t i o n was t o p r o v i d e a Redlich-Kwong e q u a t i o n o f s t a t e d e s c r i p t i o n which would be r e p r e s e n t a t i v e o f a l l t h e o i l s produced from an e n t i r e f o r m a t i o n . Thus, i f s u c c e s s f u l , o n l y t h e o i l c o m p o s i t i o n and r e s e r v o i r tempera t u r e would u l t i m a t e l y be r e q u i r e d t o p r e d i c t t h e i m p o r t a n t C O 2 - 0 1 I phase e q u i l i b r i a . Such an approach was t a k e n i n o r d e r t o m i n i m i z e the need f o r a d d i t i o n a l c o s t l y e x p e r i m e n t a l s t u d i e s on f l u i d s from w e l l s not a l r e a d y t r e a t e d i n the e x p e r i m e n t a l program. T h i s i s a s e v e r e t e s t o f t h e e q u a t i o n o f s t a t e . The s u c c e s s e s and shortcomi n g s o f f l u i d d e s c r i p t i o n s so developed are d i s c u s s e d below.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

Development o f t h e F l u i d D e s c r i p t i o n s :

Overview

The s t r a t e g y employed i n t h e i n v e s t i g a t i o n c o n s i s t s o f f i r s t d e v e l o p i n g an e q u a t i o n o f s t a t e d e s c r i p t i o n f o r t h e r e s e r v o i r o i l p r i o r t o c o n t a c t w i t h a d i s p l a c i n g gas. T h i s u n c o n t a c t e d o r base o i l d e s c r i p t i o n i n c l u d e s a C7+ heavy hydrocarbon d i s t r i b u t i o n and c h a r a c t e r i z i n g parameters and c u s t o m i z e d methane-C7+ i n t e r a c t i o n parame t e r s f o r t h a t d i s t r i b u t i o n . The d e s c r i p t i o n was then extended t o m i x t u r e s o f CO2 w i t h t h e base r e s e r v o i r o i l . I n a d d i t i o n , the e f f e c t o f compressing t h e C7+ d e s c r i p t i o n was examined. F o r t h e base o i l , t h e 34 components r e s u l t i n g from c h r o matographic a n a l y s i s were regrouped t o c r e a t e d e s c r i p t i o n s w i t h 14, 7, 3 and 2 C7+ pseudocomponents. F o r C 0 2 ~ r i c h systems C7+ d e s c r i p t i o n s w i t h 14 and 3 pseudocomponents were d e v e l o p e d . These cases a l l o w e d a comparison t o be made between an extended C7+ d e s c r i p t i o n and a more compressed d e s c r i p t i o n s u i t e d t o c o m p o s i t i o n a l r e s e r v o i r model c a l c u l a t i o n s where t h e t o t a l number o f components must be r e s t r i c t e d because o f computer time and memory c o n s t r a i n t s . D e s c r i p t i o n s were developed u s i n g as much o f t h e a v a i l a b l e phase e q u i l i b r i u m d a t a as p o s s i b l e . F o r t h e s e San Andres f i e l d s a l a r g e q u a n t i t y o f base o i l PVT i n f o r m a t i o n was accumulated p r i o r t o i n i t i a t i o n o f EOR s t u d i e s . T y p i c a l l y , t h i s information includes reservoir f l u i d compositional a n a l y s i s , a d i f f e r e n t i a l vaporization a n a l y s i s (DVA), and p e r t i n e n t o p e r a t i n g parameters from the w e l l . Even though much o f t h e s e d a t a were o b t a i n e d t h i r t y t o f o r t y y e a r s ago, they s t i l l were found t o form a c o n s i s t e n t d a t a s e t . In a d d i t i o n , more r e c e n t work a s s o c i a t e d w i t h enhanced gas d r i v e s t u d i e s was i n c o r p o r a t e d i n t o t h e base o i l d a t a c o l l e c t i o n . These e x p e r i m e n t a l d a t a were e v a l u a t e d f o r c o n s i s t e n c y o f r e p o r t e d c o m p o s i t i o n and f l u i d p r o p e r t i e s . Those d a t a found c o n s i s t e n t were used t o d e v e l o p a r e p r e s e n t a t i v e C7+ heavy h y d r o c a r b o n d i s t r i b u t i o n ( s i m u l a t e d t r u e b o i l i n g p o i n t a n a l y s i s ) and e q u a t i o n o f s t a t e p a r ameters i n c l u d i n g methane-C7+ i n t e r a c t i o n p a r a m e t e r s . N e x t , t h e base o i l d e s c r i p t i o n was r e f i n e d t o match p r o p e r t i e s o f C02~base o i l m i x t u r e s . O b t a i n i n g s a t i s f a c t o r y agreement o f c a l c u l a t e d and e x p e r i m e n t a l r e s u l t s n e c e s s i t a t e d adjustment o f carbon dioxide-C7+ and carbon dioxide-methane i n t e r a c t i o n p a r a m e t e r s . Moreover, where hydrogen s u l f i d e was p r e s e n t i n s i g n i f i c a n t amounts i n t h e d i s p l a c i n g gas, s p e c i a l i z e d H 2 S - C 7 + and H2S-CO2 i n t e r a c t i o n parameters were a l s o d e v e l o p e d . In a l l c a s e s , t h e parameter adjustments were made by t r i a l and e r r o r u s i n g an h e u r i s t i c adjustment method; t h a t i s , changes were

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

408

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

made i n t e r a c t i v e l y a t a computer c o n s o l e r a t h e r than by some f o r m a l i z e d m a t h e m a t i c a l o p t i m i z a t i o n approach. As a consequence o f the cumbersome n a t u r e of the adjustment p r o c e d u r e , o n l y base o i l and some f i r s t - c o n t a c t C O 2 - 0 1 I d a t a were used i n the d a t a a d j u s t ment p r o c e s s . The remainder of the f i r s t - c o n t a c t d a t a ( i n c l u d i n g r e s u l t s f o r three-phase (L-L-V) f l u i d e q u i l i b r i a ) were used o n l y t o t e s t the parameters. The M o d i f i e d Redlich-Kwong

Equation of State

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

Throughout the d i s c u s s i o n r e f e r e n c e i s made t o v a r i o u s p h y s i c a l p r o p e r t i e s and e q u a t i o n of s t a t e parameters, f o r example, c r i t i c a l p r e s s u r e and a c e n t r i c f a c t o r . These p r o p e r t i e s a r e i n c o r p o r a t e d i n t o the Redlich-Kwong e q u a t i o n ( E q u a t i o n 1) as shown i n Equa t i o n s 2 and 3. Ρ =

RT V - b

(1)

Vf V(V + b)

where a and b a r e e q u a t i o n parameters d e f i n e d by:

NC

NC

Σ

Σ

i=l j=l

b =

NC

NC

Σ

Σ

Ω 1

x.x.U J

Ω ,RT . bi c i

(1 • D. .)

χ. χ. j=l 1 J

i=l

a r e

The omega parameters, a t u r e and a c e n t r i c f a c t o r : Ω

a

2 2.3 R Τ ai ci Ρ ci

_ 2 2.5 Ω ,RT , aj cj Ρ . cj Ω

1/2 (2)

.RT . bj cj Ρ cj

c o r r e l a t e d w i t h reduced

(3)

temper

= f ( T ,ω) Τ £ 1 r r f(l,u>) T > 1

(4)

= g(T ,uO T

(5)

r

Q

b

r

g(l,u>)

T

r

£ 1

r

> 1

The f and g f u n c t i o n s shown i n E q u a t i o n s 4 and 5 a r e d i f f e r e n t c o r r e l a t i o n s based on reduced temperature and a c e n t r i c f a c t o r as d e s c r i b e d by Yarborough ( O . Symbols used i n t h e s e e x p r e s s i o n s a r e d e f i n e d i n the G l o s s a r y a t the end o f the t e x t . T a b l e I shows such p r o p e r t i e s and i d e n t i f i e s pseudocomponents as t h e y appear i n com p u t e r output f o r a t y p i c a l c a s e : O i l D mixed w i t h a H 2 S / C O 2 d i s p l a c i n g gas. Note t h a t o i l s a r e g i v e n l e t t e r d e s i g n a t i o n s c o n s i s t e n t w i t h those of Turek, e t a l . ( 2 ) , whenever a p p r o p r i a t e . As d e s c r i b e d by M o r r i s and Turek ( 3 ) , the Ω^ and Ω^ f o r η-butane and l i g h t e r components a r e f u n c t i o n s of temperature f o r v a l u e s o f reduced temperature above and below the c r i t i c a l t e m p e r a t u r e . Those f o r pentane and h e a v i e r components a r e a f u n c t i o n of tempera t u r e o n l y below the c r i t i c a l temperature as shown i n E q u a t i o n s 4 and 5 above.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1

Ω

1 , f

2

S

M

0.41509 0.42619 0.42826 0.43724 0.44083 0.44836 0.44672 0.45181 0.45612 0.46028 0.46395 0.46709 0.46719 0.46984 0.47200 0.47417 0.47582 0.47370 0.46662 0.43883

6 7 8 9 10 11 27 28 29 30 31 32 33 34 35 36 37 38 39 40

2

44.097 58.124 58.124 72.146 72.146 86.172 92.920 105.640 118.370 132.390 147.070 161.750 171.200 185.900 200.540 221.540 256.300 305.210 366.510 487.900

616.04 527.94 550.56 483.00 489.50 440.00 528.30 479.87 436.69 392.42 352.55 319.55 335.51 306.55 281.87 252.93 215.18 175.89 139.98 97.35

0.0105 0.2310 0.0992 0.0911 0.1523 0.1852 0.1996 0.2223 0.2539 0.3007 0.2689 0.3127 0.3542 0.3989 0.4449 0.4892 0.4952 0.5360 0.5751 0.6272 0.7081 0.8500 0.9750 1.2500

1.58450 1.51300 2.35600 1.56500 3.20400 4.15000 4.08600 4.90000 4.87000 5.93000 5.68100 6.44100 7.20900 8.13200 9.13900 10.15600 10.31500 11.34200 12.36300 13.80000 16.17500 19.48000 23.56000 31.76400

0.0 0.0 0.35630 0.78920 0.50690 0.56250 0.58380 0.62420 0.63060 0.66330 0.76245 0.78373 0.80201 0.81030 0.81460 0.81869 0.87164 0.87144 0.87254 0.87753 0.88733 0.90421 0.92769 0.96365

These parameters a r e t e m p e r a t u r e dependent.

665.73 734.13 765.29 828.69 845.19 913.79 985.43 1039.48 1088.86 1131.15 1168.41 1202.51 1248.74 1277.12 1303.39 1339.72 1392.45 1456.08 1522.75 1613.14

ACENTRIC FACTOR

CRIT VOL, FT3/LB MOL

DENSITY AT 60°F

MOLE WT

0.07987 0.08094 0.08060 0.08127 0.08049 0.07907 0.07982 0.07828 0.07670 0.07498 0.07319 0.07142 0.07095 0.06931 0.06773 0.06560 0.06237 0.05688 0.05210 0.04238

Ρ c, PSIA 16.043 44.009 30.070 34.080

Τ c, DEG R 343.00 666.45 547.58 1070.74 549.59 706.54 672.35 1306.65

Β

0.08890 0.07975 0.08141 0.08112

Ω

denotes C + pseudocomponent. and Ω a r e e v a l u a t e d a t 105°F.

7

2

Α

0.45268 0.40340 0.41017 0.41158

Ω

2 3 4 5

COMP. ID NO.

Methane Carbon D i o x i d e Ethane Hydrogen Sulfide Propane I-Butane N-Butane I-Pentane N-Pentane N-Hexane C7S C8S C9S C10S C11S C12S C13S C14S C15S C16-17S C18-20S C21-24S C25-29S C30-40S

COMPONENT

TABLE I . COMPONENT PROPERTIES FOR THE GENERALIZED REDLICH-KWONG EQUATION OF STATE FOR OIL D, SAN ANDRES FORMATION

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

410

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

In g e n e r a l the C O 2 i n t e r a c t i o n parameters a r e f u n c t i o n s o f t e m p e r a t u r e . Only those C O 2 i n t e r a c t i o n parameters developed d u r i n g the d a t a matching e x e r c i s e a r e temperature independent. T a b l e I I c o n t a i n s the u n l i k e m o l e c u l a r p a i r i n t e r a c t i o n parameters f o r t h i s example San Andres system. The parameter s e t s i n T a b l e s I and I I a p p l y d i r e c t l y o n l y t o t h e O i l D system. However, t h e pseudocomponent p r o p e r t i e s and r e s e r v o i r temperatures a r e s i m i l a r f o r the v a r i o u s San Andres o i l s ; hence, the parameters i n T a b l e I and 11 c o u l d p r o b a b l y be a p p l i e d t o San Andres o i l s of d i f f e r e n t compos i t i o n w i t h o u t i n c u r r i n g s e r i o u s e r r o r . The a p p r o p r i a t e C7+ pseudocomponent c o m p o s i t i o n would have t o be used f o r such c a s e s .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

P r e p a r a t i o n of C o m p o s i t i o n a l Data The carbon number d i s t r i b u t i o n w i t h i n the C7+ f r a c t i o n i s o b t a i n e d v i a t e m p e r a t u r e programmed gas chromatography ( s i m u l a t e d t r u e b o i l i n g p o i n t a n a l y s i s ) . D i s t r i b u t i o n s f o r s e v e r a l f i e l d s and an average used t o r e p r e s e n t the San Andres Formation a r e shown i n F i g u r e 1. The c h r o m a t o g r a p h i c d i s t r i b u t i o n i s s u b s e q u e n t l y a d j u s t e d t o match the c r y o s c o p i c a l l y determined m o l e c u l a r weight o f the C7+ f r a c t i o n . F i r s t , the C40+ carbon number i s s e t t o 50 and the C40+ molX t o 5.0. These adjustments a r e made t o r e f l e c t the i n e v i t a b l e l o s s of some h e a v i e r C7+ components d u r i n g the chromatog r a p h i c e l u t i o n . A f i n a l adjustment i s then a p p l i e d t o match the c r y o s c o p i c mole w e i g h t . T h i s adjustment a p p l i e s an e x p o n e n t i a l c o r r e c t i o n t o the d i s t r i b u t i o n which i s p r o p o r t i o n a l t o carbon number· T h i s a d j u s t e d C7+ d i s t r i b u t i o n i s mated w i t h the a p p r o p r i a t e o v e r a l l c o m p o s i t i o n , n i t r o g e n through hexanes, w i t h the C 7 + f r a c t i o n lumped as a s i n g l e component. The lumped C7+ f r a c t i o n i s d i s t r i b u t e d among the carbon number f r a c t i o n s shown i n F i g u r e 1. The development c o n t i n u e s w i t h the s e l e c t i o n of the a p p r o p r i a t e s e t o r a r r a y o f PNA ( P a r a f f i n s , A r o m a t i c s , and Naphthenes) p r o p o r t i o n s f o r the extended pseudocomponent a n a l y s i s . T h i s C7+ c h a r a c t e r i z a t i o n method i s an e x t e n s i o n o f i d e a s d e s c r i b e d i n the AGA monograph by Bergman e t a l . ( 4 ) . The subsequent adjustment o f some pseudocomponent p r o p e r t i e s t o match e x p e r i m e n t a l d a t a i s , i n e f f e c t , a f u r t h e r r e f i n e m e n t o f t h e s e PNA r a t i o s . C o n s o l i d a t i o n of the extended C7+ d e s c r i p t i o n t o 14, 7 , 3, o r 2 pseudocomponents i s c a r r i e d out n e x t . For d e s c r i p t i o n s w i t h 14 or 7 pseudocomponents t h i s c o n s o l i d a t i o n procedure i s d e s i g n e d t o g i v e a p p r o x i m a t e l y e q u a l c o m p o s i t i o n s f o r a l l pseudocomponents. Thus, a w i d e r range of l e s s abundant h e a v i e r carbon number f r a c t i o n s i s combined t o form the h e a v i e r pseudocomponents. An example showing t h e s e 14 pseudocomponents i s shown i n T a b l e I . For t h r e e pseudocomponents c o n s o l i d a t i o n o f the d i s t r i b u t i o n a t carbon number 12 and carbon number 20 i s s e l e c t e d . For two components d i v i s i o n at c a r b o n number 20 and 12 was t e s t e d f o r base o i l c a l c u l a t i o n s o n l y . As d i s c u s s e d below, d i v i s i o n a t carbon number 20 gave b e t t e r results· Development o f Base O i l Parameters In a d d i t i o n t o the base o i l d a t a p r o v i d e d by Turek et a l . ( 2 ) and

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

20.

Phase Behavior of San Andres Formation Oils

CHABACK AND TUREK

411

TABLE I I . INTERACTION PARAMETERS ( C . D . . ) FOR THE GENERALIZED REDLICH-KWONG EQUATION OF STATE FOR OIL D, SAN ANDRES FORMATION, EVALUATED AT 105°F

ID NUMBER

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5

1

2 C. .

2 D. . ij

3 *-0.09Q000 4 0.005000 5 0.084700 6 0.010000 7 0.010000 8 0.010000 0.010000 9 10 0.010000 11 0.010000 (27-40) *0.010000 4 0.156698 6 0.152101 7 0.149255 8 0.148010 0.146046 9 10 0.143313 11 0.139264 27 *0.090000 28 *0.090000 *0.090000 29 30 *0.090000 31 *0.090000 32 *0.090000 33 *0.090000 34 *0.090000 35 *0.090000 36 *0.090000 37 *0.090000 38 *0.090000 *0.090000 39 40 *0.090000 5 *0.160000 4 0.090500 6 0.084600 7 0.082000 8 0.080000 0.078000 9 10 0.075000 11 0.070539 (27-40) *0.150000

3

-0.038260 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.033048 -0.029926 -0.027993 -0.027147 -0.025813 -0.023956 -0.021206 -0.023074 -0.020501 -0.018062 -0.015436 -0.012733 -0.010130 -0.009777 -0.007380 -0.005082 -0.002021 0.002733 0.011071 0.018416 0.034575 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Component ID Nos. a r e i d e n t i f i e d w i t h s p e c i f i c compounds i n T a b l e I . D e f a u l t C.. and D. . f o r carbon d i o x i d e a r e f u n c t i o n s of temperature and

2

1J

acentric factor. 3

1J

See References ( 1 ) and ( 2 ) .

A n * denotes g e n e r a l i z e d San Andres C.. V a l u e . values.

D. .'s a r e a l l d e f a u l t

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

412

F i g u r e 1. Comparison o f Heavy Hydrocarbon D i s t r i b u t i o n s . Results f o r F l u i d s from t h e San Andres F o r m a t i o n : A n a l y s e s f o r S e v e r a l F i e l d s and Smoothed Average.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

CHABACK AND TUREK

20.

Phase Behavior of San Andres Formation Oils

413

Yarborough ( 5 ) , e x t e n s i v e r e s u l t s f o r San Andres f l u i d s have been accumulated from o t h e r enhanced gas d r i v e s t u d i e s as w e l l as from h i s t o r i c a l measurements d u r i n g the p r e s s u r e d e p l e t i o n phase of r e s e r v o i r o p e r a t i o n . T a b l e I I I compares the f l u i d c o m p o s i t i o n s used

TABLE I I I . COMPARISON OF CANDIDATE DATABASE COMPOSITIONS A l l Compositions are i n M o l % Database Case Comp

2

DSX

C+

0.55 15.61 0.61 7.43 0.00 6.32 0.48 4.62 1.18 2.93 4.69 55.58

C+

Propertie s

N Ci C0 C HS 2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

B3

2

2

2

c

3

1C4

nC

4

1C5

nC

c

5

6 7

7

1.08 9.63 1.72 4.60 0.41 5.41 1.29 4.11 1.96 2.66 6.48 60.65

Β 1.35 18.19 0.29 5.25 0.00 4.41 1.10 3.57 1.77 3.34 3.39 57.34

Fl 0.47 19.03 2.63 7.86 0.00 7.60 1.29 3.91 1.65 1.71 3.44 50.41

F

A

Bl

D

0.54 16.12 4.68 13.50 0.00 10.03 1.92 6.52 1.47 1.95 3.16 40.11

2.15 17.78 4.74 8.57 1.66 7.71 0.94 4.71 1.58 3.21 3.17 43.79

0.52 15.43 0.56 7.26 0.00 6.14 0.49 4.10 1.36 2.69 3.61 57.84

0.00 15.24 0.00 3.69 0.00 4.66 0.00 3.89 1.49 1.80 2.37 66.86

Mol Wt 250 253 232 242 243 215 Sp Gr 0.8822 0.8880 0.8861 0.8805 0.8839 0.889

B2 0.41 13.06 0.49 7.04 0.04 7.18 1.59 4.57 2.26 2.72 3.21 57.48

240 232 236 0.8870 0.8874 0.8870

d a t a b a s e cases a r e g i v e n l e t t e r d e s i g n a t i o n s c o n s i s t e n t w i t h used by Turek, e t a l . (2) B 3 r e f e r s t o an a l t e r n a t e a n a l y s i s of O i l B l o b t a i n e d by an independent l a b o r a t o r y .

those

2

i n these s t u d i e s . These d a t a were screened, as d e s c r i b e d below, f o r i n c l u s i o n i n a San Andres Formation base o i l database by making f l u i d p r o p e r t y p r e d i c t i o n s u s i n g d e f a u l t e q u a t i o n of s t a t e parame t e r s developed from p r e v i o u s e x p e r i e n c e w i t h h y d r o c a r b o n m i x t u r e s . Phase e q u i l i b r i a were c a l c u l a t e d w i t h the Redlich-Kwong e q u a t i o n of s t a t e ; the S t a n d i n g - K a t z c o r r e l a t i o n (6) was used t o c a l c u l a t e the l i q u i d d e n s i t i e s . As such, t h i s method has been found t o g e n e r a l l y p r o v i d e a c c u r a t e r e s u l t s f o r o i l bubble p o i n t s and d e n s i t i e s but cannot be used f o r C 0 - o i l m i x t u r e s . I t s v a l u e here l i e s i n e s t a b l i s h i n g c o n s i s t e n c y between measured f l u i d c o m p o s i t i o n and f l u i d properties· I n g e n e r a l , a case was r e j e c t e d f o r i n c l u s i o n i n the database i f the r e p o r t e d l i q u i d d e n s i t y was i n s e r i o u s disagreement w i t h the S t a n d i n g - K a t z v a l u e ( t h a t i s , g r e a t e r than 3%) o r i f the bubble p o i n t p r e s s u r e c o u l d not be matched w i t h r e a s o n a b l e adjustment of 2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

414

APPLICATIONS

the C7+ a c e n t r i c f a c t o r s ( t h a t i s , l e s s than 2 0 % ) . Only r e s e r v o i r o i l d e n s i t y d a t a f o r O i l B2 were r e j e c t e d on t h e s e grounds, and, as a consequence, O i l B2 was not i n c l u d e d i n the d a t a b a s e . R e f i n e d C7+ d e s c r i p t i o n s f o r each of the s e l e c t e d database cases were then p r e p a r e d . These d e s c r i p t i o n s were based upon C7+ d i s t r i b u t i o n , m o l e c u l a r w e i g h t , and PNA d i s t r i b u t i o n . F u r t h e r a d j u s t m e n t s by t r i a l and e r r o r were made to o p t i m a l l y match the measured base o i l p r o p e r t i e s . These d a t a i n c l u d e d s i n g l e phase d e n s i t i e s , u s u a l l y o b t a i n e d a t 1500-2000 p s i a and r e s e r v o i r temper a t u r e and the bubble p o i n t p r e s s u r e a l s o at r e s e r v o i r t e m p e r a t u r e . To b r i n g about agreement of c a l c u l a t e d and e x p e r i m e n t a l r e s u l t s , the p s e u d o c r i t i c a l p r e s s u r e s and methane i n t e r a c t i o n parameters w i t h s e l e c t e d C7+ h y d r o c a r b o n f r a c t i o n s were a d j u s t e d . Changes i n the c r i t i c a l p r e s s u r e a f f e c t p r i m a r i l y the d e n s i t y w h i l e changes i n the methane-heavy hydrocarbon i n t e r a c t i o n param e t e r a f f e c t p r i m a r i l y the bubble p o i n t p r e s s u r e . These two parame t e r s a r e n e a r l y independent i n t h e i r e f f e c t ; t h u s , agreement of c a l c u l a t e d and e x p e r i m e n t a l v a l u e s f o r these d a t a i s e a s i l y a c h i e v e d . As noted above, a d j u s t i n g the c r i t i c a l p r e s s u r e of a C7+ pseudocomponent i s tantamount to changing i t s PNA d i s t r i b u t i o n t o r e f l e c t the s p e c i f i c c h a r a c t e r i s t i c s of the f l u i d . The r e s u l t i n g a d j u s t e d parameter s e t s f o r each database e n t r y were a r i t h m e t i c a l l y averaged to p r o v i d e a g e n e r a l San Andres Formation d e s c r i p t i o n f o r the base o i l . T a b l e IV shows t h e s e averaged parameters f o r 7, 3 and 2 C7+ pseudocomponents. Those f o r 14 pseudocomponents a r e g i v e n i n T a b l e I and i n c l u d e a c e n t r i c f a c t o r and p s e u d o c r i t i c a l p r e s s u r e v a l u e s f o r components Nos. 38, 39 and 40. To c r e a t e a C7+ f r a c t i o n w i t h

TABLE IV.

ADJUSTED EQUATION OF STATE PARAMETERS FOR SAN FLUIDS USING 7, 3 and 2 C + PSEUDOCOMPONENTS

ANDRES

7

C. . χ 1000 IJ

7

3

Ρ 's, p s i a ~ 3

= 7 2

C22-27

C28+

C13-20

C21+

5 2 C7-20

AVE 10 12 16 156.3 103.3 257.99 125.55 334.0 C^ . denotes i n t e r a c t i o n parameters f o r methane w i t h each of the

C21+ 127.7 14

pseudocomponents. P 's c

a r e c r i t i c a l p r e s s u r e s f o r the pseudocomponent c a r b o n number

grouping.

two pseudocomponents, the C7+ d i s t r i b u t i o n was s p l i t a t carbon number 20 or 12. R e s u l t s f o r d i v i s i o n a t carbon number 20 a r e shown i n T a b l e IV. One e f f e c t of r e d u c i n g the number of pseudocom ponents i s t o i n c r e a s e the s i z e of the i n t e r a c t i o n parameter w i t h methane. T h i s i n c r e a s e i s a consequence of the i n c r e a s e d s o l u b i l i t y of methane i n the more s e v e r e l y compressed and, therefore,

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

20.

CHABACK AND TUREK

415

Phase Behavior of San Andres Formation Oils

l i g h t e r pseudocomponents. The change i n pseudocomponent c h a r a c t e r i s a l s o r e f l e c t e d by t h e i n c r e a s e i n c r i t i c a l p r e s s u r e . Table V compares r e s u l t s f o r t h e v a r i o u s C7+ c o m p r e s s i o n s . Note t h e i n s e n s i t i v i t y o f t h e AAD (average a b s o l u t e d e v i a t i o n ) t o changes i n C7+ compression and t h a t f o r o n l y two pseudocomponents d i v i s i o n a t carbon number 20 g i v e s b e t t e r r e s u l t s . I n a d d i t i o n t o i n t e r p o l a t i o n w i t h i n t h e database cases t h e m s e l v e s , t h e averaged parameter s e t s were a l s o t e s t e d by p r e d i c t i o n o f h i s t o r i c a l meas urements of r e s e r v o i r o i l p r o p e r t i e s . T a b l e VI l i s t s t h e

TABLE V.

COMPARISON OF EXPERIMENT WITH CALCULATIONS USING AVERAGE PARAMETERS FOR THE DATABASE CASES USING 14, 7, 3 AND 2 C + PSEUDOCOMPONENTS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

7

Bubble P o i n t P r e s s u r e , p s i a OIL Bl DSX Β Fl F A Bl D AAD

Exp 14

7

3

2A

1

2B

S i n g l e Phase D e n s i t i e s , gm/cc 1

Exp

14

7

3

2B

2A

800 801 800 804 819 810 0.811 0.817 0.817 0.820 0.829 628 582 580 579 572 575 0.832 0.825 0.823 0.821 0.817 1064 1008 1005 1007 1013 1019 0.815 0.815 0.814 0.816 0.819 1033 1039 1036 1025 999 1003 0.788 0.785 0.784 0.779 0.762 1000 990 988 989 995 989 0.776 0.777 0.777 0.778 0.781 1190 1247 1244 1251 1267 1262 0.792 0.796 0.790 0.799 0.810 787 784 0.815 0.815 0.815 0.814 0.813 749 788 787 788 620 669 668 667 671 663 0.827 0.830 0.830 0.828 0.827 0.44 0.40 0.65 1.37 % 4.0 4.1 4.2 5.0 4.6

0.818 0.819 0.817 0.768 0.779 0.805 0.813 0.825 0.94

2A denotes d i v i s i o n o f t h e C7+ d i s t r i b u t i o n i n t o pseudocomponents a t carbon number 12; 2B r e f e r s t o d i v i s i o n a t carbon number 20.

screened h i s t o r i c a l d a t a and compares t h e r e p o r t e d c o m p o s i t i o n s . Cases No. 558, 238, 241 and 444 were a c c e p t e d as r e p o r t e d . C a l c u l a t e d r e s u l t s f o r t h e s e cases a r e shown i n T a b l e V I I . Note t h e e x c e l l e n t agreement between c a l c u l a t e d and e x p e r i mental v a l u e s . R e c a l l t h e s e d a t a were o b t a i n e d 30 t o 40 y e a r s ago; agreement i s p r o b a b l y w i t h i n e x p e r i m e n t a l u n c e r t a i n t y . Of s p e c i a l i n t e r e s t a r e r e s u l t s f o r o i l f o r m a t i o n volume f a c t o r ( F V F ) . I t i s w e l l r e p r e s e n t e d a t b o t h t h e bubble p o i n t and 2000 p s i a . T h i s agreement suggests t h a t t h e average d e s c r i p t i o n f i t s not o n l y f l u i d d e n s i t y a t a p o i n t but f l u i d c o m p r e s s i b i l i t y o v e r a range o f p r e s sure as w e l l . Note f u r t h e r t h a t t h e w e l l s c i t e d i n t h e s e cases span a range o f temperatures from 94° t o 114 F w h i l e t h e parameters were developed from d a t a a t 105° and 106°F. Thus, t h e averaged parameters may be e x t r a p o l a t e d w i t h o u t l o s s o f p r e c i s i o n a t l e a s t 10 F above o r below t h e temperature a t which t h e y were d e v e l o p e d . Development o f F l u i d D e s c r i p t i o n s f o r Carbon D i o x i d e - O i l M i x t u r e s The p r e c e d i n g development

o f base o i l d e s c r i p t i o n s f o r t h e forma-

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2

2

5

5

4

4

420 103 365 326 394 328 384

320

149

175

150

162

204

GOR a t Bubble P t . (SCF/bbl)

1493 175 1198 1380 1288

1072 1478

1122

284

436

264

320

419

Bubble P t . (psia)

108 102 110 108 109

107

114

108

93

94

94

94

95

R e s e r v o i r T, <°F)

7.61 2.18 4.72 1.22 2.60 3.14 51.40

7.72

19.41

642

230 0.8888

0.75 2.74 0.52 5.15 0.52 6.11 1.37 3.91 2.15 2.24 2.50 72.04

561

223 0.8849

1.58 18.24 0.23 8.61 0.00 7.31 1.04 4.38 1.47 1.89 1.95 53.30

558

223 0.8905 271 0.8990

Mol Wt Sp Gr

2.16 16.70 0.09 7.77 0.03 7.25 1.35 4.38 2.03 1.71 1.23 55.30

553

225 0.8899

0.80 22.61 4.04 7.60 0.92 6.98 1.22 4.14 1.65 2.22 3.46 44.36

444

236 0.8805

0.13 18.07 3.86 6.75 1.33 5.88 1.46 3.78 1.48 2.81 3.31 51.64

244

242 0.8816

2.87 22.94 0.15 8.20 0.00 6.88 1.34 4.25 1.67 2.07 3.39 46.24

241

233 0.8888

9.13 3.45 5.32 0.73 1.87 5.18 63.88

4.22

2.33 18.30 0.66 8.20 0.08 7.37 1.15 4.19 1.94 2.44 3.48 49.86

238

219 0.8822

11.07 2.06 5.92 0.12 5.19 4.46 64.75

3.62

3.22

...

...

2.81

217

216

216 0.8852

8.40 2.83 6.89 0.23 6.28 4.74 63.36

4.06

3.21

215

223 0.8775

10.54 1.99 6.41 0.15 6.23 4.58 62.16

3.94

4.00

214

225 0.8778

Properties

0.39 6.64 0.50 7.78 1.05 9.67 1.34 5.59 2.26 2.57 2.70 59.51

213

238 0.8794

7

C+

7

6

3

2

2

C0 C H S c iC nC iC nC c C+

Ci

N

Case

A l l Compositions a r e i n M o l %

TABLE V I . COMPARISON OF HISTORICAL DATABASE COMPOSITIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

20.

C H A B A C K A N D TUREK

Phase Behavior of San Andres Formation Oils

TABLE V I I . SAN ANDRES RESULTS FOR HISTORICAL CASES: CALCULATED AND EXPERIMENTAL RESULTS

417

COMPARISON OF

AAD% C+ Compression 7

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

14 7 3 2

BPP

GOR

5.44 5.31 5.79 6.61

6.56 6.56 5.99 6.62

FVF(BPP) 1.59 1.59 1.47 1.53

FVF(2000)

DVA(resid)

1.62 1.63 1.51 1.58

1.13 1.17 1.11 1.36

where BPP denotes bubble p o i n t p r e s s u r e , p s i a , GOR denotes g a s / o i l r a t i o , SCF/bbl o f r e s i d u a l o i l , FVF(BPP) denotes f o r m a t i o n volume f a c t o r a t t h e bubble p o i n t , bbl reservoir f l u i d / b b l residual o i l FVF(2000) denotes f o r m a t i o n volume f a c t o r a t 2 0 0 0 p s i a , DVA denotes DVA r e s i d u a l o i l s p e c i f i c g r a v i t y .

t i o n s o f i n t e r e s t suggests t h a t hydrocarbon phase b e h a v i o r i s w e l l d e s c r i b e d by t h e m o d i f i e d Redlich-Kwong e q u a t i o n of s t a t e . Thus, e x t e n s i o n of those r e s e r v o i r o i l d e s c r i p t i o n s t o carbon d i o x i d e - o i l m i x t u r e s ought t o be brought about by changing o n l y c a r b o n d i o x i d e - r e l a t e d i n t e r a c t i o n parameters. Parameters r e l a t e d t o h y d r o c a r b o n pseudocomponents ought t o remain unchanged. E a r l y i n t h e i n v e s t i g a t i o n i t was r e c o g n i z e d t h a t t h e p h y s i c a l s i t u a t i o n was t o o complex t o be r e p r e s e n t e d by such an approach. Attempts t o f i t t h e C O 2 - 0 1 I m i x t u r e d a t a o b t a i n e d by Turek, e t a l . ( 2 ) w i t h o n l y C O 2 parameter adjustments proved u n s u c c e s s f u l . I n p a r t i c u l a r , t h e l i q u i d - l i q u i d branch o f the s a t u r a t i o n p r e s s u r e c o m p o s i t i o n (P-X) l o c u s was p o o r l y r e p r e s e n t e d . O f t e n s i n g l e phase b e h a v i o r was p r e d i c t e d a t c o m p o s i t i o n s where two-phase b e h a v i o r was observed. However, p r i o r t o t h e p r e s e n t i n v e s t i g a t i o n , c a l c u l a t i o n s w i t h O i l B 2 - C O 2 mixtures l e d to the r e a l i z a t i o n that large a c e n t r i c f a c t o r s f o r t h e h e a v i e s t pseudocomponents ( c h a r a c t e r i s t i c o f t h e E d m i s t e r c o r r e l a t i o n ( 7 ) ) used i n c o n j u n c t i o n w i t h t h e c h a r a c t e r i z a t i o n procedure d e s c r i b e d by Yarborough (_1) c o u l d b r i n g t h e l i q u i d - l i q u i d branch of t h e P-X l o c u s i n t o b e t t e r agreement w i t h e x p e r i m e n t . See F i g u r e 2 . F o r t h i s i n v e s t i g a t i o n t h e d e c i s i o n was made t o a d j u s t t h e C 7 + a c e n t r i c f a c t o r s a s s i g n e d t h r o u g h t h e c h a r a c t e r i z a t i o n procedure d i s c u s s e d e a r l i e r and g i v e up some p r e c i s i o n i n t h e c a l c u l a t i o n of base o i l d e n s i t y i n o r d e r t o enhance t h e p r e d i c t i o n o f phase b e h a v i o r . Note t h a t E d m i s t e r - l i k e a c e n t r i c f a c t o r s a r e t y p i c a l l y l a r g e r by some 5 - 1 0 % than those p r o v i d e d by our C 7 + c h a r a c t e r i z a t i o n procedure u t i l i z i n g PNA d i s t r i b u t i o n s . The phase e q u i l i b r i u m d a t a f o r O i l s B l , B 2 , D, F and A form the f i r s t - c o n t a c t d a t a s e t f o r matching. The c o m p o s i t i o n s o f t h e base o i l s used i n these carbon d i o x i d e m i x t u r e s t u d i e s a r e r e p o r t e d i n T a b l e V I . The matching procedure was s t a r t e d w i t h O i l B 2 - C Û 2 m i x t u r e s because o f e a r l i e r success i n matching c a l c u l a t e d and

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

418

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

Legend: Δ

E x p e r i m e n t a l data Adj. Parms.: 14 C7+ Comps

0 4000

Adj. Parms.. 3 C7+ Comps Edmister Parameters

Ψ

Ι

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

/ /

3000

a, Q> U

2

/

a,

ΔΔ *Δ

40

60

Mole Percent C 0

a

F i g u r e 2. C O 2 - O 1 I B2 System: Comparison o f S a t u r a t i o n L o c i a t 106°F. E x p e r i m e n t a l V a l u e s Versus C a l c u l a t i o n s f o r 14 and 3 C 7 + Pseudocomponents·

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

20.

CHABACK AND TUREK

Phase Behavior of San Andres Formation Oils

419

e x p e r i m e n t a l r e s u l t s f o r t h a t system. An o v e r a l l compromise f i t t o the f i r s t - c o n t a c t d a t a f o r O i l B2 was a c h i e v e d through t r i a l and e r r o r adjustments o f the a c e n t r i c f a c t o r s f o r the C20+ pseudocomponents as w e l l as through adjustments to the carbon dioxide-C7+ and carbon dioxide-methane i n t e r a c t i o n parameters. R e s u l t s f o r O i l B2 a r e c o l l e c t e d i n Table V I I I f o r d e n s i t y , phase d i s t r i b u t i o n and c o e x i s t i n g phase c o m p o s i t i o n . The 2.5% e r r o r i n base o i l d e n s i t y i n c u r r e d by the new parameters i s s t i l l w i t h i n a c c e p t a b l e l i m i t s . Moreover, they now a l l o w t h e carbon d i o x i d e - r i c h phase e q u i l i b r i a t o be c a l c u l a t e d w i t h s a t i s f a c t o r y p r e c i s i o n as w e l l . F u r t h e r r e s u l t s w i t h these new parameters a r e g i v e n i n F i g u r e 2 which shows a comparison of e x p e r i m e n t a l s a t u r a t i o n l o c i w i t h t h a t c a l c u l a t e d i n t h i s i n v e s t i g a t i o n and u s i n g t h e E d m i s t e r c o r r e l a t i o n . Note the e x c e l l e n t agreement f o r a l l p r o p e r t i e s except three-phase c o n s o l u t e p r e s s u r e l o c i . Carbon d i o x i d e c o m p o s i t i o n s a r e w e l l r e p r e s e n t e d . Only f o r the m i x t u r e w i t h 95 mol% carbon d i o x i d e i s t h e r e s e r i o u s d i s a g r e e ment between c a l c u l a t e d and e x p e r i m e n t a l K - v a l u e s . However, comp a r i s o n w i t h o t h e r o i l - r i c h f l u i d c o m p o s i t i o n s from San Andres 01I-CO2 systems shows t h a t the r e p o r t e d CO2 c o m p o s i t i o n o f 81.4 mol% i n the o i l - r i c h phase i s p r o b a b l y i n e r r o r . See T a b l e V I I I . Note a l s o the s i m i l a r i t y of r e s u l t s f o r c a l c u l a t i o n s w i t h o n l y 3 pseudocomponents t o those w i t h 14 which once a g a i n u n d e r s c o r e s the i n s e n s i t i v i t y of these r e s u l t s t o C7+ breakdown. T a b l e s IX and X extend the a p p l i c a t i o n o f t h e s e new a c e n t r i c f a c t o r s t o a l l of the r e m a i n i n g f i r s t - c o n t a c t d a t a f o r San Andres f l u i d s o b t a i n e d by Turek, e t a l . ( 2 ) . Because these a r e now p r e d i c t i o n s r a t h e r than i n t e r p o l a t i o n s , some l o s s of a c c u r a c y i s e x p e c t e d . T y p i c a l l y , Table I X shows the base o i l d e n s i t y t o be i n e r r o r by a p p r o x i m a t e l y 3%. On the o t h e r hand, the c a l c u l a t e d o i l r i c h phase d e n s i t i e s remain i n e x c e l l e n t agreement w i t h e x p e r i m e n t . Note the t r e n d i n c a l c u l a t e d r e s u l t s f o r o i l - r i c h and C02-rich l i q u i d d e n s i t y f o r the O i l D-CO2 system. There i s a s u b s t a n t i a l improvement i n d e n s i t y p r e d i c t i o n as the carbon d i o x i d e c o m p o s i t i o n i s i n c r e a s e d t o v a l u e s c o n s i s t e n t w i t h the f l o o d i n g p r o c e s s . C02~rich l i q u i d d e n s i t i e s , on the o t h e r hand, a r e s t i l l i n e r r o r by 5-10%. A g a i n , the t a b l e shows the s i m i l a r i t y of r e s u l t s f o r 3 and 14 C7+ components. Except f o r the O i l F c a s e , Table X shows t h a t t h e CO2 K - v a l u e s and i n d i v i d u a l carbon d i o x i d e phase c o m p o s i t i o n s a r e i n r e a s o n a b l e agreement w i t h e x p e r i m e n t a l v a l u e s . For most cases t h e y agree w i t h i n 5 o r 6%. For the O i l F case the r e p o r t e d c o m p o s i t i o n of 60 mol% i s s u s p e c t ; i t i s p o s s i b l e t h a t the i n d i v i d u a l phase compos i t i o n s are a l s o i n e r r o r . W h i l e F i g u r e s 3 and 4 show t h a t the P-X l o c u s i s w e l l r e p r e s e n t e d a l o n g two-phase b o u n d a r i e s , three-phase e q u i l i b r i a a r e p o o r l y r e p r e s e n t e d . The three-phase r e g i o n f a i l s t o extend a c r o s s the e n t i r e e x p e r i m e n t a l l y observed c o m p o s i t i o n range. Furthermore, the p r e s s u r e range f o r the three-phase r e g i o n i s too narrow, t h a t i s , the upper c o n s o l u t e p r e s s u r e i s too near the lower c o n s o l u t e p r e s s u r e . As shown i n F i g u r e 4, f o r m i x t u r e s c o n t a i n i n g O i l A t h i s f a i l u r e i s e s p e c i a l l y pronounced. S i m i l a r l y , phase d i s t r i b u t i o n r e s u l t s a r e not i n good a g r e e ment w i t h e x p e r i m e n t . U s u a l l y too much o i l - r i c h l i q u i d i s p r e -

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

0.820 0.865 0.892 0.919

1

1

0.841 0.865 0.886 0.918

75.3 38.1 9.6

75.5 39.9 9.2

74.2 40.1 9.6

t

0.766 0.785 0.786

2

2

0.756 0.769 0.771

0.752 0.766 0.769

2

2

ADJUSTMENT

66.7 67.7 81.4 2

69.3 69.5 66.2

68.6 68.9 66.1

86.0 92.0 98.1

86.6 91.0 96.3

87.0 91.2 96.3

L i q u i d - L i q u i d Phase E q u i l i b r i a Phase C o m p o s i t i o n ( M o l % CQ ) O i l - R i c h Phase CQ -Rich Phase Exp't C a l c ( 1 4 ) CalcTÏ) Exp't Calc(14) Calc(3)

0.842 0.867 0.888 0.917

e n s

e s t i m a t e d from S t a n d i n g - K a t z c o r r e l a t i o n . V a l u e b e l i e v e d t o be i n e r r o r .

75 85 95

7

p £ y ^ gm/cc Oil-Rich Liquid CQ -Rich L i q u i d Exp t C a l c ( 1 4 ) C a l c ( 3 ) Exp't Calc(14) Calc(3)

Phase D i s t r i b u t i o n (Vol% Oil-Rich Liquid) Exp't C a l c ( 1 4 ) CalcTT)

Base O i l 75 85 95

2

Composition, M o l % CQ

2

TABLE V I I I . FIRST CONTACT C 0 - 0 I L DATA USED FOR PARAMETER RESULTS FOR 14 AND 3 C + PSEUDOCOMPONENTS BASIS: OIL B2 DATA AT 106°F AND 2200 PSIA

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

1.29 1.36 1.21

1

Exp 1

2

1.25 1.31 1.46

1.27 1.32 1.46

CQ K-Value Calc(14) Calc(3)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986. 0.823 0.833 0.870 0.920

2015 p s i a , 105°F 0.792 0.819 0.831 0.813 0.864 0.866 0.928 0.918

O i l A at Base O i l 50 75 95

1

2

0.749 0.767

0.737

0.743 0.773 0.782

0.766 0.767

2

2

C a l c u l a t e d r e s u l t f o r 71.98 mol% C0 . Compositions are i n mol% added C0 . C a l c u l a t e d r e s u l t s f o r 63.5 mol% added C0 .

2

3

0.801 0.811 0.817 0.828

2015 p s i a , 105°F 0.776 0.799 0.789 0.809 0.790 0.815 0.811 0.825

O i l F at Base O i l 30 50 60

2

0.853 0.857 0.861 0.869 0.880 0.888 0.913

O i l D at Base O i l 25 50 71.2 80 85 95

2000 p s i a , 105°F 0.827 0.854 0.835 0.857 0.840 0.861 0.861 0.867 0.878 0.880 0.892 0.888 0.924 0.915

0.838 0.848 0.871 0.914

O i l Bl at 2015 psia, 106°F Base O i l 0.815 0.837 50 0.827 0.848 78 0.873 0.870 0.920 95 0.913

Mol% C02

0.705 0.742

0.686

0.733 0.752 0.758 0.757

0.738 0.751

2

0.703 0.740

0.683

0.731 0.748 0.754 0.754

0.734 0.748

1

40.8 7.3

73.6

68.5 40.3 31.1 8.1

46.2 8.5

56.5 8.5

91

88 61 44 11

62.2 10.0

56.1 7.62

88

1

84 59.9 43.4 11.0

61.1 9.4

Phase D i s t r i b u t i o n ( V o l % O i l - R i c h Phase) Exp* t Calc(14) Calc(3

COMPARISON OF CALCULATED AND EXPERIMENTAL FIRST CONTACT DATA FOR SAN ANDRES FLUIDS

Density, gm/cc ~ O i l - R i c h Phase C0 -Rich Calc(14) Calc(3) Exp't Calc(14) Calc(3)

TABLE IX.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2

65.1 64.4

64.8 64.1

1

2

2

82.3 92.6

68.0

86.5 91.0 93.9

85.1 95.3

2

C a l c u l a t e d r e s u l t f o r 71.98 molZ C0 . Compositions are i n mol% added C0 . C a l c u l a t e d r e s u l t s f o r 63.5 molZ added C0 .

75 95

O i l A at 2015 p s i a , 105°F Base O i l

2

3

64.3 63.8

62.9

O i l F at 2015 p s i a , 105°F Base O i l 30 50 59.2 60 63.5

2

67.4 68.6 68.7 66.2

:::

:::

67.8 65.7

O i l D at 2000 p s i a , 105°F Base O i l 25 50 71.2 65.5 67.8 80 69.2 85 69.3 65.9 95 66.2 66.4

O i l Bl at 2015 p s i a , 106°F Base O i l 50 78 65.1 68.4 95 68.3 66.2

Mol% CQ

2

85.3 95.9

77.8

86.8 90.0 91.9 96.5

88.0 96.0

2

85.5 96.3

78.0

87.3 90.3 92.1 96.6

88, 96,

1

Phase Composition, mol% CQ O i l - R i c h Phase CQ -Rich Phase Exp't Calc(14) Calc(3) Exp't Calc(14) Calc(3)

1.26 1.44

1.15

1.42

1.32

1.31 1.40

T

Exp t

TABLE X. COMPARISON OF CALCULATED AND EXPERIMENTAL CONTACT DATA FOR SAN ANDRES FLUIDS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

2

1.32 1.50

1.23

1.28 1.30 1.33 1.45

29 45

1.33 1.51

1.24

1.30 1.32 1.34 1.46

1.30 1.47

C0 K-Value Calc(14) Calc(3)

FIRST

20. C H A B A C K A N D T U R E K

Phase Behavior of San Andres Formation Oils

4000-1

Legend: Smoothed Expt Data Δ

L-L-V Expt Data Calc: 14 C7+ Comps

ο

Calc: 3 C7+ Comps

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

3000

0)

U CL

1000

so

60

Mole Percent C 0

a

F i g u r e 3. C O 2 - O 1 I B l System: Comparison o f S a t u r a t i o n L o c i a t 106 F, E x p e r i m e n t a l V a l u e s V e r s u s C a l c u l a t i o n s f o r 14 and 3 C 7 + Pseudocomponent s.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

20.

CHABACK AND TUREK

Phase Behavior of San Andres Formation Oils

425

d i e t e d . E r r o r s a r e l a r g e e s p e c i a l l y a t the lower carbon d i o x i d e c o m p o s i t i o n s where the premature t e r m i n a t i o n of the t h r e e phase r e g i o n o c c u r s . Poor phase volume p r e d i c t i o n s may be a t t r i b u t e d , at l e a s t i n p a r t , t o t h i s f a i l u r e t o c a l c u l a t e three-phase b e h a v i o r a c r o s s the e n t i r e e x p e r i m e n t a l l y observed c o m p o s i t i o n range. Note t h a t the p r e d i c t i o n s improve somewhat a t h i g h e r carbon d i o x i d e comp o s i t i o n s where three-phase b e h a v i o r i s p r e d i c t e d . See Table IX. A d d i t i o n a l i n s i g h t i n t o t h i s d i s c r e p a n c y i s o b t a i n e d through comparison of the c a l c u l a t e d and measured P-X l o c i . I n the l i q u i d - l i q u i d r e g i o n the q u a l i t y l i n e s a r e v e r y s t e e p l y s l o p e d and c l o s e l y spaced as they emanate from the upper c o n s o l u t e l o c u s . See F i g u r e 5. Because the q u a l i t y l i n e s a r e so s t e e p l y s l o p e d i n the l i q u i d - l i q u i d r e g i o n , s m a l l c o m p o s i t i o n a l e r r o r s can l e a d t o l a r g e e r r o r s i n the c a l c u l a t e d P-X l o c u s and i n the l o c a t i o n s of q u a l i t y l i n e s . Such e r r o r s can l e a d t o poor r e s u l t s f o r c a l c u l a t e d phase d i s t r i b u t i o n s as noted i n Table IX. Moreover, the l i k e l i h o o d of e r r o r s i s i n c r e a s e d f o r cases where the d i s t o r t i o n i n q u a l i t y l i n e s i s made worse because of a poor f i t t o the e x p e r i m e n t a l s a t u r a t i o n p r e s s u r e l o c u s . The r e s u l t s f o r O i l F, shown i n F i g u r e 5, a r e an example of t h i s b e h a v i o r . A f u r t h e r example of t h i s b e h a v i o r i s shown i n F i g u r e 3. The c a l c u l a t e d s a t u r a t i o n l o c u s i s a g a i n more g e n t l y s l o p e d than the e x p e r i m e n t a l l o c u s and, so a g a i n , c u t s i n t o a r e g i o n t h a t e x p e r i m e n t a l l y c o n t a i n s two c o e x i s t i n g phases. Thus, too much o i l - r i c h l i q u i d i s expected. Less u n d e r s t a n d a b l e i s the same p r e d i c t i o n f o r O i l A. F i g u r e 4 shows t h a t the c a l c u l a t e d P-X l o c u s l i e s t o the r i g h t of the e x p e r i m e n t a l one. Too l i t t l e o i l - r i c h l i q u i d i s expected. Hence, the e r r o r i n the s l o p e of the q u a l i t y l i n e s e x i s t s even when the shape of the s a t u r a t i o n l o c u s i s c o r r e c t . E x t e n s i o n of the San Andres D e s c r i p t i o n t o A c i d G a s - O i l M i x t u r e s Because of i n t e r e s t i n hydrogen s u l f i d e mixed w i t h carbon d i o x i d e as a d i s p l a c i n g medium, t h i s s e c t i o n e x p l o r e s c a l c u l a t i o n of experi m e n t a l d a t a o b t a i n e d from m i x t u r e s of carbon d i o x i d e and hydrogen s u l f i d e i n p r o p o r t i o n s of 80-20 or 70-30 mol%. T a b l e XI r e v i e w s the a v a i l a b l e base o i l c o m p o s i t i o n s . Yarborough (5) and Core Labor a t o r i e s ( i n a p r i v a t e study f o r Amoco) have o b t a i n e d d a t a on 80-20 CO2/H2S mixtures. The phase b e h a v i o r observed by Core L a b o r a t o r i e s i s s h i f t e d s l i g h t l y toward h i g h e r a c i d gas c o m p o s i t i o n s due i n p a r t t o a s l i g h t d i f f e r e n c e i n the p r e p a r a t i o n of the a c i d gas; t h e i r a c i d gas was found t o c o n t a i n 21 mol% H 2 S w h i l e Yarborough had 20 mol%. I n an e a r l i e r Core L a b o r a t o r i e s study d a t a were o b t a i n e d f o r a 70-30 p r o p o r t i o n of C O 2 and H 2 S . Because t h i s Core L a b o r a t o r y study r e p o r t e d no c o m p o s i t i o n , an o i l of c o m p o s i t i o n s i m i l a r t o O i l DS i s assumed. As i n the carbon d i o x i d e - o i l s t u d i e s , the p h i l o s o p h y of the p r e s e n t approach was t h a t parameter adjustments f o r 0 1 I - H 2 S i n t e r a c t i o n s were a l l t h a t would be needed t o b r i n g about agreement between e x p e r i m e n t a l and c a l c u l a t e d v a l u e s . T a b l e s I and I I show the recommended parameters f o r 14 pseudocomponents; a t h r e e pseudocomponent d e s c r i p t i o n was not p r e p a r e d . Through t r i a l and e r r o r a d j u s t m e n t s , an o v e r a l l f i t t o the P-X l o c u s was o b t a i n e d f o r an 80-20 r a t i o . E x p e r i m e n t a l and c a l c u l a t e d r e s u l t s a r e compared i n

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

20.

CHABACK AND TUREK

Phase Behavior of San Andres Formation Oils

427

TABLE X I . COMPARISON OF SAN ANDRES OILS USED IN SOUR GAS STUDIES

Component N Ci C0 C HS c 2

2

2

2

3

1C4 nC4

iC nC c

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

5 5

6

C7+

Oil

O i l DSX

DS _

10.87 1.10 3.66 0.10 4.35 1.16 4.36 1.82 2.73 2.77 67.08

1.08 9.63 1.72 4.60 0.41 5.41 1.29 4.11 1.96 2.66 6.48 60.65

223 0.8863

232 0.8880

C7+ P r o p e r t i e s Mol Wt Sp Gr

F i g u r e 6. C a l c u l a t i o n s f o r F i g u r e 6 were made u s i n g t h e composit i o n o f O i l DS shown i n T a b l e X I . These parameters were then used to p r e d i c t t h e P-X l o c u s f o r t h e 70-30 m i x t u r e . Comparison o f c a l c u l a t e d and e x p e r i m e n t a l r e s u l t s a r e shown i n F i g u r e 7, u s i n g O i l DS c o m p o s i t i o n . Note t h a t f o r t h e 80-20 a c i d gas t h e three-phase r e g i o n i s found t o extend a c r o s s t h e e n t i r e c o m p o s i t i o n range from t h e s a t u r a t i o n l o c u s t o v i r t u a l l y pure a c i d gas. T h i s b e h a v i o r c o n t r a s t s w i t h t h e v e r y narrow r e g i o n o f three-phase e q u i l i b r i a observed f o r carbon d i o x i d e systems. F o r t h e 70-30 C0 /H S m i x t u r e three-phase b e h a v i o r i s a l s o n e a r l y as w e l l r e p r e s e n t e d w i t h t h e three-phase r e g i o n e x t e n d i n g almost t o t h e l i q u i d - l i q u i d b r a n c h . T a b l e X I I compares two-phase e q u i l i b r i u m r e s u l t s f o r a compos i t i o n near t h e d r o p l e t p o i n t o f about 71 mol% added 80/20 a c i d gas. Note t h e e x c e l l e n t agreement f o r phase s p l i t and e q u i l i b r i u m phase c o m p o s i t i o n s . The o i l - r i c h l i q u i d d e n s i t y i s i n e r r o r by about 2%, c h a r a c t e r i s t i c o f San Andres Formation o i l - C 0 m i x t u r e calculations generally. F i g u r e s 8 and 9 compare c o n s t a n t c o m p o s i t i o n v o l u m e t r i c expans i o n (CCVE) d a t a f o r t h e 80-20 and 70-30 C0 /H S m i x t u r e s . F o r t h e 80-20 c a s e , F i g u r e 8 d i s p l a y s CCVE b e h a v i o r f o r an a c i d gas concent r a t i o n below t h e observed m i x t u r e c r i t i c a l p o i n t . The c a l c u l a t i o n s f o r 25 and 50 mol% added a c i d gas ( u s i n g the O i l DSX c o m p o s i t i o n ) a r e i n good agreement w i t h e x p e r i m e n t a l l y observed v a l u e s . Howe v e r , f o r c o m p o s i t i o n s r i c h e r than t h e c r i t i c a l p o i n t (about 72 mol% added a c i d gas) CCVE b e h a v i o r i s n o t so w e l l r e p r e s e n t e d . As shown i n F i g u r e 6, t h e c a l c u l a t e d c r i t i c a l p o i n t i s t o o h i g h by 1800 p s i a . A s i m i l a r e r r o r i s expected f o r t h e 70-30 c a s e . T h i s l a r g e e r r o r d i s t o r t s t h e p r e d i c t e d q u a l i t y l i n e s away from 2

2

2

2

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND

428

APPLICATIONS

Legend: Calc Oil DS: 14 C7+ Comps • Β

Calc Critical Point: Oil DS Oil DS (Yarborough) Data Smoothed Data (DS a n d DSX) Smoothed Data (DS and DSX)

•

Expt Critical Point

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

Δ

I 20

I 40

I 60

I Θ0

Mole Percent Added Acid Gas F i g u r e 6. A c i d G a s - O i l DS and DSX Systems: Comparison o f C a l c u l a t e d V e r s u s E x p e r i m e n t a l S a t u r a t i o n P r e s s u r e L o c i a t 105 F f o r 20% H S/80% C 0 i n t h e A c i d Gas. 2

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Phase Behavior of San Andres Formation Oils

CHABACK AND TUREK

Legend:

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

Oil DS: 14 C7+ ComDS

%

•

Calc Critical Point: Oil DS

Δ

Core Lab Expt Data

2400-1

Δ

i

ι 0

1 Πτ^·—

;

ι

ι

ι

ι

20

40

60

80

τ 100

Mole Percent Added Acid Gas

F i g u r e 7. A c i d G a s - O i l DS System: Comparison o f C a l c u l a t e d Versus E x p e r i m e n t a l S a t u r a t i o n P r e s s u r e L o c i a t 105 F f o r 30% H S/70% C 0 i n t h e A c i d Gas. 2

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

430 TABLE X I I .

LIQUID-VAPOR EQUILIBRIUM COMPOSITIONS FOR OIL DS ACID GAS MIXTURE: COMPARISON OF EXPERIMENT WITH CALCULATIONS AT 105°F AND 1220 PSIA. BASIS: 71% ACID GAS: 80/20 C0 /H S RATIO 1

2

Component Ci C0 C HS c 2

2

2

3

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

1C4

nC4 c nC c C+ 5

5

6

7

Liquid Exp't Calc 3.26 52.20 1.59 15.13 1.79 0.41 1.44 0.44 0.54 1.72 21.48

2.94 56.04 1.06 14.28 1.30 0.35 1.31 0.55 0.83 0.84 20.50

223 0.8841

223 0.8863

0.826 0.824

0.844 0.839

2

Vap( Dr Exp't C a l c 9.54 76.90 1.11 9.79 0.83 0.17 0.50 0.18 0.20 0.24 0.57

7.14 76.05 1.00 14.08 0.65 0.12 0.37 0.11 0.13 0.08 0.27

7

7.7

C7+ P r o p e r t i e s Mole Wt. Sp. G r . L i q u i d Phase D e n s i t y (gm/cc) a t 1800 p s i a 1500 p s i a Volume P e r c e n t Liquid experimental

93

92.3

d a t a a r e taken from Yarborough ( 5 ) .

those o c c u r r i n g e x p e r i m e n t a l l y because t h e q u a l i t y l i n e s converge at the c r i t i c a l p o i n t . For example, i n F i g u r e 9 r e s u l t s f o r t h e 70-30 m i x t u r e a r e shown a t 80 and 90 mol% added a c i d gas. A l t h o u g h no c r i t i c a l p o i n t was determined e x p e r i m e n t a l l y f o r t h e 70-30 c a s e , based on t h e c r i t i c a l p o i n t l o c a t i o n f o r t h e 80-20 m i x t u r e t h e s e c o m p o s i t i o n s a r e l i k e l y t o l i e on t h e dew p o i n t s i d e o f t h e c r i t i c a l composit i o n . Because t h e c a l c u l a t e d c r i t i c a l p o i n t o c c u r s a t a composit i o n r i c h e r i n a c i d gas than expected e x p e r i m e n t a l l y , i t i s n o t s u r p r i s i n g t h a t a t 80 mol% added a c i d gas t h e c a l c u l a t e d r e s u l t s shown i n F i g u r e 9 t r e n d toward a bubble p o i n t , t h a t i s , 100 volume percent l i q u i d , w h i l e t h e e x p e r i m e n t a l d a t a show a t r e n d toward a dew p o i n t , t h a t i s , z e r o volume percent l i q u i d . A t 90 mol% added a c i d gas both t h e e x p e r i m e n t a l and c a l c u l a t e d c r i t i c a l compositions a r e exceeded, and a dew p o i n t i s now c a l c u l a t e d . But t h e c a l c u l a t e d dew p o i n t p r e s s u r e remains t o o h i g h .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

C H A B A C K A N D TUREK

Phase Behavior of San Andres Formation Oils

431

Le gend: 2 5 % Added Acid Gas: 14 C7+ Comps 5 0 % Added Acid Gas: 14 C7+ Comps 1200-4

•

25%

ι\dde d Aci d Ga s: Es[pt D ata

•

50%

i\dde d Ac id Ga s: E Jcpt D ata

i

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

1y

800

Ck

6 u 3

•

IX

8ooH

0

20

40

60

80

100

Volume Percent Liquid F i g u r e 8. A c i d G a s - O i l DSX System: Comparison o f C a l c u l a t e d Versus E x p e r i m e n t a l Phase D i s t r i b u t i o n s a t 105°F f o r 25% and 50% Added A c i d Gas w i t h 20% H S/80% C 0 i n t h e A c i d Gas. 2

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

432

Legend: 9 0 % Added Acid Gas: 14 C7+ Comps

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

8 0 % Added Acid Gas: 14 C7+ Comps •

9 0 % Added Acid Gas: Expt Data

Δ

8 0 % Added Acid Gas: Expt Data

• U

3 w

to a> u a,

m

/ •

Δ

•

0

/

Δ

20

40

80

Θ0

100

Volume Percent L i q u i d F i g u r e 9. A c i d G a s - O i l DS System: Comparison o f C a l c u l a t e d Versus E x p e r i m e n t a l Phase D i s t r i b u t i o n s a t 105 F f o r 80% and 90% Added A c i d Gas w i t h 30% H S/70% C 0 i n t h e A c i d Gas. 2

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

CH ABACK AND TUREK

20.

Phase Behavior of San Andres Formation Oils

433

Conclusions W i t h a p p r o p r i a t e parameter adjustments o f t h e type d i s c u s s e d h e r e , the m o d i f i e d Redlich-Kwong e q u a t i o n o f s t a t e i s c a p a b l e o f r e p r o d u c i n g t h e v o l u m e t r i c p r o p e r t i e s and complex phase e q u i l i b r i a e x h i b i t e d by San Andres Formation o i l - a c i d gas systems. The a d j u s t e d parameters l e a d t o p r e d i c t i o n o f l i q u i d - v a p o r c r i t i c a l i t y , a l t h o u g h the c a l c u l a t e d c r i t i c a l p r e s s u r e i s t o o h i g h by a s i g n i f i c a n t amount. Two-phase s a t u r a t i o n l o c i ( L - L o f L-V) a r e more a c c u r a t e l y reproduced than those f o r t h r e e phases ( L - L - V ) . F i n a l l y , changing the r e p r e s e n t a t i o n o f t h e C7+ f r a c t i o n from 14 t o 3 pseudocomponents does not s i g n i f i c a n t l y a f f e c t t h e p r e c i s i o n o f the c a l c u l a t e d results. G l o s s a r y o f Symbols Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch020

C

ij'

D

i j

=

i

n

t

e

r

a

c

t

i

o

n

parameters; 1 £ i £ NC, 1 £ j £ NC

C. .= C . and D. . = D.. iJ J !J J i NC = number o f components R = gas c o n s t a n t Ρ = pressure of f l u i d P £ = c r i t i c a l p r e s s u r e o f component i Τ = temperature o f f l u i d T ^ = c r i t i c a l temperature o f component i 1

c

V = s p e c i f i c volume o f f l u i d x^ = mole f r a c t i o n o f component i

Literature Cited 1. Yarborough, L. In "Equations of State in Engineering and Research"; Chao, K. C.; Robinson, R. L., Eds.; ADVANCES IN CHEMISTRY SERIES No. 182, American Chemical Society: Washington D. C., 1979, pp. 385-439. 2. Turek, Ε. Α.; Metcalfe, R. S.; Fishback, R. Ε. AIME Preprints 1984, Society of Petroleum Engineers Paper No. 13117. 3. Morris, R. W.; Turek, E. A. In "Equations of State Theories and Applications"; Chao, K. C.; Robinson, R. L., Eds.; ACS SYMPOSIUM SERIES, To be published, American Chemical Society: Washington, D. C., 1985. 4. Bergman, D. F.; Tek, M. R.; Katz, D. L. "Retrograde Condensa tion in Natural Gas Pipelines"; AGA Project PR-26-69, 1975; pp. 143-148. 5. Yarborough, L. Internal Amoco Production Co. report. 6. Standing, M. B.; Katz, D. L. AIME Pet. Div. Tech. 1942, 146, 159-165. 7. Edmister, W. C. Pet. Refiner 1958, 37, 173-178. RECEIVED November

8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

21 Application of Cubic Equations of State to Polar Fluids and Fluid Mixtures Gus K. Georgeton, Richard Lee Smith, Jr., and Amyn S. Teja

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100

The limitations of cubic equations of state for phase equilibrium predictions involving polar fluids have been widely attributed to the inability of these equations to correlate the vapor pressures and densities of these fluids. This work examines the behavior of two and three constant cubic equations (in particular, those of Peng and Robinson, and Patel and Teja) in calculations of vapor-liquid equilibria, liquid-liquid equilibria and critical states of mixtures containing polar components. I n s p i t e of t h e i r l i m i t a t i o n s , cubic equations of state are widely used i n phase e q u i l i b r i u m c a l c u l a t i o n s s i n c e they r e p r e s e n t a satisfactory compromise between accuracy and speed of computation. The t h e o r e t i c a l and p r a c t i c a l l i m i t a t i o n s o f these e q u a t i o n s have been d i s c u s s e d by a number o f workers i n c l u d i n g A b b o t t ( 1 ) and V i d a l (2). I n p a r t i c u l a r , the i n a b i l i t y o f twoc o n s t a n t c u b i c e q u a t i o n s to a c c u r a t e l y p r e d i c t l i q u i d d e n s i t i e s has been documented i n some d e t a i l . V i d a l ( 2 ) has reviewed a number o f three-cons t a n t c u b i c e q u a t i o n s and shown t h a t the a d d i t i o n o f a t h i r d c o n s t a n t i n g e n e r a l l e a d s t o improved d e n s i t y predictions. I n t h i s work, we have compared r e p r e s e n t a t i v e s o f two and three c o n s t a n t e q u a t i o n s f o r t h e i r a b i l i t y to c o r r e l a t e d e n s i t i e s , vapor p r e s s u r e s , v a p o r - l i q u i d e q u i l i b r i a , l i q u i d - l i q u i d e q u i l i b r i a and c r i t i c a l s t a t e s o f m i x t u r e s c o n t a i n i n g p o l a r components. Pure F l u i d C a l c u l a t i o n s As shown by A b b o t t , follows:

a g e n e r a l c u b i c e q u a t i o n may be w r i t t e n as α "~2

V

+ <5V + ε

0097-6156/86/0300-0434$06.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(1)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

21.

GEORGETON E T A L .

Polar Fluids and Fluid Mixtures

435

where p a r t i c u l a r c h o i c e s f o r a, 3, δ and ε f o r two of the more p o p u l a r two c o n s t a n t c u b i c e q u a t i o n s ( t h e Redlich-Kwong-Soave and the Peng-Robinson e q u a t i o n s ) a r e g i v e n i n T a b l e I . A l s o shown i n T a b l e I a r e these c o n s t a n t s f o r some r e c e n t t h r e e - c o n s t a n t c u b i c e q u a t i o n s , namely those proposed by Schmidt and Wenzel ( 3 ) , Harmens and Knapp (A), Hey en (_5) and Pa t e l and T e j a (6). These three c o n s t a n t e q u a t i o n s a f f o r d a means f o r g e n e r a l i z i n g the Soave and Peng-Robinson e q u a t i o n s and o f i n t e r p o l a t i n g among them. Thus, the Schmidt-Wenzel equation may be viewed as an i n t e r p o l a t i o n between the Soave e q u a t i o n (which g i v e s good l i q u i d d e n s i t y p r e d i c t i o n s f o r argon and methane, when ω * 0) and the Peng-Robinson equation (which gives good liquid density p r e d i c t i o n s f o r η-heptane, when ω * 1/3). S i m i l a r l y , the P a t e l T e j a e q u a t i o n reduces to the Soave e q u a t i o n when c = 0, to the Peng-Robinson e q u a t i o n when c = b and to the Schmidt-Wenzel e q u a t i o n when c = 3oib. Good d e n s i t y p r e d i c t i o n s can be o b t a i n e d because of t h i s added f l e x i b i l i t y and, as shown below, these p r e d i c t i o n s a l s o extend to p o l a r components. As was demonstrated by Soave ( 7_), c u b i c e q u a t i o n s may be used s u c c e s s f u l l y i n phase e q u i l i b r i u m c a l c u l a t i o n s i f the c o n s t a n t ' a i s made temperature dependent and i f t h i s temperature dependence i s o b t a i n e d from the vapor p r e s s u r e of the pure component. Two of the common temperature f u n c t i o n s proposed f o r 'a' a r e d i s c u s s e d below. The f i r s t i s a q u a d r a t i c , o r i g i n a l l y proposed by Soave and l a t e r used s u c c e s s f u l l y i n the Peng-Robinson and o t h e r e q u a t i o n s of s t a t e . T h i s i s g i v e n by: 1

α [T ]

= a {l + F(l - T

R

c

1 / 2 R

)}

2

(2)

Both Soave and Peng and Robinson, however, c o r r e l a t e d F w i t h the a c e n t r i c f a c t o r ω and t h e i r e q u a t i o n s t h e r e f o r e worked b e s t f o r n o n p o l a r substances. P a t e l and T e j a (6) l a t e r showed t h a t i f F i s determined from the a c t u a l vapor p r e s s u r e s of the pure components, then the e q u a t i o n may a l s o be used f o r p o l a r s u b s t a n c e s . The functional form of the temperature dependence i s , however, i n c o r r e c t a t h i g h reduced temperatures — s i n c e <*[T ] must decrease monotonously w i t h T . Heyen suggested an e x p o n e n t i a l form f o r a [ T ] which i s g i v e n by: R

R

R

a [T ] = a R

c

exp

n

{Κ (1 - T ) }

(3)

R

T h i s form i s u s e f u l f o r s u p e r c r i t i c a l gases such as H and He and, i n d e e d , because of the a d d i t i o n a l c o n s t a n t a v a i l a b l e f o r f i t t i n g , i t c o r r e l a t e s vapor p r e s s u r e s of many p o l a r compounds e x t r e m e l y w e l l ( T a b l e I I ) . However, as shown i n F i g u r e 1 f o r ammonia, the two f u n c t i o n a l forms ( i n c l u d i n g t h e i r f i r s t , second and third d e r i v a t i v e s w i t h temperature) show s i m i l a r b e h a v i o r over most p r a c t i c a l ranges of temperature. Moreover, as w i l l be shown below, any s m a l l improvement i n pure component vapor p r e s s u r e r e p r e s e n t a t i o n i s not always apparent when m i x t u r e phase b e h a v i o r is calculated. For most p r a c t i c a l purposes, t h e r e f o r e , the two forms a r e e q u i v a l e n t i n t h e i r a b i l i t y to p r e d i c t phase e q u i l i b r i a . In order to compare the correlative and predictive c a p a b i l i t i e s of the two and three c o n s t a n t c u b i c s we show below 2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

436

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

TABLE I . C h o i c e o f C o n s t a n t s f o r the General Cubic Equation

2-constant SRK

a[T]

b

b

0

PR

a[T]

b

2b

- b

SW

a[T]

b

b + 3ajb

HK

a[T]

b

be

H

a[T]

b[T]

PT

a[T]

b

2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

3-constant

b[T] + c

b + c

2

- 3u>b

(c-l)b

2

- b[T] c

- be

TABLE I I . The E f f e c t o f D i f f e r e n t Forms of α on Vapor P r e s s u r e s

Compound

PT-1

PT-2

Cons t a n t *

Cons t a n t * *

PR-AP(AAD%)

ΔΡ(ΑΑΡ%)

ΔΡ(ΑΑΡ%)

1-Bu t a n o l

10.81

7.13

1.33

1-Pentanol

14.70

10.17

0.47

1-Octanol

17.96

9.64

2.98 1.32

Acetic Acid

6.69

2.36

Buta note A c i d

7.69

8.56

4.54

Octanoic Acid

12.71

5.87

1.33

*

P a t e l - T e j a e q u a t i o n w i t h Soave-type f u n c t i o n f o r ct[T J c o n t a i n i n g 1 c o n s t a n t .

**

P a t e l - T e j a e q u a t i o n w i t h Heyen-type f u n c t i o n f o r ot[T ] c o n t a i n i n g 2 c o n s t a n t s .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

G E O R G E T O N ET A L .

F i g u r e 1.

Polar Fluids and Fluid Mixtures

Comparison of the Two Forms o f α f o r Ammonia.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

438

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

the r e s u l t s o f our c a l c u l a t i o n s o f l i q u i d densities, vapor p r e s s u r e s , v a p o r - l i q u i d and l i q u i d - l i q u i d e q u i l i b r i a , and c r i t i c a l p o i n t s u s i n g the Peng-Robinson (PR) and P a t e l - T e j a (PT) e q u a t i o n s . The P a t e l - T e j a e q u a t i o n may ρ _lî V - b

be w r i t t e n a s :

a[T] V(V + b) + c(V - b)

=

The e q u a t i o n of s t a t e c o n s t a n t s may p o i n t by s e t t i n g :

φ

τ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

=0 ;!p= e

c

4 v

be o b t a i n e d from

=0 ; c

(

?

the

)

'

critical

(5)

c

The t h i r d c o n s t a n t i n the e q u a t i o n of s t a t e a l l o w s ζ to be chosen f r e e l y , u n l i k e the case of the two c o n s t a n t e q u a t i o n s . Acceptable p r e d i c t i o n of both low and h i g h p r e s s u r e d e n s i t i e s r e q u i r e s t h a t , in g e n e r a l , ζ be g r e a t e r than the experimental critical c o m p r e s s i b i l i t y ( A b b o t t ( 1 ) ) and we have c o n f i r m e d t h i s f o r a l a r g e number of s u b s t a n c e s . A p p l i c a t i o n of e q u a t i o n (5) l e a d s t o : 2

2

a

- Ω (R Τ /Ρ ) c a c e

(6)

b

= Q (RT /P )

(7)

c

- Ω (RT /Ρ ) c c e

(8)

« 1 - 3ζ„ c

(9)

b

c

c

where Β c Ω

and

3ζ

a

2 β

+ 3(1 - 2ζ

+ Ω^

+ 1 - 3ζ

β

(10)

Ω^ i s the s m a l l e s t p o s i t i v e r o o t o f :

S

3

+

<

2

-

3

ç

c

>

a

b

2

+

3

ç

c

2

a

b -

Ç

c

3

=

0

It s h o u l d be noted here t h a t i f ζ = 0.3074, these equations reduce t o the Peng-Robinson e q u a t i o n and i f ζ = 0.3333, they reduce to the Soave e q u a t i o n . The P a t e l - T e j a e q u a t i o n t h e r e f o r e r e t a i n s many of the u s e f u l f e a t u r e s of the Soave and Peng-Robinson equations.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

21.

GEORGETON E T A L .

439

Polar Fluids and Fluid Mixtures

I f , f o r c o n v e n i e n c e , we choose the Soave-type f u n c t i o n f o r <*[T ] ( e q u a t i o n 2) then the P a t e l - T e j a e q u a t i o n c o n t a i n s f o u r substance-specific parameters Τ ,Ρ , ζ and F ( f o r nonpolar substances, and F c a n be c o r r e l a t e d w i t h ω, so t h a t the e q u a t i o n c o n t a i n s o n l y t h r e e subs t a n c e - s p e c i f i c parameters, as shown elsewhere ( 6 ) ) . I n g e n e r a l , ζ and F a r e o b t a i n e d from a f i t of vapor p r e s s u r e and s a t u r a t e d l i q u i d d e n s i t y d a t a . D e t a i l s o f the o b j e c t i v e f u n c t i o n s and f i t t i n g procedures a r e g i v e n elsewhere (6). The c o n s t a n t s a r e g i v e n i n T a b l e I I I f o r 27 s u b s t a n c e s , i n c l u d i n g p o l a r substances such as a l c o h o l s and a c i d s , a l o n g w i t h a b s o l u t e average d e v i a t i o n s i n vapor p r e s s u r e s and s a t u r a t e d l i q u i d d e n s i t i e s f o r both the PT and PR e q u a t i o n s o f s t a t e . T a b l e IV c o n t a i n s v a l u e s of F and ζ f o r compounds determined p r e v i o u s l y (6_,8_). I t i s i n t e r e s t i n g to note t h a t ζ i s found to be c l o s e to the SRK v a l u e o f 0.3333 f o r s m a l l s p h e r i c a l m o l e c u l e s such as a r g o n , methane, oxygen and n i t r o g e n ; and that i t I s c l o s e to the PR v a l u e of 0.3074 f o r moderately l a r g e hydrocarbons such as nheptane. What i s perhaps s u r p r i s i n g i s t h a t a v a l u e c l o s e to 0.3074 i s o b t a i n e d f o r many moderately s i z e d m o l e c u l e s , i n c l u d i n g those t h a t a r e p o l a r . The PR and PT e q u a t i o n s t h e r e f o r e g i v e v e r y s i m i l a r p r e d i c t i o n s of l i q u i d d e n s i t i e s (and o f d e n s i t y - d e r i v e d p r o p e r t i e s such as p a r t i a l molar volumes) f o r moderately s i z e d m o l e c u l e s . A s i g n i f i c a n t improvement i n l i q u i d d e n s i t y p r e d i c t i o n i s seen when the PT e q u a t i o n i s used to c a l c u l a t e d e n s i t i e s o f s u b s t a n c e s whose ζ i s v e r y d i f f e r e n t from 0.3074. T h i s i s shown i n F i g u r e 2 f o r wa 8er and i n F i g u r e 3 f o r methanol-water m i x t u r e s . T a b l e I I I shows comparisons between the vapor p r e s s u r e s and l i q u i d d e n s i t i e s o f p o l a r m o l e c u l e s p r e d i c t e d by the PR and PT e q u a t i o n s . Not u n e x p e c t e d l y , a l l o w i n g F to be determined from the a c t u a l vapor p r e s s u r e , r a t h e r than the vapor p r e s s u r e a t T =0.7 ( i . e . from the a c e n t r i c f a c t o r ) and an n-alkane c o r r e l a t i o n , l e a d s to s i g n i f i c a n t improvements i n p r e d i c t i o n s , e s p e c i a l l y f o r p o l a r f l u i d s . Vapor p r e s s u r e p r e d i c t i o n s can be f u r t h e r improved i f the e x p o n e n t i a l form ( w i t h two s u b s t a n c e - s p e c i f i c c o n s t a n t s ) i s used, as shown i n T a b l e I I . However, f o r many substances o v e r p r a c t i c a l ranges o f reduced temperature, the two forms o f a [ T ] g i v e equivalent results. A l l c a l c u l a t i o n s shown below therefore u t i l i z e the Soave-type temperature f u n c t i o n w i t h one s u b s t a n c e s p e c i f i c c o n s t a n t F. One d i s a d v a n t a g e o f the PT e q u a t i o n i s t h a t F (and ζ ) must be known f o r a l l pure f l u i d s o f i n t e r e s t . For nonpolar f l u i d s , these c o n s t a n t s can be r e l a t e d to the a c e n t r i c f a c t o r as f o l l o w s :

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

0

R

R

2

F

= 0.452413 + 1.30982ω -0.295937ω

ζ - 0.329032 - 0.0767992ω + 0.0211947ω c

(12)

(13)

For these f l u i d s , the PT e q u a t i o n t h e r e f o r e uses the same i n p u t i n f o r m a t i o n as the PR e q u a t i o n ( b u t i s b e t t e r a b l e to p r e d i c t densities of large molecules f o r w h i c h ζ < 0.3074). These c o r r e l a t i o n s do n o t , however, h o l d f o r p o l a r f l u i d s . We have t h e r e f o r e attempted to develop such c o r r e l a t i o n s f o r c l a s s e s o f

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

440

TABLE I I I .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

Substance

Substance Dependent Constants ζ and £

*c

F

F for Several Compounds

ρ-Τ ΔΡ(AAD%) ~Δρ(ΑΑΌ%)

P-R AP(AAD%) ~Δρ(ΑΑΌ%) 15.12

Ammonia

0.283

0.642740

1.80

3.13

0.74

Benzene

0.311

0.698911

1.70

3.30

1.97

3.45

Toluene

0.306

0.753893

0.72

3.37

0.93

3.05

0-Xylene

0.305

0.812845

0.70

3.40

0.76

3.66

m-Xylene

0.301

0.816962

0.87

3.44

0.83

5.49

p-Xylene

0.300

0.807023

0.82

3.44

0.92

Me thanol

0.274

0.965347

1.96

3.87

5.39

21.73

Ethanol

0.292

1.171714

1.13

3.99

1.12

10.89

1.211304

4.95

3.79

5.23

7.13

3.41

10.81

3.98

14.70

2.88

1-Propanol 1-Butanol

0.302 0.305

1.221182

5.73

5.46

1-Pentanol

0.308

1.240459

10.17

3.08

1-Hexanol

0.330

1.433586

8.42

8.29

14.64

13.02

1-Heptanol

0.301

1.215380

13.62

3.31

22.16

5.95

1-Octanol

0.308

1.270267

9.64

3.44

17.96

2.70

Acetic Acid

0.258

0.762043

2.36

3.32

6.69

32.62

Propanoic Acid

0.295

1.146553

5.64

4.80

5.25

42.81

Butanoic Acid

0.329

1.395151

8.56

5.30

7.69

8.88

Pentanoic Acid

0.292

1.174746

5.58

3.04

7.54

10.48

Hexanoic Acid

0.291

1.272986

5.56

3.44

36.73

Octanoic Acid

0.292

1.393678

5.87

2.93

12.71

10.96

Decanoic Acid

0.290

1.496554

3.83

4.65

18.33

11.74

Acetone

0.283

0.701112

1.34

3.23

2.66

0.308

0.787322

0.60

3.67

0.65

0.314

0.694866

1.16

3.92

1.32

4.33

3.45

1.60

8.49

Diethyl Ether

8.47

15.73 3.19

Carbon Tetrachloride Ethyl Acetate

0.296

0.842965

2.12

nPropyl Acetate

0.295

0.882502

2.70

3.17

2.35

8.39

1.39

2.95

1.27

8.79

Diethyl Ketone

0.294

0.826046

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

21. GEORGETON ET AL.

Polar Fluids and Fluid Mixtures

441

TABLE IV. C o m p i l a t i o n o f P r e v i o u s l y Determined V a l u e s of ζ and F c

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

Compound Argon Nitrogen Oxygen Me thane Ε thane Ε thylene Propane Propylene Acetylene n- Buta ne i-Bu tane 1-Butene n-Pentane i-Pentane n-Hexane η-Hep tane n-Oc tane n-Nonane n-Decane n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Heptadecane n-Octadecane n-Eicosane Carbon D i o x i d e Carbon Monoxide Sulfur Dioxide Hydrogen S u l f i d e Water Cyclohexane Quinoline m-Cresol D i p h e n y l Methane 1-Methyl Naphthalene

Ç

c

0.328 0.329 0.327 0.324 0.317 0.313 0.317 0.324 0.310 0.309 0.315 0.315 0.308 0.314 0.305 0.305 0.301 0.301 0.297 0.297 0.294 0.295 0.291 0.283 0.276 0.277 0.309 0.328 0.310 0.320 0.269 0.303 0.310 0.300 0.305 0.297

F 0.450751 0.516798 0.487035 0.455336 0.561567 0.554369 0.648049 0.661305 0.664179 0.678389 0.683133 0.696423 0.746470 0.741095 0.801605 0.868856 0.918544 0.982750 1.021919 1.080416 1.115585 1.179982 1.188785 1.297054 1.276058 1.409671 0.707727 0.535060 0.797391 0.583165 0.689803 0.665434 0.859036 1.000087 1.082667 0.827417

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

F i g u r e 3. Comparison o f the S a t u r a t e d L i q u i d D e n s i t i e s f o r Methanol-Water M i x t u r e s a t 298.15K.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

21.

Polar Fluids and Fluid Mixtures

GEORGETON ETAL.

443

p o l a r f l u i d s by examining the d i f f e r e n c e s between F and ζ of these f l u i d s and those of n-alkanes c o n t a i n i n g the same number of carbon atoms. ( T h i s approach i s s i m i l a r i n p r i n c i p l e to the homomorph concept of Bondi and S i m k i n ( 9 ) ) . The normal a l k a n e s show a g r e a t r e g u l a r i t y i n t h e i r b e h a v i o r w i t h the number of carbon atoms. P r o p e r t i e s such as T , P , F and c can t h e r e f o r e be c o r r e l a t e d a c c u r a t e l y w i t h the number o f c a r b o n atoms i n the m o l e c u l e . T h i s has been demonstrated f o r Τ and P elsewhere ( 10) and i s shown f o r F and ζ i n F i g u r e s A and 5. P r o p e r t i e s of r e l a t e d f l u i d s — such as i s o p a r a f f i n s , o l e f i n s , n a p t h e n i c compounds — can then be determined from the n-alkane c o r r e l a t i o n s i f an e f f e c t i v e carbon number (ECN) i s used. The ECN i s the number of carbon atoms i n an n-alkane h a v i n g the same normal b o i l i n g p o i n t as the f l u i d o f i n t e r e s t . The ECN c o n c e p t has been used s u c c e s s f u l l y by Ambrose (11) and more r e c e n t l y by Chase (12) and W i l l m a n and T e j a (10) to p r e d i c t vapor p r e s s u r e s . I t s h o u l d be added t h a t the ECN can be, and u s u a l l y i s , noni n t e g r a l f o r compounds o t h e r than the n - a l k a n e s . S i n c e p o l a r f l u i d s do n o t f o l l o w the n-alkane c o r r e l a t i o n s , we have p l o t t e d F and ζ f o r c l a s s e s of these f l u i d s ( a l c o h o l s , c a r b o x y l i c a c i d s ) a g a i n s t carbon number and these c o r r e l a t i o n s a r e a l s o shown i n F i g u r e s A and 5. Not s u p r i s i n g l y , the n-alkane and p o l a r s e r i e s c o r r e l a t i o n s e x h i b i t the l a r g e s t d i f f e r e n c e s a t low carbon numbers. As the carbon number i n c r e a s e s (beyond n-hexanol i n the case of the a l c o h o l s e r i e s and η-butyric a c i d i n the case of the c a r b o x y l i c a c i d s e r i e s ) , the c u r v e s approach the n-alkane correlation. In general, the d i f f e r e n c e between F and ζ f o r p o l a r s u b s t a n c e s and those f o r the n-alkanes c o n t a i n i n g the same number of carbon atoms decreases as the hydrocarbon p a r t of a p o l a r m o l e c u l e becomes more dominant. These d i f f e r e n c e s can a l s o be correlated with the ECN. Figures 6 and 7 show plots of ψ and θ where these a r e d e f i n e d by: C

C

c

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

c

* =

( F

θ

Ç

/ F

polar HC> "

1

( 1 4 )

and ( 1 5 )

- c,HC * " «c.polar

As can be r e a d i l y seen from the diagrams, ψ and θ c o r r e l a t e w e l l w i t h the ECN. T h e r e f o r e , f o r any pure component, the f o u r parameters of the PT e q u a t i o n a r e o b t a i n e d w i t h o n l y a knowledge o f the ECN, w h i c h i s o b t a i n e d from the b o i l i n g p o i n t . T and P are obtained directly for each compound, as in Reference 10, w i t h ζ and F o b t a i n e d d i r e c t l y from c o r r e l a t i o n s o n l y f o r the alkanes. F o r a l c o h o l s and a c i d s , the base v a l u e s ( a l k a n e s ) a r e determined a l o n g w i t h the c o r r e c t i o n s due to the polarity effects. These v a l u e s a r e c a l c u l a t e d from c o r r e l a t i o n s w i t h ECN, and used to g e n e r a t e the ζ and F f o r the p o l a r compounds u s i n g e q u a t i o n s 1A and 15. I t i s expected t h a t the c o n s t a n t s f o r secondary and t e r t i a r y a l c o h o l s , f o r example, may be o b t a i n e d from c

c

C

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

444

COMPOUND CLASSES _AUCANES ALCOHOLS ACIDS

,

—Γ

Η

8

F i g u r e 4.

ECN

C o r r e l a t i o n o f F f o r S e v e r a l C l a s s e s o f Compounds.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

G E O R G E T O N ET A L .

Polar Fluids and Fluid Mixtures

COMPOUND CLASSES ALCOHOLS ACIDS 2

4

β

8

10

12

14

ECN F i g u r e 6. C o r r e l a t i o n o f the C o r r e c t i o n Parameter ψ f o r A l c o h o l s and A c i d s .

CD

!

COMPOUND CLASSES ALCOHOLS ACIDS 2

4

6

8

10

12

14

ECN F i g u r e 7. C o r r e l a t i o n o f the C o r r e c t i o n Parameter θ f o r A l c o h o l s and A c i d s .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

446

those of the η-alcohols u s i n g the ECN concept. T h i s approach t h e r e f o r e a l l o w s g e n e r a l i z e d c o n s t a n t s to be developed f o r c l a s s e s of p o l a r f l u i d s . Mixture Calculations E q u a t i o n 4 can be used f o r the c a l c u l a t i o n of m i x t u r e p r o p e r t i e s and phase e q u i l i b r i a i f the m i x t u r e c o n s t a n t s > » > c a l c u l a t e d u s i n g the m i x i n g r u l e s : a

b

m

a

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

a r e

m

m

- ΣΣ x , x . a . i j i j

(16)

m

m

= Σ x,b. i i

(17)

m

= Σ x.c. I i

(18)

m

b

c

c

m

4

The c h o i c e of t h i s model i s , of c o u r s e , c o m p l e t e l y a r b i t r a r y — p a r t i c u l a r l y i n the case of c . We have chosen a l i n e a r m i x i n g r u l e f o r c because i t i s s i m p l e and a l s o because, s e t t i n g c = b a l l o w s the e q u a t i o n s to reduce to the PR case. T h i s would n o t be p o s s i b l e i f a d i f f e r e n t m i x i n g r u l e i s used f o r c . The c r o s s - i n t e r a c t i o n term a ^ j i n e q u a t i o n (16) i s e v a l u a t e d u s i n g the combining r u l e : m

m

·«

-hi

(

3

ϋ

3

^

)

1

/

2

m

( 1 9 )

where ξ i s a b i n a r y i n t e r a c t i o n parameter which i s e v a l u a t e d from e x p e r i m e n t a l d a t a . The PT and PR e q u a t i o n s were used to c a l c u l a t e v a p o r - l i q u i d e q u i l i b r i a f o r a number of b i n a r y m i x t u r e s c o n t a i n i n g p o l a r components. A s i n g l e b i n a r y i n t e r a c t i o n c o e f f i c i e n t was used i n each c a s e . O v e r a l l , the e r r o r s i n the c o r r e l a t i o n of bubble p o i n t p r e s s u r e s and vapor phase c o m p o s i t i o n s c o u l d be reduced to s i m i l a r v a l u e s f o r these two e q u a t i o n s w i t h the use of b i n a r y i n t e r a c t i o n coefficients. However, as shown i n F i g u r e 8, the VLE curve c a l c u l a t e d u s i n g the PR e q u a t i o n f o r a system such as t o l u e n e + 1p e n t a n o l i s d i s t o r t e d because of the i n a b i l i t y of the PR e q u a t i o n to p r e d i c t the vapor p r e s s u r e of 1-pentanol a c c u r a t e l y . The d i s t o r t i o n i n the P-x-y diagram i s reduced c o n s i d e r a b l y when the PT e q u a t i o n i s used and o b v i o u s l y depends on how well the function a[T ] represents the vapor pressure of the pure components. Thus the g r e a t e s t d i s t o r t i o n w i l l o c c u r when the PR e q u a t i o n i s used to c a l c u l a t e VLE f o r systems c o n t a i n i n g 1h e x a n o l , 1 - h e p t a n o l , 1 - o c t a n o l , hexanoic a c i d , d e c a n o i c acid, o c t a n o i c a c i d , e t c . as can r e a d i l y be determined from T a b l e 3. T h i s i s a l s o shown i n F i g u r e 9 f o r the cyclohexane + 1-hexanol system. Not s u r p r i s i n g l y , improved d e n s i t y r e p r e s e n t a t i o n has little e f f e c t on the VLE calculations. However, a t h i g h p r e s s u r e s , d e n s i t y e f f e c t s s h o u l d become s i g n i f i c a n t a l t h o u g h v e r y few h i g h p r e s s u r e data a r e a v a i l a b l e f o r m i x t u r e s c o n t a i n i n g p o l a r components w i t h ζ « 0.3074. R

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Polar Fluids and Fluid Mixtures

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

G E O R G E T O N ET A L .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

448

The PT and nt e q u a t i o n s were a l s o used to c a l c u l a t e l i q u i d l i q u i d e q u i l i b r i a f o r s e v e r a l b i n a r y systems c o n t a i n i n g p o l a r components. A g a i n , the e f f e c t of improved d e n s i t i e s was not apparent i n the c a l c u l a t e d LLE c u r v e s . However, both e q u a t i o n s were v e r y poor i n t h e i r a b i l i t y to p r e d i c t LLE b e h a v i o r w i t h the van der Waals m i x i n g r u l e s , as can be seen i n F i g u r e 10 f o r the methanol-heptane system and i t is entirely possible that s i g n i f i c a n t d i f f e r e n c e s would be observed w i t h , f o r example, d e n s i t y - d e p e n d e n t m i x i n g r u l e s . Note t h a t both e q u a t i o n s of s t a t e p r e d i c t one l i q u i d phase which i s e s s e n t i a l l y pure methanol. The e f f e c t s of improved d e n s i t i e s can be r e a d i l y observed when c r i t i c a l l o c i a r e c a l c u l a t e d . T h i s has been d i s c u s s e d i n d e t a i l elsewhere ( 1 3 ) . We show here our c a l c u l a t i o n s f o r the argon + water system ( F i g u r e s 11-12). Both the PR and the PT e q u a t i o n c o r r e l a t e the T vs x^ and the P vs T b e h a v i o r i n t h i s h i g h l y n o n - i d e a l system f a i r l y w e l l up to p r e s s u r e s of 50 MPa. The V vs x^ b e h a v i o r i s , however, c o n s i d e r a b l y b e t t e r p r e d i c t e d by the PT e q u a t i o n , as e v i d e n c e d i n F i g u r e 11. One consequence o f t h i s i s obvious i f the P ~ V ~ T c u r v e i s examined ( F i g u r e 12). As can be seen from the diagram, the P ~ V ~ T c u r v e s (and hence the PVTx s u r f a c e s ) c a l c u l a t e d u s i n g the PR and PT e q u a t i o n s a r e d i s t o r t e d near the pure component ends of the c u r v e s due to t h e i r poor r e p r e s e n t a t i o n of pure-subs tance c r i t i c a l volumes. This d i s t o r t i o n i s g r e a t e r f o r the PR e q u a t i o n than f o r the PT e q u a t i o n and cannot be overcome s i m p l y by s e t t i n g ζ = Ζ ^ (13) s i n c e t h i s d i s t o r t s the r e s t of the c u r v e . Thus, h i g h p r e s s u r e phase e q u i l i b r i a — p a r t i c u l a r l y i n the c r i t i c a l region — must r e f l e c t t h i s d i s t o r t i o n i n the PVTx surface. U n f o r t u n a t e l y , few d a t a e x i s t i n t h i s r e g i o n f o r p o l a r m i x t u r e s . N e v e r t h e l e s s , we can say w i t h some j u s t i f i c a t i o n t h a t a three-cons t a n t c u b i c e q u a t i o n i s b e t t e r a b l e to p r e d i c t the p r o p e r t i e s of m i x t u r e s a t h i g h p r e s s u r e s when d e n s i t y e f f e c t s a r e significant. I t may a l s o be advantageous to use t h r e e c o n s t a n t cubics w i t h density-dependent m i x i n g r u l e s s i n c e the o v e r a l l r e p r e s e n t a t i o n of the PVTx b e h a v i o r i s more r e a l i s t i c than i n the case o f two-constant c u b i c s .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

c

c

c

c

C

C

C

C

C

C

fc

e x

Conclusions I n t h i s work, the Peng-Robinson (PR) and P a t e l - T e j a (PT) e q u a t i o n s were used as r e p r e s e n t a t i v e s of the c l a s s of c u b i c e q u a t i o n s o f state. Pure component vapor p r e s s u r e s and d e n s i t i e s , and VLE and LLE f o r m i x t u r e s of p o l a r compounds were examined a l o n g w i t h the a p p l i c a b i l i t y of the e q u a t i o n s i n the c r i t i c a l region. In g e n e r a l , c u b i c e q u a t i o n s of s t a t e can be extended r e a d i l y to p o l a r components i f actual vapor pressure data are used to obtain a [ T ] . However, the c h o i c e of f u n c t i o n a l form of α i s unimportant a t moderate c o n d i t i o n s a t which most d a t a e x i s t . A s i g n i f i c a n t improvement i n d e n s i t i e s i s o b t a i n e d i f the c u b i c e q u a t i o n uses a t h i r d c o n s t a n t (as i n the PT e q u a t i o n ) , which i s related to a c a l c u l a t e d c r i t i c a l compressibility. Separate c o r r e l a t i o n s f o r d e t e r m i n i n g substance dependent parameters are necessary f o r p o l a r compounds, but these parameters can be R

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

21.

G E O R G E T O N ET A L .

Polar Fluids and Fluid Mixtures

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

LU Ο DC d

PT EQUATION f-0.923 PR EQUATION f-0.928 Δ EXPERIMENTAL (16) 0.0

0.2

0.4

0.6

O.fi

METHANOL COMPOSITION F i g u r e 1 0 . LLE i n the Methanol-Heptane System.

• ο *

CD

, Ε

« q c ο

_PT EQUATION £-0.56 PR EQUATION £-0.65 EXPERIMENTAL (17) ,

,

!

,

,

,

,

1

0.026 0.060 0.075 0.1O0 0.125 0.150 0.176 0.200 0.225

ARGON MOLE FRACTION F i g u r e 1 1 . V ~ x P r o j e c t i o n o f the Argon-Water System i n the C r i t i c a l Region. c

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

449

450

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

P

e

(MPa)

F i g u r e 12. PVT Diagram o f the Argon-Water System i n the C r i t i c a l Region.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

21.

GEORGETON ETAL.

Polar Fluids and Fluid Mixtures

451

o b t a i n e d w i t h o n l y a knowledge o f the b o i l i n g p o i n t . The e f f e c t of improved d e n s i t i e s i s n o t r e f l e c t e d i n low p r e s s u r e VLE and LLE p r e d i c t i o n s i n terms o f a b s o l u t e d e v i a t i o n s . However, f o r VLE, the curves p r e d i c t e d w i t h the PT e q u a t i o n a r e l e s s skewed than those p r e d i c t e d w i t h the PR e q u a t i o n . An improvement due t o better density representation might be n o t i c e d i f density dependent m i x i n g r u l e s are used. The c r i t i c a l l o c u s , n e v e r t h e l e s s shows the d r a m a t i c e f f e c t o f improved d e n s i t i e s , p a r t i c u l a r l y i f viewed i n three dimensions.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch021

Literature Cited 1. Abbott, M. M. ADVANCES IN CHEMISTRY SERIES No.182, American Chemical Society; 1979; p. 47. 2.

Vidal, J. Fluid Phase Equilibria, 1983, 13, 15.

3.

Schmidt, G.; Wenzel, H. Chem. Eng. Sci., 1980, 35, 1503.

4.

Harmens, Α.; Knapp, H. Ind. Eng. Chem., Fund. 1980, 19, 291.

5.

Heyen, G. "Phase Equilibria and Fluid Properties in the Chemical Industry", 2nd International Conference Berlin (West), Proceedings, Part 1 1980.

6.

Patel, N. C.; Teja, A. S. Chem. Eng. Sci. 1982, 37, 463.

7.

Soave, G. Chem. Eng. Sci. 1972 27, 1197.

8.

Han, C. H. M.S. Thesis, Georgia Institute of Technology, Atlanta, 1984.

9.

Bondi, Α.; Simkin, D. J. AIChE J. 1957, 3, 473.

10. Willman, B. T.; Teja, A. S. Ind. Eng. Chem., Proc. Des. Dev. In Press. 11. Ambrose, D. NPL Report Chem. 57, Teddington, England, 1976. 12. Chase, J. D. Chem. Eng. Prog. 1984, 80 (4), 63. 13. Teja, A. S.; Smith, R. L.; Sandler, S. J. Fluid Phase Equilibria 1983, 14, 265. 14.

Carr, C.; Riddick, J. A. Ind. Eng. Chem. 1951, 43, 692.

15. Gmehling, J . ; Onken, U.; Arlt, W. "Vapor-Liquid Equilibrium Data Collection", Vol I, Part 2b, DECHEMA, Frankfurt, Germany, 1978. 16.

Sorensen, J. M.; Arlt, W. "Liquid-Liquid Equilibrium Data Collection", Vol V, Part 1, DECHEMA, Frankfurt, Germany, 1979.

17. Hicks, C. P.; Young, C. L. Chem. Rev. 1975, 75, 119. RECEIVED

November 4, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

22 Parameters from Group Contributions Equation and Phase Equilibria in Light Hydrocarbon Systems 1

2

Ali I. Majeed and Jan Wagner 1

Nawazish International, Wetumka, OK 74883 School of Chemical Engineering, Oklahoma State University, Stillwater, OK 74078

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

2

The Parameters From Group Contribution equation of state is applied to pure fluids and mixtures with emphasis on the representation of phase equilibria. Group parameters and group-interaction parameters were derived from published pure component data using a nonlinear, multiproperty fitting program. Comparisons of calculated and experimental vapor-liquid equilibrium phase compositions, volumetric properties, and enthalpy departures demonstrated the applicability of the PFGC equation to systems of hydrocarbons and hydrocarbons with water. Phase behavior of mixtures of light hydrocarbons with acid gases is also described over a fairly wide range of temperature and pressure. The use of e q u a t i o n s of s t a t e t o d e s c r i b e t h e phase b e h a v i o r and thermodynamic p r o p e r t i e s of l i g h t hydrocarbon systems i s w e l l e s t a b l i s h e d . Among t h e more w i d e l y used c o r r e l a t i o n s are t h e Soave-Redlich-Kwong (1), Peng-Robinson (2), and S t a r l i n g - B e n e d i c t Webb-Rubin (3) e q u a t i o n s . These e q u a t i o n s were developed t o d e s c r i b e t h e b e h a v i o r of nonpolar o r weakly p o l a r s u b s t a n c e s . When r e s t r i c t e d t o t h e s e types of systems a l l t h r e e e q u a t i o n s y i e l d phase b e h a v i o r p r e d i c t i o n s s u i t a b l e f o r many a p p l i c a t i o n s i n process design. At l e a s t two o f t h e s e e q u a t i o n s , t h e SoaveRedlich-Kwong and t h e Peng-Robinson e q u a t i o n s , have been extended t o hydrocarbon-water systems i n an e f f o r t t o d e s c r i b e v a p o r l i q u i d - l i q u i d phase b e h a v i o r . Many problems encountered i n t h e gas p r o c e s s i n g i n d u s t r y i n v o l v e nonideal l i q u i d s o l u t i o n s . F o r example, d e h y d r a t i o n and h y d r a t e i n h i b i t i o n processes may i n v o l v e m i x t u r e s of l i g h t hydrocarbons w i t h aqueous methanol or g l y c o l s o l u t i o n s . The Parameters From Group C o n t r i b u t i o n s , o r PFGC, e q u a t i o n i s an e q u a t i o n of s t a t e analogy t o an a c t i v i t y c o e f f i c i e n t e q u a t i o n which i s c a p a b l e of d e s c r i b i n g v a p o r - l i q u i d - l i q u i d e q u i l i b r i u m i n systems e x h i b i t i n g nonideal l i q u i d b e h a v i o r . As t h e name i m p l i e s , t h e parameters i n t h i s e q u a t i o n a r e d e r i v e d from group c o n t r i b u t i o n t e c h n i q u e s r a t h e r than c o r r e l a t i o n s w i t h c r i t i c a l p r o p e r t i e s . T h i s approach a l s o o f f e r s p o t e n t i a l advantages i n 0097-6156/B6/0300-0452$06.50/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

22.

M A J E E D A N D WAGNER

453

Parameters from Group Contributions Equation

a p p l i c a t i o n s t o systems i n v o l v i n g undefined m i x t u r e s of petroleum and s y n t h e t i c l i q u i d s . The f u n c t i o n a l groups i n these types of m i x t u r e s can be i d e n t i f i e d by modern a n a l y t i c a l t e c h n i q u e s such as NMR s p e c t r o s c o p y . Reduced r e l i a n c e on c r i t i c a l p r o p e r t i e s c o r r e l a t i o n s i n terms of s p e c i f i c g r a v i t y , average b o i l i n g p o i n t , c h a r a c t e r i z a t i o n parameters, e t c . , should l e a d t o improved p r e d i c t i o n s of thermodynamic p r o p e r t i e s of s y n t h e t i c and n a t u r a l hydrocarbon systems. In t h i s paper we would l i k e t o share some of our e x p e r i e n c e s and r e s u l t s i n d e v e l o p i n g and e v a l u a t i n g the PF6C e q u a t i o n f o r l i g h t hydrocarbon systems. Our primary emphasis has been on the p r e d i c t i o n of phase b e h a v i o r v i a e q u a t i o n s of s t a t e f o r a p p l i c a t i o n s t o process d e s i g n .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

The PFGC Equation of S t a t e The Parameters From Group C o n t r i b u t i o n s (PFGC) e q u a t i o n of s t a t e was i n t r o d u c e d by Cunningham and Wilson (4) i n 1974. The d e t a i l s of the model f o r m u l a t i o n and d e r i v a t i o n s of the e q u a t i o n are g i v e n by Cunningham ( 5 ) . A b r i e f summary i s i n c l u d e d here f o r convenience. The b a s i s f o r the PFGC e q u a t i o n of s t a t e l i e s i n the assumption t h a t the form of e m p i r i c a l r e l a t i o n s h i p s which have s u c c e s s f u l l y d e s c r i b e d the excess Gibbs f r e e energy of m i x i n g can a l s o be used as the b a s i s f o r modeling the Helmholtz f r e e energy. In a d d i t i o n , the v o i d spaces between molecules i n a m i x t u r e are assumed t o be i d e n t i f i a b l e as an a d d i t i o n a l component d e s i g n a t e d as " h o l e s " . The volume i s e v a l u a t e d as the m o l e c u l a r volume occupied by the component d i v i d e d by the t o t a l volume. I n c l u d i n g an a r b i t r a r y parameter f o r one mole of h o l e s , volume f r a c t i o n s can be converted t o mole f r a c t i o n s . Using these r e l a t i o n s h i p s f o r mole f r a c t i o n s permits the Helmholtz f r e e energy of m i x i n g t o be expressed as a f u n c t i o n of c o m p o s i t i o n and volume. The Helmholtz f r e e energy of mixing i s composed of two contributions: 1.

A m o d i f i e d F l o r y - H u g g i n s e q u a t i o n t o account f o r entropy e f f e c t s due t o d i f f e r e n c e s i n m o l e c u l a r s i z e , and A m o d i f i e d Wilson e q u a t i o n which r e p r e s e n t s the i n d i v i d u a l groups i n a m i x t u r e .

2.

Molecular a c t i v i t y c o e f f i c i e n t s corrected for differences in m o l e c u l a r s i z e are c a l c u l a t e d as the sum of group a c t i v i t y c o e f f i c i e n t s w h i c h , i n t u r n , are determined by group c o m p o s i t i o n r a t h e r than by m o l e c u l a r c o m p o s i t i o n . The Helmholtz f r e e energy i s w r i t t e n as

A

n

ΤΓΓ-

J

N

I

T

b

T

I n - T

1

1

* ^ V

-nb (£-) I ψ. In (Dj

.

ι

nb)

I

In

(L^.)

nb + nb Υ ΤλΤΤ

λ...

— - )

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(1)

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

454

where upper case s u b s c r i p t s r e f e r t o m o l e c u l a r p r o p e r t i e s , lower case s u b s c r i p t s r e f e r t o group p r o p e r t i e s , and η = total

number of moles

b = total

m o l e c u l a r volume =

^ Xj bj

b j = volume of one mole of molecules of type I =

| mj^ b^

b-j = volume of one mole of groups of type i mj.j = number of groups of type i i n molecule b^ = volume of one mole of

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

V = total

I

holes

volume

C = a universal

constant

i n the m o d i f i e d Wilson

s = I Xj Sj = e x t e r n a l degrees I mixture

equation

of freedom parameter f o r the

s j = £ mj.j s.j = e x t e r n a l degrees of freedom parameter f o r ι molecules of type I Sj = e x t e r n a l degrees type i

of freedom parameter f o r groups o f

E^j = i n t e r a c t i o n energy between groups i and j λ ·,· = EXP(-Ε ·j/kT) and j Ί

Ί

I

x

ψ.,· =

i

m

i i

b

= i n t e r a c t i o n parameter between groups i

i

jjj

= group f r a c t i o n of type i

The PFGC e q u a t i o n of s t a t e f o l l o w s d i r e c t l y from the e x p r e s s i o n f o r the Helmholtz f r e e energy of m i x i n g by using f o l l o w i n g thermodynamic r e l a t i o n s h i p :

7ΓΓ

=

" W

the

^ Τ , η

Thus, i n terms of c o m p r e s s i b i l i t y f a c t o r , the PFGC e q u a t i o n of s t a t e can be w r i t t e n as

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

( 2 )

22.

MAJEED AND WAGNER

Parameters from Group Contributions Equation

455

(3)

where ν i s the molar volume. T a k i n g c / b as a " u n i v e r s a l " c o n s t a n t , t h e r e are t h r e e b a s i c parameters i n the PFGC e q u a t i o n : H

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

(1) (2) (3)

Ε ·,·, the i n t e r a c t i o n energy between groups b j , the volume of one mole of groups of type i S j , a parameter p r o p o r t i o n a l t o the e x t e r n a l degrees freedom per group i Ί

of

Note t h a t t h r e e parameters must be known f o r groups i n s t e a d of m o l e c u l e s . The i n t e r a c t i o n energy i s e v a l u a t e d as

where a^,- i s an i n t e r a c t i o n c o e f f i c i e n t . When a^- i s u n i t y , the m i x t u r e p r o p e r t i e s are n e a r l y i d e a l . If a-jj i s l e s s than one, d e v i a t i o n s from i d e a l i t y are i n t h e p o s i t i v e d i r e c t i o n ; f o r a^j g r e a t e r than one, d e v i a t i o n s from i d e a l i t y are i n the n e g a t i v e direction. The group i n t e r a c t i o n energy i s s l i g h t l y temperature dependent. For c o n v e n i e n c e , the f o l l o w i n g form i s used (6)

(5) Thus, t h e PFGC e q u a t i o n of s t a t e i n . i t s f i n a l form has f i v e a d j u s t a b l e parameters: S j , b j , E j j ( ° ' , E j j ' ' , and E j ^ ' . In a d d i t i o n , t h e r e i s one b i n a r y i n t e r a c t i o n c o e f f i c i e n t f o r each p a i r of groups. A p p l i c a t i o n of the e q u a t i o n of s t a t e t o t h e p r e d i c t i o n of thermodynamic p r o p e r t i e s i s s t r a i g h t - f o r w a r d once a p p r o p r i a t e group parameters are a v a i l a b l e . The PFGC e q u a t i o n of s t a t e w r i t t e n i n terms of compressibility factor e x h i b i t s cubic-type behavior. Several i t e r a t i v e schemes can be a p p l i e d t o f i n d the l i q u i d - l i k e and/or v a p o r - l i k e r o o t s of E q u a t i o n 3. Moshfeghian, et a l . (6) used a d i r e c t s u b s t i t u t i o n method. However, a t h i r d - o r d e r i t e r a t i o n method u s i n g a Richmond convergence scheme has been found t o improve both the speed and r e l i a b i l i t y of the c a l c u l a t i o n s . The b a s i c a l g o r i t h m i s given by Lapidus (7) as 1

2f(Xj)

2

fix,)

(6) 2Cf

(Xi ) ]

2

-

f ( X j )

f t x j )

S e t t i n g χ = b/v, E q u a t i o n 3 can be w r i t t e n as

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

456

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

f(x)

• {J -

(1 • S)X

-

• b ( f ) I *, " H i

(1-X)

S 1(1

(1 - l λ j

1

)

1 J

J i — 1 - χ + χ l

(7) λ..

+ J

J

The f i r s t and second d e r i v a t i v e s f (x) = 1 + s

* — + b (£-) H

are

Σ «. [ * Ρ - 1] i (1 - χ + χ I * j ^ j )

b

1

(8)

2

J

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

and ς f"(x)

=

r

+ 2b (
7

Z

b

1 ~ Σ ψ, λ. . I ψ. i (1 - x + x *j J

1

ι

1 J

3

(9)

J

Other thermodynamic f u n c t i o n s and p r o p e r t i e s can a l s o be d e r i v e d from the e q u a t i o n of s t a t e . For phase e q u i l i b r i u m c a l c u l a t i o n s , the f u g a c i t y c o e f f i c i e n t f o r component I i n a multicomponent m i x t u r e i s given by

(10) V

l

u

Using t h i s d e f i n i t i o n w i t h Equation 3 l e a d s In

Φ ι

=

[^1

ν - ^ b

b j ν - S j ] In

b

+

b. - Σ m b, λ , . - b l ψ. — y _ J _ J l

}

+ (f-) H

ί I m i

n 1 1

b

v

In (·

i

-

-

(1

b

to

^)

l * j

x

iU

1

n

i

·

ν - b + b

I

(11)

x..

E x p r e s s i o n s have a l s o been d e r i v e d t o the chemical p o t e n t i a l and the i s o t h e r m a l e f f e c t of p r e s s u r e on e n t h a l p y ( 6 , 8 , 9 ) . To o b t a i n values f o r the group parameters i n the PFGC e q u a t i o n of s t a t e , s e v e r a l steps have been f o l l o w e d . F i r s t , groups were s e l e c t e d t o r e p r e s e n t t h e components of i n t e r e s t . Data f o r those components which were i d e n t i f i e d by a s i n g l e g r o u p , such as methane and carbon d i o x i d e , were used t o o b t a i n pure group parameters f o r these groups/components. Parameters f o r o t h e r t y p e s of groups are more d i f f i c u l t t o o b t a i n and r e q u i r e a

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

22.

M A J E E D A N D WAGNER

Parameters from Group Contributions Equation

457

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

c o n s i d e r a b l e amount of judgement i n the f i t t i n g p r o c e d u r e s . For example, p r e l i m i n a r y e s t i m a t e s of the methyl and methylene group parameters were o b t a i n e d from thermodynamic data f o r ethane and e t h y l e n e , r e s p e c t i v e l y . The f i n a l methyl and methylene group p a r a m e t e r s , as w e l l as the group b i n a r y i n t e r a c t i o n c o e f f i c i e n t , were d e r i v e d from a simultaneous r e g r e s s i o n u s i n g data f o r b u t a n e , hexane, o c t a n e , and heptadecane. Parameters were f i t t o m i n i m i z e the average a b s o l u t e e r r o r i n pure component vapor p r e s s u r e . However, c a r e was e x e r c i s e d t o m a i n t a i n r e a s o n a b l e q u a l i t y f o r v o l u m e t r i c and e n t h a l p y d e p a r t u r e p r e d i c t i o n s . M i x t u r e s of d i f f e r e n t components were used t o o b t a i n group b i n a r y i n t e r a c t i o n c o e f f i c i e n t s . These i n t e r a c t i o n c o e f f i c i e n t s were s e l e c t e d t o m i n i m i z e average a b s o l u t e percentage e r r o r s i n e q u i l i b r i u m K-values f o r each component i n the m i x t u r e . Group parameters are l i s t e d i n Table I. Applications Vapor P r e s s u r e . A b a s i c requirement of any e q u a t i o n of s t a t e which w i l l be used f o r phase e q u i l i b r i u m c a l c u l a t i o n s i s t h a t pure component vapor p r e s s u r e s must be p r e d i c t e d a c c u r a t e l y . T a b l e II p r o v i d e s a comparison of p r e d i c t e d and e x p e r i m e n t a l vapor pressures f o r several p a r a f f i n hydro-carbons. E r r o r s i n vapor p r e s s u r e p r e d i c t i o n s u s i n g the Soave-Redlich-Kwong (SRK) e q u a t i o n o f s t a t e are i n c l u d e d f o r comparison. Percentage e r r o r s i n c a l c u l a t e d vapor p r e s s u r e of isopentane as a f u n c t i o n of reduced t e m p e r a t u r e are shown i n F i g u r e 1. T h i s type of e r r o r d i s t r i b u t i o n i s t y p i c a l of the PFGC e q u a t i o n of s t a t e and i s very s i m i l a r t o t h a t of the c u b i c SRK e q u a t i o n . However, average a b s o l u t e percentage e r r o r s i n pure component vapor p r e s s u r e are on t h e o r d e r of two t o ten times g r e a t e r u s i n g the PFGC e q u a t i o n . One l i m i t a t i o n of the PFGC e q u a t i o n w i t h the c u r r e n t s e t o f groups i s t h e i n a b i l i t y of t h i s group c o n t r i b u t i o n method t o d i s t i n g u i s h the e f f e c t of m o l e c u l a r s t r u c t u r e on p h y s i c a l p r o p e r t i e s . T h i s i s e v i d e n t i n the average a b s o l u t e e r r o r s i n p r e d i c t e d vapor p r e s s u r e s f o r 2-methylpentane and 3 - m e t h y l pentane. In terms of group c o m p o s i t i o n , these two components are identical. The PFGC e q u a t i o n does p r e d i c t pure component vapor p r e s s u r e s f a i r l y w e l l , e s p e c i a l l y c o n s i d e r i n g the number of parameters available. For example, o n l y e l e v e n parameters are used t o d e s c r i b e the pure component p r o p e r t i e s of a l l the normal a l k a n e s - - two s e t s of f i v e group parameters plus one b i n a r y i n t e r a c t i o n coefficient. In c o n t r a s t , the SRK e q u a t i o n of s t a t e i n c l u d e s t h r e e parameters f o r each pure component. Molar Volumes. A b a s i c problem w i t h c u b i c e q u a t i o n s of s t a t e i s t h e a b i l i t y t o p r e d i c t both phase b e h a v i o r and v o l u m e t r i c properties well (17). The PFGC e q u a t i o n e x h i b i t s t h i s same b e h a v i o r . A b s o l u t e average percentage e r r o r s i n c a l c u l a t e d l i q u i d and vapor volumes are summarized i n T a b l e III f o r t y p i c a l p a r a f f i n s , o l e f i n s , a r o m a t i c s , and nonhydrocarbons. Errors in c a l c u l a t e d volumes u s i n g the SRK e q u a t i o n of s t a t e are i n c l u d e d f o r comparison.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

0.2100 0.0343 0.8857

-19.1556 -23.0062 -28.1383

-55.6278 -81.5054 -133.2944

.9260

8.8889 476.9444 26.7175 .0006 .0092

-31.7222 -96.0556 -1544.0556 -69.8889 -56.3144 -9.7502

-75.7056 -338.6667 -1472.9444 -153.7611 -0.4167 -22.2261

2.3659 3.0920 2.5992 3.4335 2.2000 .1938 -.6500 -.4428

.02777 .02079 .02530 .02527 .01248 .01689 .03972 .03801

N

co

CO

HS H0

2

2

>C=

=C=

>C-

12.

13.

14.

15.

16.

17. 23.

24.

2

-.0046 -61.1222 -18.4444 92.0556 -65.7222

-.2419

.01940

-C=

11.

2

-62.3567

10.

17.2222 32.4755 -107.6667 -142.8333 -174.0336

.3471 .2272

.01610 .01634

9.

2

-320.1728

-94.5814

0.07422 3.6022

1.6667

0.1697

.6333

.02336

>CH *CH=C<

8.

-42.0811

1.3235

.02093

-110.3583

-2.6435

.01614

2

>C<

=CH

7.

0.2599

6.

-43.2389

3.3501

.01505

2

-150.0167

-32.5722

-CH >CH-

4.2153 -32.5201

5.

.4956

.02078 .01665

-176.5500

4.

3

-71.3501

E(2) Κ

κ

E(D

-CH

1.9780

(0) Κ

3.

1.8982

E

4

.03727

m^/kg-mo le

CH

Group

2.

No

TABLE I. TYPICAL GROUP PARAMETERS FOR THE PFGC EQUATION

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

22.

M A J E E D A N D WAGNER

Parameters from Group Contributions Equation

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

I

ο ο

00 ^

ο

ο

ο

ο

ο

ο

ο

ο

ο

ο

cn

to

ο

ο

ο

Ο O

ο

O

L

Ο Ο

O

00

C

ο ο

t-4 O

σ m

O

O

C

M

ο ο

m <τ>

O r O C T i O O C V J i — ι Ο Ο

O O O C O O C O O O O O

ο ο ο CO CM If) o o o u 3 c r > o o c \ ] o o o

ο

ο

CM CM O O O O C O C T i O O O O O O

ο Ο

LO

Ο

ο Ο

O O O O O C T i

ο

ο

ο

LO Ο

«3-

LO CO CT>

ο

en

ο

ο

ο

LO

τ—I

<—·

CO

Ο

CO

ο

CO CM CO

Η

ο

ο

ο

ο

CM LO CT>

Ο LO

00

σι

οο

οο

σι

ο CTi

0 0 CO Ο

CTi Ο

ο

Ο

ο

Ο

Ο

Ο

ο

ο

ο

CO

r-t

Ο

LO

Ο

Ο

Ο

ο

ο

CM

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

460

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

TABLE I I .

COMPARISON OF VAPOR PRESSURE PREDICTIONS

Average A b s o l u t e P e r c e n t E r r o r

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

Component

Methane Ethane Propane i-Butane n-Butane i-Pentane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane 2-Methylpentane 3-Methylpentane 2,3-Dimethylbutane Nitrogen Carbon D i o x i d e Hydrogen S u l f i d e

PFGC

SRK

No. of Points

Ref,

2.38 3.60 5.32 1.89 5.66 1.16 6.46 1.95 4.47 3.45 3.72 2.15 5.78 7.86 9.90 13.16 14.80 6.43 13.10

1.50 1.59 1.06 .86 1.62 1.13 1.55 1.33 1.05 1.22 1.16 1.15 2.31 2.94 2.29 4.07 1.07 1.40 1.13

21 38 45 25 52 38 55 44 61 62 27 27 27 27 27 27 58 58 59

10 10 10 11 10 12 10 11 10 10 13 13 13 13 13

1.86 6.94 0.94

0.78 .47 .91

60 48 30

14 15 16

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

10 10 13

22. M A J E E D A N D WAGNER

Parameters from Group Contributions Equation

461

61 LU

ce D CO CO 4I LU *| LU £

Ο PFGG • SRK

S* Η

Ο

LU < ϋ

>

^LU Ζ-

...

2

. C

1

ο •••φ

Ο , Ρ 0

< >

···

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

LU Ο

0.5

0.6

Figure 1 Deviations pentane ( 1 6 ) .

TABLE I I I .

Component Methane Ethane iso-Pentane n-Hexane Ethylene Propylene 1-Butene Benzene Toluene Nitrogen Carbon Monoxide Carbon D i o x i d e Hydrogen S u l f i d e Sulfur Dioxide

0.7 0.8 0.9 REDUCED TEMPERATURE

1.0

i n vapor p r e s s u r e p r e d i c t i o n s f o r

iso-

AVERAGE ABSOLUTE PERCENT ERRORS IN MOLAR VOLUMES

PFGC 7.11 3.25 14.14 2.86 3.81 1.76 2.48 6.55 16.05 1.77 17.40 12.20 4.86 7.83

Liquid SRK No. P t s . 4.51 6.59 12.42 19.44 8.11 8.60 12.96 14.16 16.98 3.91 4.67 15.08 6.85 19.54

62 58 38 44 63 63 27 48 61 81 25 60 27 34

PFGC

3.79 4.94 6.19 8.40 9.50 4.72 27.50 12.33 5.04 9.32 6.31 13.30 4.95 14.2

SRK

2.19 3.01 7.48 3.02 3.22 2.56 3.19 1.85 1.74 1.30 2.36 4.05 5.24 7.61

Vapor No. P t s . R e f . 95 77 83 69 108 64 27 48 61 116 67 71 66 64

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

18 19 12 11 20 21 11 11 10 14 11 15 16 11

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

462

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

F i g u r e 2 i l l u s t r a t e s the performance of the two e q u a t i o n s f o r p r e d i c t i n g the molar volumes of s a t u r a t e d i s o p e n t a n e . The SRK e q u a t i o n p r e d i c t s l i q u i d volumes t h a t are t o o h i g h , e s p e c i a l l y near the c r i t i c a l p o i n t . In c o n t r a s t , the PFGC e q u a t i o n p r e d i c t s molar l i q u i d volumes t h a t are t o o low. In g e n e r a l , a b s o l u t e e r r o r s i n l i q u i d volumes are about the same f o r the two e q u a t i o n s . E r r o r s i n p r e d i c t e d vapor volumes are of t h e same o r d e r of magnitude f o r the two e q u a t i o n s . P r e d i c t i o n s of molar volumes can be improved somewhat, but o n l y at t h e expense of vapor p r e s s u r e or phase b e h a v i o r p r e d i c t i o n s . In f i t t i n g group parameters we have t r i e d t o m i n i m i z e e r r o r s i n vapor p r e s s u r e p r e d i c t i o n s w h i l e m a i n t a i n i n g reasonable accuracy of v o l u m e t r i c p r e d i c t i o n s . V a p o r - L i q u i d E q u i l i b r i u m . The c a p a b i l i t i e s of the PFGC e q u a t i o n of s t a t e i n p r e d i c t i n g the phase b e h a v i o r and phase c o m p o s i t i o n s has been e v a l u a t e d f o r a v a r i e t y of systems, i n c l u d i n g t h o s e c o n t a i n i n g both hydrocarbon and nonhydrocarbon components. Examples of e r r o r s i n p r e d i c t e d K-values and l i q u i d volume f r a c t i o n s are presented i n Tables IV and V. I n s p e c t i o n of t h e s e t a b l e s shows a f a i r l y good agreement between e x p e r i m e n t a l and c a l c u l a t e d phase b e h a v i o r f o r both hydrocarbon-hydrocarbon and hydrocarbon-non-hydrocarbon b i n a r y systems. One of the d i f f i c u l t i e s i n u s i n g the group c o n t r i b u t i o n t e c h n i q u e i n v o l v e s the use of t h e same group b i n a r y i n t e r a c t i o n c o e f f i c i e n t s f o r both pure components and m i x t u r e s . For example, t h e b i n a r y i n t e r a c t i o n c o e f f i c i e n t between the -CH3 and =CH2 groups i n t o l u e n e was o p t i m i z e d u s i n g vapor p r e s s u r e data f o r t o l u e n e . However, t h e same i n t e r a c t i o n c o e f f i c i e n t i s a key v a r i a b l e i n o b t a i n i n g good p r e d i c t i o n s of t o l u e n e K-values i n m i x t u r e s of p a r a f f i n s . T h i s problem i s c h a r a c t e r i s t i c of a group c o n t r i b u t i o n t e c h n i q u e where the number of groups are much s m a l l e r t h a n the number of components. The group parameters and t h e b i n a r y i n t e r a c t i o n c o e f f i c i e n t s r e p r e s e n t a d e l i c a t e balance between t h e a b i l i t y t o p r e d i c t pure component thermodynamic p r o p e r t i e s and the r e p r e s e n t a t i o n of multicomponent v a p o r - l i q u i d equilibria. The performance of the PFGC e q u a t i o n i n p r e d i c t i n g the phase b e h a v i o r and v o l u m e t r i c of multicomponent n a t u r a l gas and r e t r o g r a d e condensate types of systems has been d i s c u s s e d by Wagner et a l . ( 5 9 ) . For h e a v i e r m i x t u r e s , the phase b e h a v i o r of a s y n t h e t i c o i l w i t h high carbon d i o x i d e c o n t e n t s r e p r e s e n t s an extreme t e s t of the PFGC e q u a t i o n . Using group i n t e r a c t i o n c o e f f i c i e n t s d e r i v e d from carbon d i o x i d e - hydrocarbon b i n a r y systems, t h e PFGC e q u a t i o n p r e d i c t e d reasonably w e l l t h e s a t u r a t i o n p r e s s u r e s of the m i x t u r e w i t h carbon d i o x i d e c o n t e n t s up t o 97 mole percent (69) as shown i n F i g u r e 3. Aqueous L i g h t Hydrocarbon Systems For a p p l i c a t i o n s of the PFGC e q u a t i o n t o aqueous systems, two group i n t e r a c t i o n c o e f f i c i e n t s were d e f i n e d f o r the v a r i o u s phases p r e s e n t . One b i n a r y group i n t e r a c t i o n c o e f f i c i e n t i s used f o r both the vapor and h y d r o c a r b o n - r i c h l i q u i d phases; a second group i n t e r a c t i o n c o e f f i c i e n t i s used f o r t h e aqueous l i q u i d phase.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Parameters from Group Contributions Equation

M A J E E D A N D WAGNER

Ϊ ••

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

-

fi^ft" eggQ On m •••

nnn

A

n

nn-

SSH^Q

• • ° - , ιΠΗ°° 0 0

D

^

D D

Ο

D

n

U

OO

U

LU O -20

t-

• •

LIQUID VAPOR • - SRK • ο PFGC · 0.5

0.6

0.7

• • • •

0.8

0.9

1.0

REDUCED TEMPERATURE Figure 2 Deviations i s o - p e n t a n e (16)

i n s a t u r a t e d molar volume p r e d i c t i o n s f o r

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986. 460.0 200.0

-> -*·

79.0

100.0

0.0

150.0

CH (]D - n C H ( 2 )

C H ( ]D " n C H ( 2 )

C H ( ]D " n C H ( 2 )

2 2

1 8

1 4

2

6

CH
1 0

2 2

1 4

nC H (2)

6

50.0 •>

->»

150.0

D - nC H (2)

1 2

·*·

5

1 0

40.0

H

6

nC H (2)

C

2

4

8

2 6< D -

3

- C H (2)

1 0

8

6

1 2

nC H (2)

H

2 6< D

4

4

4

5

C H ( D -

C

-131.24 -• 32.0 302.0

-140.0

4

1 0

25.0 100.0

460.0

50.0 ->

21.99 5.22 6.18

7.92

1715.0

1146.0

13.16 22.66

7.94 14.53

39 112

68

955.0

33

32

31

30 5.56

19

805.0

16.93 4.07

3.30

•¥

509.0

29

9.24 2.78

2.11 78

752.0

•>

100.0

28 10.89

25.42 157

5000.0

9.13

27

8.76 12.51

26

25

24

23

22

Reference No.

11.50

11.02

18.14

8.44

9.16

11.60

17.63

22.42

17.80

9.98

32.05

6.42

8.33

7.32

4.21

2

P e r c e n t Abs. Avg. E r r o r K L/F

7.44

105

64

105

81

118

40.0

2675.0

2200.0

1822.0

1450.0

748.0

35

·>

·>

•¥

•>

-•

-*·

No. of Points

3865.0

146.9

19.9

50.0

20.1

50.0

28.0

Pressure Range ( p s i a )

350.0

340.0

250.0

50.0

40.0

32.0

CH (1D - n C H ( 2 )

4

8

-200.0

4

3

-99.8

CH (]D - n C H ( 2 )

4

·>

-176.0

6

CH (1D - C H ( 2 )

2

-225.0

4

Temperature Range (°F)

DEVIATIONS IN K-VALUE PREDICTIONS FOR HYDROCARBON-HYDROCARBON BINARIES

CH (1D - C H ( 2 )

System

TABLE IV.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

Parameters from Group Contributions Equation

M A J E E D A N D WAGNER

£ co

^ co

co

LO

00

CT>

CVJ

CO

CVJ rH

i—1 ι—1

ο

i—1

o

CVJ r—1

LO

ΙΟ CVJ

00

P-. CVJ

CVJ

^ co

oo oo

Ο LO

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

er> oo

co

CVJ

co co

ο

uo co

CVJ

oo

co

CO

VO CO

r—

σ> O r-H

•

00 CVJ

Γ

er>

CVJ CO

CO CVJ

Ο rr H H

CT>

00 LO

ι—1 P-

p-

00 i-H

cvj oo

·

i—·

r-H 00

Ο

c o c o ^ a -

•

CO

00

oo

CO

CVJ VO LO

CO 00 i-H

LO P-

CVJ

00 CVJ

P-

CVJ CVJ

00 i-H

CO P-

P-

rH

oo

CVJ

·

*

ο

•

UO CVJ

ι—

«3-

4-

ο Ο LO

·

4-

ρ-

p-

ρ-

«3r—I

«3i-H

f—I

Ο

00

Ο Ο VO

LO LO Ρ-

Ο

^

*

·

CVJ

Ο

CVJ

^

LO

4·

4-

4-

•

•

4-

LO

P-

• ο

LO 00

CO CVJ

i—1 CO

rH

ο

ο

00 00 CVJ

ο

ο

•

_ ο CVJ

•

o

•

CVJ CVJ

o

I—«

i-H

+

4·

ο

Ο

ο

Ο

LO

ο

CO

LO

ο ο

4-

·

LO ι—I

4-

Ο

·

• rH LO CO

ο

•

4-

ο

4·

LO

.

4-

Ο

Ο

r**

·

ι—1

ο

LO

rH

Ο Ο CVJ

CVJ CVJ 00 CT> CVJ

LO CO 4-

o ο ο

ι—I

Ο ο ο

^j-

4-

ο ο

4-

ο ο 4-

Ο

ο

Ο ο

ο ο

rH

rH I

^ ^ CVJ ^ w

cvj 31

ο «-< ο C

• ^ «—< v

co 31

co ο

CVJ CVJ I

ο *—I

ο c

I Ο r Ο

ι ,— ^ >>CVJ χι ^ 4->

>> ο

eu eu χ c rcs

I

^

rH Ο rH

•= LO

ο c

I CVJ

^

—^

I

^

4->

CU

rH Q .

=c eu

ΖΠ >>

LOX:

ο

CVJr— ,—ICJ LOU

ο c

χ: cvj +->

I

1

>>

>>

SZ eu ε

eu

Σ.

eu c Π3

c

i-H Ο) — - C CVJitJ H X

ο c

ι r—

1

C V J ^ CVJ

CVJ CVJ »

χ: 4->

eu E co -—^ CVJ

χ

O)

i-HX: — CVJ— H U

ο

ΠΖ

LOO

ο c

—ι—I eu

t-H C — - fCJ 004-> r-HC

^

•c eu Ο

ο00

00 C L

c

CVJ

ο c

ο

1

I

I

rH

rH

rH

ο

eu

C ' (O 00+J H C •= φ

c

rH

00 CO

CL

eu c eu

Ν c

eu

CÛ

eu c eu

Ν c

eu

0Û

eu eu =5

'ο

h-

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

60.0 33.0

100.0 + 280.0

40.1 + 220.0

nC H (2)

nC H (2)

nC H (2)

-

-

-

co (i;

C0 (1)

co (i]

3 4

- Toluene (2)

2

co (i;

2

-

2

N (2)

2

C0 (1) - H S(2)

2

co (i;

2

C 0 ( 1 ) - Benzene (2)

1 6

nC H (2)

2 2

1 4

1 2

1 0

-

2

co (i;

2

1 0

6

5

4

8

nC H (2)

3

C0 (1] -

2

2

2

2

6

4-

194.0

-67.0 + 32.0

-2.34

100.6 + 399.0

77.0 •> 104.0

372.9 • 735.08

40.0 + 460.0

255.0

293.9

48.4

129.6

284.5

50.0

113.0

100.0

40.0+ 160.0

C H (2)

-

co (i]

2

104.0 4- 248.0

90.0

-58.0 + 68.0

2

- CH (2)

C0 (1 1 - C H (2)

co (i;

4»

4·

•>

·>

•>

-»•

4-

4·

4·

4-

1907.0

1175.7

2218.0

1124.1

749.5

2732.0

1682.0

1397.0

1150.0

950.0

914.0

1146.0

Pressure Range ( p s i a )

161.0

4

Temperature Range (°F)

30

85

34

17

31

88

40

47

54

67

54

45

No. of Points

4.34

3.53

10.63

17.87

8.30

19.20

24.82

12.21

11.21 2.07

20.80

15.90

12.20

7.07

5.95

19.80

4.89

2

11.32

9.48

8.98

7.45

4.49

3.01

8.51

χ

12.43

21.42

11.76

7.77

8.16

7.78

5.94

6.31

9.11

9.90

20.41

9.61

P e r c e n t Abs. Avg. E r r o r Κ K L/F

SYSTEM DEVIATIONS IN K-VALUE PREDICTIONS FOR NONHYDROCARBON-HYDRQCARBON BINARIES

-100.0 • 29.0

2

System

TABLE V.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

52

51

50

49

48

47

46

45

44

43

42

41

Reference No.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

4

6

1 0

13.34 16.02 5.29 21.66

4.68 6.62 6.04 28.82

2.08 3.21 4.66 8.96

-

-

-

D

:D

H S(

HS

2

2

1 0

2 2

1 2

nC H (2)

5

1 0

nC H (2)

4

nC H (2)

6

D

2

2

H S(

2

C H (2)

-

D

H S(

40.0 + 340.0

40.0 4- 340.0

100.0 • 250.0

-99.80 4. 50.0

20.0 4- 1935.0

20.0 4- 1302.0

69.4 • 1150.0

9.45 + 442.0

50

60

77

45

7.49

1600.0

12.46

200.0

5.08

-120.0 + 200.0

59

- CH (2)

2

D

2

H S(

2

4

12.43

4.34

17.87

30

255.0 + 1907.0

- 6 7 . 0 +• 32.0

N (]D - C0 (2)

2

57

56

55

54

53

52

58 6.07

10.10

10.60

17

900.86 + 4454.4

167.0 4- 257.0

N ( ] D - Benzene (2)

2 2

57

1 0

12.96

2

18.58

56

55

54

53

27.62

13.24

17.25

10.19

11.03

92

26.14

13.61

13.74

6.19

80.0 4- 5000.0

1 0

22.05

14.23

13.70

4.74

21

28

31

101

350.8 + 4506.5

236.0 + 3402.0

152.7 + 1913.7

40.5 • 710.0

100.0 4- 280.0

5

-200.0 + 0.0

100.0 •> 280.0

-99.67 + 62.33

-240.0 + 130.0

N ( ] L) " n C H ( 2 )

2

N ( lD - n C H ( 2 )

2

2

- C H (2)

4

N ( l L) " n C H ( 2 )

2

N 0D

2

N (]D - CH (2)

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

S' 3

-Ci

S' S

S

Ci

ο

î

Ο Ζ m

0

> Ζ D

2 >

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

468

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

F i g u r e 3 E x p e r i m e n t a l and p r e d i c t e d bubble p o i n t p r e s s u r e s f o r a s y n t h e t i c o i l w i t h v a r y i n g carbon d i o x i d e content (140)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

22.

M A J E E D A N D WAGNER

Parameters from Group Contributions Equation

469

Data f o r m i x t u r e s of water w i t h l i g h t h y d r o c a r b o n s , carbon d i o x i d e , hydrogen s u l f i d e , n i t r o g e n , and carbon monoxide were used t o d e r i v e b i n a r y group i n t e r a c t i o n c o e f f i c i e n t s f o r the aqueous phase. These i n t e r a c t i o n c o e f f i c i e n t s were f i t t o m i n i m i z e t h e average a b s o l u t e e r r o r s i n t h e c o m p o s i t i o n of each phase. Some of the nonhydrocarbon-water group i n t e r a c t i o n c o e f f i c i e n t s were found t o be l i n e a r l y dependent on a b s o l u t e t e m p e r a t u r e . T a b l e VI summarizes the e r r o r s i n p r e d i c t e d phase c o m p o s i t i o n s f o r s e v e r a l b i n a r y w a t e r - h y d r o c a r b o n and w a t e r nonhydrocarbon systems. For p a r a f f i n s and o l e f i n s , t h e PFGC e q u a t i o n g i v e s e x c e l l e n t p r e d i c t i o n s of t h e vapor, hydrocarbon l i q u i d , and aqueous l i q u i d phases up t o a p p r o x i m a t e l y 9000 p s i a . V a p o r - l i q u i d - l i q u i d p r e d i c t i o n s f o r t h e b i n a r y systems of water w i t h carbon d i o x i d e and hydrogen s u l f i d e are good up t o 3000 psia. The a b i l i t y of the PFGC e q u a t i o n t o handle t h e s e nonideal aqueous m i x t u r e s i s a r e s u l t of u s i n g an a c t i v i t y c o e f f i c i e n t model as a t h e o r e t i c a l b a s i s f o r the e q u a t i o n of s t a t e .

Other A p p l i c a t i o n s We have r e c e n t l y l i n k e d the PFGC e q u a t i o n of s t a t e w i t h P a r r i s h and P r a u s n i t z (68) hydrate model as m o d i f i e d by Menton, P a r r i s h and Sloan ( 6 9 ) . The p r e l i m i n a r y r e s u l t s of t h i s e f f o r t are very p r o m i s i n g ( 5 9 ) , and we are c o n t i n u i n g our e v a l u a t i o n of the p r e d i c t i o n of h y d r a t e f o r m a t i o n c o n d i t i o n s below the i c e p o i n t . Procedures are a l s o being developed t o extend the PFGC e q u a t i o n t o m i x t u r e s of u n d e f i n e d components i n n a t u r a l gas condensates and crude o i l s . Summary The PFGC e q u a t i o n of s t a t e has the c a p a b i l i t y t o p r e d i c t v a p o r l i q u i d e q u i l i b r i u m and v o l u m e t r i c p r o p e r t i e s f o r a v a r i e t y of l i g h t hydrocarbon and hydrocarbon-water systems. In g e n e r a l , the q u a l i t y of the p r e d i c t i o n s i s q u i t e good c o n s i d e r i n g the s m a l l number of group parameters a v a i l a b l e t o d e s c r i b e a l a r g e number of components. In more p r a c t i c a l t e r m s , the PFGC e q u a t i o n i s s u i t a b l e f o r process d e s i g n / s i m u l a t i o n c a l c u l a t i o n s f o r many l i g h t hydrocarbon systems. Although the Soave-Redlich-Kwong and PengRobinson e q u a t i o n s of s t a t e are more r e l i a b l e f o r " n o r m a l " hydrocarbon systems, the PFGC e q u a t i o n has an advantage i n aqueous systems w i t h both hydrocarbons and nonhydrocarbons. Acknowledgment The authors a p p r e c i a t e t h e f i n a n c i a l support of t h e Mobil Foundation and the School of Chemical E n g i n e e r i n g , Oklahoma S t a t e U n i v e r s i t y , which made t h i s work p o s s i b l e .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986. 65 65

16.6 18.67

68

984.8 1305.4 1446.0 50759.0 2000.0 2000.0 200.0

413.4 301.7 264.0 2900.0 50.0 50.0 50.0

473.0 536.0 564.8 662.0 600.0 600.0 600.0

H 0(1) - n C H ( 2 ) 392.0

H 0(1) - n C H ( 2 ) 392.0

H 0(1) - n C H ( 2 ) 392.0

122.0

100.0

100.0

100.0

H 0(1) - C0 (2)

H 0(1) - H S(2)

H 0(1] - N (2)

H 0(1) - C0(2)

2

2

2

2

2

2

2

2

2

2

10

9

7

6

22

20

16

14

67 65 65 65

3.55 3.81 14.90

12.01 7.84 9.68

18 16 18

65 24.52

21.43

18.50

12.46

7

5

65

24.2

3

789.0

536.6

437.0

2

12

66

H 0(1) - n C H ( 2 ) 392.0

5

10

31.66

9.81

32

3000.0

120.0

600.0

2

4

65

100.0

H 0(1) - n C H ( 2 )

2

4

21.80

115

934.7

52.2

280.0

100.0

H 0(1) - n C H ( 2 )

64 11.21

6.60

6.98

7

491.6

52.2

280.0

99.9

2

10

H 0(1) - n C H ( 2 )

63 10.51

12.90

7.16

240

3000.0

72.0

310.0

8

42.3

2

3

H 0(1) - C H (2)

62

4.71

711.3 130

460.0

100.0

2

6

H 0(21D - C H (2)

2

680.0

302.0

4

H 0(1) - CH (2) 10000.0

Reference No.

200.0

Percent Abs. Avg. Error i n Smaller Component Cone. Vapor Hydrocarbon Water Phase Liquid 61

No. of Points

7.33

Pressure Range (psia)

45

Temperature Range (°F)

14226.0

2

System

2

TABLE VI. H 0 BINARY SYSTEMS DEVIATIONS IN PHASE CONCENTRATION PREDICTIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

22.

MAJEED AND WAGNER

Parameters from Group Contributions Equation 47

Literature Cited 1. 2. 3. 4. 5.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21.

Soave, G., Chem. Eng. Sci., 1972, Vol. 27, 1197 Peng, D. Y. and Robinson, D. B. Canadian Jour. Chem. Engr., 1976, Vol. 54, December, 595. Starling, Κ. E. and Han, M. S. Hydrocarbon Processing, 1972, Vol. 51, No. 5, 129. Cunningham, J. R. and Wilson, G. M. Paper presented at GPA Meeting, Tech. Section F, Denver, Colorado, March 25, 1974. Cunningham, J. R., M.S. Thesis, Brigham Young University, Provo, Utah, 1974. Moshfeghian, M., Shariat, Α., and Erbar, J. Η., Paper presented at the AIChE National Meeting, Houston, Texas, April 1-5, 1979. Lapidus, L . , "Digital Computation for Chemical Engineers", McGraw-Hill, New York, 1962. Moshfeghian, Μ., Shariat, Α., and Erbar, J. H. Paper presented at NBS/NSF Symp. on Thermo. of Aqueous Sys., Airlie House, Virginia, 1979. Moshfeghian, M., Shariat, Α., and Erbar, J . H. ACS Symposium Series 113. Engineering Sciences Data Unit, Chemical Engineering Series, Physical Data, Volume 5, 251-259 Regent Street, London WIR 7AD, August, 1975. Canjar, L. and Manning, F. "Thermodynamic Properties and Reduced Correlations for Gases," Gulf Publishing Corporation, Houston, Texas, 1966. Arnold, E. W., Liou, D. W., and Eldridge, J. W., J. Chem. Eng. Data, 1965, Vol. 10, No. 2, 88. Selected Values of Properties of Hydrocarbons and Related Compounds, API Research Project 44, Texas A&M University, Updated, 1979. Strobridge, T. R., "The Thermodynamic Properties of Nitrogen from 64 to 300 Κ between 0.1 and 200 Atmospheres", Nat. Bur. Stand., Tech. Note 129, 1962. IUPAC "International Thermodynamic Tables of the Fluid State", Chem. Data Series No. 7, Carbon Dioxide, Pergamon Press, London, 1980. West, J. R., Chem. Engr. Prog., 1948, 44(4), 287. Videl, J., Proceedings of the Third International Conference on Fluid Properties and Phase Equilibria for Process Design, Callaway Gardens, Georgia, April 10-15, 1983. Goodwin, R. D., "The Thermophysical Properties of Methane from 90 to 500 Κ at Pressures 700 Bar", Nat. Bur. Stand. (U.S.), Tech. Note 653, 1974. Goodwin, R. D., "The Thermophysical Properties of Ethane from 90 to 600 Κ at Pressures 700 Bar", Nat. Bur. Stand. (U.S.), Tech. Note 684, 1976. ASHRAE Handbook, Fundamentals 1981, American Society of Heating, Refrigerating and Air Conditioning Engineers, Third Printing, 1982, 17. IUPAC "International Thermodynamic Tables of the Fluid State", Chem. Data Series No. 25, Propylene, Pergamon Press, London, 1980.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

472 22. 23. 24. 25. 26. 27. 28.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS Wichterle, I. and Kobayashi, R. J. Chem. Eng. Data, 1972, 17(1), 9. Akers, W. W., Burns, J. F, and Fairchild, W. R. Ind. Eng. Chem., 46(12), 1954, 2531. Douglas, G. E., Chen, R. J. J., Chappelear P. S., and Kobayashi, R. J. Chem. Eng. Data, 1974. 19(1), 71. Kahre, L. C., J. Chem. Eng. Data, 1975, 20(4), 363. Lin, Y. Ν., Chen, R. R. J., Chappelear, P. S., and R. Kobayashi, J. Chem. Eng. Data, 1977, 22(4), 402. Kohn, J. P. and Bradish, W. F., J. Chem. Eng. Data, 1964, 9(1), 5. Reamer, Η. Η., Olds, R. H., Sage, Β. H., and Lacey, W. N. Ind. Eng. Chem., 1942, 34(12), 1526. Matschke, D. E. and Thodos, G. J. Chem. Eng. Data, 1962, 7(2), 232. Thodos, G. and Mehra, V. S. J. Chem. Eng. Data, 1965, 10(4), 307. Reamer, H. H., Sage, Β. Η., and Lacey, W. N. Ind. Eng. Chem., 1940, 5(1), 44. Zais, E. J. and Silberberg, I. L. J. Chem. Eng. Data, 1970, 15(2), 253. Reamer, H. H. and Sage, B. H. J. Chem. Eng. Data, 1962, 7(2), 161. Sage, Β. H. and Lacey, W. N. Ind. Eng. Chem., 1940, 32(7), 992. Reamer, H. H. and Sage, B. H., J. Chem. Eng. Data, 1966, 11(1), 17. Reamer, H. H. and Sage, B. H., J. Chem. Eng. Data, 1964 9(1), 24. Myers, H. S., Petroleum Refinery, 1957, 36(3), 175. Liu, Ε. K. and Davison, R. R., J. Chem. Eng. Data, 1981, 26(1), 85. Lin, H., Sebastion, H. M., Simnick, J. J., and Chao, K. C. J. Chem. Eng. Data, 1979, 24(2), 146. Glanville, J. W., Sage, Β. H., and Lacey, W. N. Ind. Eng. Chem., 1950, 42(3), 508. Chang, H. L. and Kobayashi, R. J. Chem. Eng. Data, 1967, 12(4), 517. Donnelly, H. G. and Katz, D.L. Ind. Eng. Chem., 1954, 46(3), 511. Fredenslund, A. and Mollerup, J. J. Chem. Soc., Faraday Transactions I, 1974, 70, 1653. Reamer, H. H. and Sage, Β. H. Ind. Eng. Chem., 1951, 43(11), 2515. Besserer, G. J. and Robinson, D. B. J. Chem. Eng. Data, 1973, 18(3), 298. Besserer, G. J. and Robinson, D. B. J. Chem. Eng. Data, 1975, 20(1), 93. Robinson, D. B., Kalra, H., Ng, H. J., and Kubota, H. J. Chem. Eng. Data, 1978, 23(4), 317. Sage, Β. H. and Reamer, H. H. J. Chem. Eng. Data, 1963, 8(4), 508. Yorizane, M., Yoshimura S., and Masuoka, H. Kagaku Kogaku, 1966, 30(12), 1093. Katayama, T. and Ohgaki, K. J. Chem. Eng. Data, 1976, 21(1), 53. In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

22.

51. 52. 53. 54. 55. 56. 57. 58. Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch022

59. 60. 61. 62. 63. 64. 65. 66. 67.

MAJEED AND WAGNER

Parameters from Group Contributions Equation

Robinson, D. B. and Ng, H. J. J. Chem. Eng. Data, 1976, 23(4), 325. Katz, D. L. and Carson, D. B. Trans. AIME, 1942, 146, 150. Zenner, H. G. and Dana, L. I. Chem. Engr. Progr. Symp. Ser. No. 44, 1963, 59, 36. Kobayashi, R., Chappealear, P. S., and Stryjek, R. J. Chem. Eng. Data, 1974, 19(4), 334. Schlinder, D. L . , Swift, G. W., and Kurata, F. Hydrocarbon Processing, 1966, 45(11), 205. Robinson, D. B., Kalra, Η., Ng, H. J., and Miranda, R. D. J. Chem. Eng. Data, 1978, 23(4), 321. Akers, W. W., Kehn, D. M., and Kilgore, C. H. Ind. Eng. Chem., 1954, 46(12), 2536. Yorizane, M., Sadamota, S., and Yoshimura, S. Kagaku Kogaku, 1968, 32(3), 257. Wagner, J., Erbar, R. C., and Majeed, A. I. Proceedings of the 64th Annual GPA Convention, Houston, Texas, March 18-20, 1985, 129. Turek, Ε. Α.; Metcalfe, R.S.; Yarborough, L.; and Robinson, R. L . , Jr. Soc. of Pet. Engr. Jour. 1984, 308. Robinson, D. B., and Besserer, G. J . Gas Processors Associations, Research Report RR-7, 1972. Robinson, D. B., Kalra, Η., Ng, H. J., and Kubota, H. J. Chem. Eng. Data, 1980, 25(1), 51. Reamer, H. H., Selleck, F. T., Sage, Β. H., and Lacy, W. N. Ind. Eng. Chem. 1953, 45, 1810. Yarborough, L. and Vogel, J . L. Chem. Engr. Prog., Symp. Series, No. 81, 1964, Vol. 63, 1. Robinson, D. B., Kalra, H., and Rempis, H. GPA Research Report, RR-31, May, 1981. Reamer, H. H., Sage, Β. H., and Lacey, W. N. Ind. Eng. Chem., 1953, 45(8), 1805. Kobayashi, R. and Katz, D. L. Ind. Eng. Chem., 1953, 45(2), 440.

RECEIVED

November 8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

47

23 Convergence Behavior of Single-Stage Flash Calculations 1

Marinus P. W. Rijkers and Robert A. Heidemann

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2C 1R4, Canada

Flash calculation procedures have been investigated to determine when a flash routine (1) converges to the trivial solution or (2) converges very slowly or not at a l l . The two mixtures used as the basis for the study are a CO containing natural gas and a CO -H S-CH mixture that shows liquid-liquid separations. The Peng-Robinson equation was used to describe all the equilibrium phases. The computation method is the successive substitution procedure with four different approaches for updating the Κ values. 2

2

2

4

When a s i n g l e e q u a t i o n o f s t a t e i s used t o model a l l t h e e q u i l i b r i u m phases i n a f l a s h c a l c u l a t i o n , a t r i v i a l s o l u t i o n can be reached i n w h i c h t h e c a l c u l a t e d phases a r e i d e n t i c a l and have a l l t h e p r o p e r t i e s of t h e feed stream. The n e c e s s a r y c o n d i t i o n s f o r e q u i l i b r i u m , i . e . , g

±

= In f

V ±

L

- in f . = 0

;

i=l,...,N

(1)

are o b v i o u s l y s a t i s f i e d by such a s o l u t i o n . Most o f t h e p r a c t i c a l e q u a t i o n s o f s t a t e have a t most threevolume r o o t s a t any p r e s s u r e and temperature. Whenever t h r e e volume r o o t s a r e obtained f o r the feed mixture i t i s p o s s i b l e t o avoid the t r i v i a l s o l u t i o n s i n c e a " l i q u i d " f e e d and "vapor" f e e d c a n be assigned different volumes at the flash conditions and t h e f u g a c i t i e s i n t h e l i q u i d - l i k e and v a p o r - l i k e phases w i l l s u r e l y be different. The problems w i t h t r i v i a l s o l u t i o n s can occur o n l y a t Τ and Ρ c o n d i t i o n s where t h e e q u a t i o n o f s t a t e f o r t h e f e e d m i x t u r e has o n l y one p o s i t i v e r e a l volume r o o t . There have been s e v e r a l s u g g e s t i o n s made f o r o b t a i n i n g from an e q u a t i o n o f s t a t e a " l i q u i d - l i k e " o r " v a p o r - l i k e " volume, as needed, i n r e g i o n s where t h e r e i s o n l y one volume r o o t ( 1 - 4 ) . These i d e a s are reported t o be h e l p f u l i n avoiding t r i v i a l solutions i n 7

Current address: Technische Hogeschool Delft, The Netherlands. 0097-6156/ 86/ 0300-0476506.00/ 0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

23.

Single-Stage Flash Calculations

RIJKERS A N D H E I D E M A N N

All

s i m u l a t i o n s where many e q u i l i b r i u m c a l c u l a t i o n s a r e performed i n sequence. An a l t e r n a t i v e approach i n d e a l i n g w i t h s u c c e s s i v e e q u i l i b r i u m computations has been to use converged s o l u t i o n s a t one o r more p o i n t s t o i n i t i a t e a new c o m p u t a t i o n ( 5 - 7 ) . I t i s assumed t h a t i n i t i a t i n g near the s o l u t i o n has the advantage of r e d u c i n g the number of i t e r a t i o n s r e q u i r e d t o r e a c h convergence and of i n c r e a s i n g the p r o b a b i l i t y of convergence t o the c o r r e c t , n o n - t r i v i a l , s o l u t i o n . I t has a l s o been r e p o r t e d t h a t , i n g e n e r a l , f l a s h r o u t i n e s can converge v e r y s l o w l y i n some p a r t s o f the P-T c o - e x i s t e n c e r e g i o n . M i c h e l s e n (8,9) has p r e s e n t e d an a n a l y s i s of the c o n v e n t i o n a l s u c c e s s i v e s u b s t i t u t i o n method and has shown v e r y c l e a r l y that r a p i d convergence c o u l d not be expected a t c o n d i t i o n s c l o s e t o the c r i t i c a l p o i n t of the m i x t u r e b e i n g f l a s h e d . In t h i s p a p e r , a s t u d y has been performed t o determine the e f f e c t o f the i n i t i a t i o n p r o c e d u r e on whether o r not the t r i v i a l s o l u t i o n i s reached. Also, several alternative flash calculation a l g o r i t h m s have been examined t o see what convergence b e h a v i o r can be expected i n a l l p a r t s of the phase diagram. Initiation The c a l c u l a t i o n s performed here a r e a l l based on use of Κ v a l u e s . F o r s u b s t a n c e i , p r e s e n t i n the l i q u i d and vapor phases w i t h mole f r a c t i o n s x^, and y^, r e s p e c t i v e l y , the phase d i s t r i b u t i o n c o e f f i c i e n t s a r e d e f i n e d by (2)

K. = y./x. The K. a r e e v e n t u a l l y t o be found from the Peng-Robinson state (10).

e q u a t i o n of

1

The

trivial

s o l u t i o n i s c h a r a c t e r i z e d by K_ = l f o r a l l i .

t i a t i n g computations w i t h

Ini

v a l u e s near u n i t y would appear t o r i s k

convergence t o the t r i v i a l s o l u t i o n . A w i d e l y used p r o c e d u r e i s t o i n i t i a t e from R a o u l t ' s l a w , K^=P^ /P where P_^ i s the vapor p r e s s u r e of s u b s t a n c e i a t the e q u i l i b r i u m t e m p e r a t u r e . A p r o p o s a l a s c r i b e d t o G.M. W i l s o n p e r m i t s e s t i m a t i o n of t h e s e R a o u l t ' s Law Κ v a l u e s from the pure component c r i t i c a l p r o p e r t i e s and a c e n t r i c f a c t o r s , i . e . S

S

Κ:. = ^

in

in(? C

/P) + 5.37 i

(1 + ω.)(1-Τ 1

/Τ)

(3)

i

I t would be i m p o s s i b l e t o examine a l l p o s s i b l e ways f o r i n i t i a t i n g the K^. We have chosen t o v a r y the i n i t i a l Κ v a l u e s from u n i t y through the W i l s o n v a l u e s a c c o r d i n g t o

in

Κ

= α

where α i s a s c a l a r .

£n The o v e r l i n e s a r e meant t o show t h a t in Κ and

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

478

in

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

are vector q u a n t i t i e s .

The Κ v a l u e s o b t a i n e d from (4) a r e f

u n i t y when α = 0 and a r e the W i l s o n R a o u l t s law v a l u e s when α= 1. We have found i t u s e f u l t o v a r y α i n t h e i n t e r v a l 0 < α £ 1.5 and t o observe t h e converged f l a s h c a l c u l a t i o n s t o see f o r what v a l u e s o f α the t r i v i a l s o l u t i o n i s o b t a i n e d . Flash

Calculations

G i v e n t h e Κ v a l u e s i t i s p o s s i b l e t o compute how a g i v e n f e e d m i x t u r e s e p a r a t e s i n t o two phases. The mass b a l a n c e e q u a t i o n s a r e o r g a n i z e d as d e s c r i b e d , f o r i n s t a n c e , by N u l l ( 1 1 ) , i . e . , f o r one mole o f feed w i t h mole f r a c t i o n s z^, (1-ν)χ + V K x Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

±

±

= z

±

(5)

±9

hence,

x

« z /[l + (K -1)V]

(6)

and

y

- Κ ζ / [ l + (K -1)V]

(7)

±

±

The l i q u i d and vapor mole f r a c t i o n s must sum t o u n i t y , o r h(v) - I (y.-χ,.) - I ( K i ) r

Z j

/[i+(K i)v] r

=o

(8)

F i n a l l y , e q u a t i o n (8) i s an e q u a t i o n i n one unknown, V, which can be s o l v e d by t h e Newton-Raphson p r o c e d u r e . The r e s u l t i n g v a l u e o f V, when s u b s t i t u t e d i n t o e q u a t i o n s (6) and ( 7 ) , g i v e s t h e e q u i l i b r i u m phase c o m p o s i t i o n s . The b e h a v i o r o f e q u a t i o n (8) has been a n a l y z e d by Nghiem e t a l . (12). They p o i n t o u t t h a t h(V) = 0 has one and o n l y one m e a n i n g f u l r o o t , 0 < V < 1, i f and o n l y i f h(0) £ 0 and h ( l ) < 0. I f e i t h e r o f these i n e q u a l i t i e s i s v i o l a t e d , t h e m i x t u r e w i l l be i n o n l y one phase. In p a r t i c u l a r , i f h ( 0 ) < 0, then V=0 and the m i x t u r e i s a l l i n the l i q u i d phase w i t h mole f r a c t i o n s x^ = ζ . The c a l c u l a t e d mole f r a c t i o n s i n the vapor phase, y^ = K^x^, w i l l n o t sum t o u n i t y . I f f u r t h e r c a l c u l a t i o n s a r e r e q u i r e d , i t i s necessary t o normalize; i.e. to set y, = K.x./ I K.x. i i ι J jJ S i m i l a r l y , i f h ( l ) > 0, then V = l , y mole f r a c t i o n s a r e found from x

i

=

(9) = z^ and m e a n i n g f u l

7

<νν Σ (y^V j

liquid 0 1

<>

In a l l t h e f l a s h c a l c u l a t i o n s performed i n t h i s s t u d y the above procedures were f o l l o w e d each time t h e were s p e c i f i e d . A f t e r ob t a i n i n g the phase mole f r a c t i o n s from e q u a t i o n s (6) and (7) ( o r (9) o r (10) i f n e c e s s a r y ) i t remained t o update the K^. I n e q u a t i o n o f s t a t e computations

the

a r e c e r t a i n l y c o m p o s i t i o n dependent.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

23.

Updating

479

Single-Stage Flash Calculations

RIJKERS A N D H E I D E M A N N

the Κ Values

Having o b t a i n e d t h e phase mole f r a c t i o n s from t h e Κ v a l u e s and mass balance equations, the f u g a c i t i e s are calculated from t h e Peng-Robinson e q u a t i o n , and t h e e q u i l i b r i u m c o n d i t i o n s o f e q u a t i o n (]) a r e checked. I f they a r e n o t s a t i s f i e d w i t h sufficient p r e c i s i o n the must be c o r r e c t e d . In t h e c o n v e n t i o n a l s u c c e s s i v e s u b s t i t u t i o n p r o c e d u r e , as d e s c r i b e d , f o r i n s t a n c e by Anderson and P r a u s n i t z ( 1 3 ) , t h e new Κ v a l u e s a r e o b t a i n e d from

κ. - <j>./. L

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

This equation coefficients i.e.,

i s derived

V Φ

f

i

L

and

from

the

definition

of the fugacity

V =

ί

(11)

v

(12)

/

(

y

i

P

)

(13)

L

<j>. - f . / ( P ) X i

w

and by e q u a t i n g f ^ and f ^ ~ . At

intermediate

stages

i n the c a l c u l a t i o n s ,

however, t h e

f u g a c i t i e s a r e n o t e q u a l . The v a l u e s o b t a i n e d f o r f ^ and f ^ f o l l o w L

from phase mole f r a c t i o n s t h a t , i n t u r n , a r e c a l c u l a t e d from the Κ v a l u e s used i n t h e p r e c e d i n g i t e r a t i o n . Mehra, e t a l . (14) observed t h a t e q u a t i o n (11) c a n be r e w r i t t e n i n forms t h a t emphasize i t s i t e r a t i v e character, p a r t i c u l a r l y κ<

η + 1 )

Or, s i n c e y./x. then i

ΙΪΓκ

=

[(y^HfJ/fJ)]

(14)

0 0

L

= K. and, s i n c e from e q u a t i o n (1) g. = & n ( f , ^ / f , ) , i i ι ι ι

( η + 1

>

-

Τ Γ κ ^

-I

(N)

(

1

5

)

where s u p e r s c r i p t s (n) and (n+1) i n d i c a t e t h e i t e r a t i o n l e v e l . The convergence p r o p e r t i e s o f f o u r Κ v a l u e u p d a t i n g a l g o r i t h m s were i n v e s t i g a t e d . 1.

Conventional Successive S u b s t i t u t i o n

(COSS)

Equation (15) i s t h e most w i d e l y used i n e q u a t i o n o f s t a t e c a l c u l a t i o n s t o up-date t h e Κ v a l u e s . T h i s procedure i s r e f e r r e d t o here as t h e " C o n v e n t i o n a l S u c c e s s i v e S u b s t i t u t i o n A l g o r i t h m " (COSS).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

480

2.

Accelerated Successive

S u b s t i t u t i o n 1.

(ACSS1)

Mebra e t a l (14) and Nghiem and Heidemann (15) have proposed t h a t e q u a t i o n (15) be m o d i f i e d by i n t r o d u c i n g a s c a l a r m u l t i p l i e r a t each η

i t e r a t i o n , λ^ ^, as f o l l o w s ; (n l)

(n)

+

£n Κ

=

An Κ

_ - λ "' g ν

( n )

( 1 6 )

v

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

Mehra viewed λ as a v a r i a b l e step l e n g t h i n a steep descent method f o r m i n i m i z i n g t h e Gibbs f r e e energy. Nghiem showed t h a t t h e formu l a e proposed by Mehra f o r e v a l u a t i n g λ were d e r i v a b l e from a c l a s s of Quasi-Newton a l g o r i t h m s d e s c r i b e d by Z e l e z n i k (16) as a p p l i e d t o s o l v i n g t h e s i m u l t a n e o u s e q u a t i o n s g = 0 w i t h £n Κ as the v e c t o r o f independent v a r i a b l e s . The s i m p l e s t procedure proposed by Mehra y i e l d s ( η 1 ) _ (η)

λ

+

λ

|5

which i s t o be i n i t i a t e d λ

( 0 )

= λ

<η)Τ£(η) (η)Τ £(η 1) _ /5

(

,

+

with

= 1.0

( 1 )

(18)

T h i s i s t h e procedure r e f e r r e d t o h e r e as A c c e l e r a t e d S u b s t i t u t i o n 1 (ACSS1). 3.

( l y )

A c c e l e r a t e d S u c c e s s i v e S u b s t i t u t i o n 2.

Successive

(ACSS2)

R i j k e r s (17) has proposed an a l t e r n a t i v e t o e q u a t i o n (17) f o r a c c e l e r a t i n g the successive s u b s t i t u t i o n procedure. His analysis f o l l o w s M e h r a s (14) approach but t r e a t s t h e m i n i m i z a t i o n problem i n a somewhat d i f f e r e n t manner. R i j k e r s a l g o r i t h m i s 1

1

χ

(η 1) +

=

χ

(η) | ( n ) T - ( n ) ( n ) T K

/ K

( { ;

( n l ) _ ^n) +

)}

( H > )

with λ

( 0 )

= 1.0

( 1

= λ >

The v e c t o r h i n R i j k e r s h = U"

1

(20) 1

a l g o r i t h m i s d e f i n e d by

i

(21)

where t h e m a t r i x IJ * has elements

= V ( l - V)(x y /z )[6 j

i

i

ij

+ x.y./(z.S)]

(22)

where 6 . i s t h e k r o n e k e r d e l t a and where S i s d e f i n e d by S = 1 "

I

Y

k

/ \

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(23)

23.

RIJKERS A N D H E I D E M A N N

481

Single-Stage Flash Calculations

T h i s a l g o r i t h m i s r e f e r r e d t o here as " A c c e l e r a t e d S u b s t i t u t i o n 2" (ACSS2). 4.

Newton-Raphson U p d a t i n g .

Successive

(NR)

A f o u r t h a l t e r n a t i v e f o r u p - d a t i n g the Κ v a l u e s i s the NewtonRaphson p r o c e d u r e . The two a c c e l e r a t e d s u c c e s s i v e s u b s t i t u t i o n p r o c e d u r e s a r e Quasi-Newton a l g o r i t h m s f o r s o l v i n g

iflSTK) = δ

( 2 4 )

The Newton-Raphson p r o c e d u r e i s

( 2 5 )

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

J_ ΔΪη~Κ = - g where J_ i s t h e J a c o b i a n m a t r i x . as the product J = W U"

I t i s most c o n v e n i e n t

t o express J_

1

(26)

where W has elements w..

^J

L

L

V

= ( 3 J l n f / 3 n ) + (3£nfY/3n ) i

J

î

(27)

J

and U * i s as d e f i n e d i n e q u a t i o n ( 2 2 ) . The T r i v i a l S o l u t i o n and S t a b i l i t y R e f e r e n c e was made e a r l i e r t o M i c h e l s e n ' s (8,9) a n a l y s i s o f t h e s u c c e s s i v e s u b s t i t u t i o n p r o c e d u r e . One c o n c l u s i o n t o be drawn from h i s a n a l y s i s i s t h a t s u c c e s s i v e s u b s t i t u t i o n cannot converge t o t h e trivial s o l u t i o n a t any p o i n t where the homogeneous phase i s thermodynamically unstable. There a r e many e q u i v a l e n t ways t o s t a t e t h e c r i t e r i a f o r s t a b i l i t y i n a homogeneous phase. M e c h a n i c a l s t a b i l i t y r e q u i r e s that OP/3v)

T

< 0

(28)

D i f f u s i o n a l s t a b i l i t y r e q u i r e s t h a t m a t r i x _B w i t h elements

b.. - ( n n f . / a n . ) ^ ^

(29)

s h o u l d be p o s i t i v e d e f i n i t e . The l i m i t of s t a b i l i t y i s l o c a t e d where the d e t e r m i n a n t of _B i s zero and whenever d e t ( B ) < 0 t h e m i x t u r e i s u n s t a b l e . However, f o r 15 t o be p o s i t i v e d e f i n i t e r e q u i r e s t h a t t h e d e t e r m i n a n t of a l l t h e p r i n c i p a l minors of Β s h o u l d a l s o be p o s i t i v e (Amundson ( 1 8 ) ) . F i g u r e s 1 and 2 show the phase envelopes f o r two m i x t u r e s . The f i r s t m i x t u r e i s a C O ^ - c o n t a i n i n g n a t u r a l gas. The second i s a CH^ -H^S-CO^ t e r n a r y m i x t u r e

t h a t shows l i q u i d - l i q u i d s e p a r a t i o n s .

drawn i n F i g u r e s 1 and 2 a r e t h e curves

corresponding

to (3P/3v)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Also T

=0

482

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

80.0

HO.O 150.0

ι i ι i 160.0 170.0 180.0 190.0 200.0

\ i 210.0 220.0

ι ι 230.0 240.0 250.0 260.0

TEMPERATURE (K) Figure

1.

Phase Boundary and S t a b i l i t y L i m i t s f o r M i x t u r e

I

160.0

140.0

4

120.0 H S

ν

α: <

ιοο.ο

£

80.0

COEXISTENCE CURVE DET (B) = 0 (dP/dv)--o

ZD

ω Lu

60.0

CL

20.0 0.0 160.0 F i g u r e 2.

ι 180.0

i 200.0

i ι • ι 220.0 240.0 260.0 TEMPERATURE (K)

ι 280.0

» 300.0

Phase Boundary and S t a b i l i t y L i m i t s f o r M i x t u r e

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

320.0 II

23.

RIJKERS A N D H E I D E M A N N

and d e t ( B ) = 0.

Single-Stage Flash Calculations

I n s i d e these curves

the m i x t u r e s

483

are u n s t a b l e

and,

according to Michelsen's a n a l y s i s , i t would be i m p o s s i b l e f o r s u c c e s s i v e s u b s t i t u t i o n f l a s h a l g o r i t h m s t o converge t o trivial solutions. The r e g i o n where (3P/3v> > 0 c o r r e s p o n d s t o the r e g i o n where T

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

the e q u a t i o n of s t a t e has t h r e e volume r o o t s . The boundary curve of the r e g i o n has been c o n s t r u c t e d by f i x i n g the volume, c a l c u l a t i n g the temperature t h a t makes (3Ρ/3ν)^ = 0 and then c a l c u l a t i n g the p r e s s u r e from the e q u a t i o n s of s t a t e e v a l u a t e d a t the volume and temperature. The cusp i s the " m e c h a n i c a l c r i t i c a l p o i n t " f o r the m i x t u r e and i s w e l l i n s i d e the two phase r e g i o n . The d i f f u s i o n a l s t a b i l i t y l i m i t , d e t ( B ) = 0, was e v a l u a t e d i n the same manner as the l i m i t of m e c h a n i c a l s t a b i l i t y and as d e s c r i b e d by Heidemann and K h a l i l ( 1 9 ) . The c r i t i c a l p o i n t of the m i x t u r e l i e s on t h i s c u r v e . For the n a t u r a l gas m i x t u r e , the two b o u n d a r i e s f o r the unstable region enclose o n l y a s m a l l p a r t of the two-phase r e g i o n . The u n s t a b l e r e g i o n f o r the CH^ - l ^ S - CO^ m i x t u r e i n c l u d e s most of the phase diagram. I n s i d e these u n s t a b l e r e g i o n s s o l u t i o n i s impossible with successive s u b s t i t u t i o n .

the

trivial

I n i t i a t i o n and the T r i v i a l S o l u t i o n For the two m i x t u r e s of F i g u r e s 1 and 2 a l a r g e number of f l a s h c a l c u l a t i o n s have been performed. For the m i x t u r e of F i g u r e 1, the two-phase r e g i o n was i n t e r s e c t e d a t p r e s s u r e s of 20, 65 and 70 bar and f l a s h c a l c u l a t i o n s were performed w i t h v a r y i n g i n i t i a l Κ v a l u e s over the whole temperature range w i t h i n the two-phase r e g i o n . For the m i x t u r e of F i g u r e 2, e q u i v a l e n t c a l c u l a t i o n s were performed at 40 and 100 b a r . F i g u r e s 3 through 6 and F i g u r e s 7 and 8 c o n t a i n the r e s u l t s f o r the two m i x t u r e s a t the v a r i o u s p r e s s u r e s . The i n i t i a l Κ v a l u e s were varied, as described earlier, by letting the "Phase D i s t r i b u t i o n C o e f f i c i e n t I n i t i a t i o n Parameter", a, take on v a l u e s between z e r o and 1.5. The v a l u e of α i s shown on the o r d i n a t e of the F i g u r e s and the temperature i s shown on the a b s c i s s a . The a r e a s of the Τ - α p l a n e are l a b e l l e d t o show the s o l u t i o n reached. For the f i r s t m i x t u r e a t 20 b a r , the two-phase r e g i o n extends from 170 to 254 K. Between 170 and 194 Κ the m i x t u r e i s e i t h e r mechanically or d i f f u s i o n a l l y unstable or both. As e x p e c t e d , the c o n v e n t i o n a l s u c c e s s i v e s u b s t i t u t i o n a l g o r i t h m d i d not converge t o the t r i v i a l s o l u t i o n w i t h i n t h i s temperature i n t e r v a l f o r any α v a l u e . T h i s i s i n d i c a t e d by the c o r r e s p o n d i n g c r o s s - h a t c h e d r e g i o n of F i g u r e 3. A l s o shown i n F i g u r e 3 i s the r e g i o n of low α v a l u e s , o u t s i d e the u n s t a b l e temperature i n t e r v a l , where COSS converged t o the trivial solution. To a v o i d the t r i v i a l s o l u t i o n i t i s only n e c e s s a r y t o have α l a r g e enough. Near the phase boundary, 254 K, l a r g e r α values are r e q u i r e d . However, no t r i v i a l s o l u t i o n s were found a t any temperature so l o n g as α was g r e a t e r than about 0.45. I n p a r t i c u l a r , i f the W i l s o n Κ v a l u e s are used (a = 1 ) , the t r i v i a l s o l u t i o n would n e v e r be reached a t 20 bar and any temperature.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

484

EQUATIONS OF

STATE: T H E O R I E S A N D

APPLICATIONS

F i g u r e 4 shows convergence b e h a v i o r w i t h COSS a p p l i e d t o the f i r s t m i x t u r e a t 65 b a r . The two-phase r e g i o n i s c o n t a i n e d between 215 and 248 K. (The p r e s s u r e i s above the " m e c h a n i c a l c r i t i c a l p r e s s u r e " , hence the m i x t u r e i s not m e c h a n i c a l l y u n s t a b l e a t any temperature.) Figure 4 i n d i c a t e s that t r i v i a l s o l u t i o n s were reached a t temperatures between the phase boundary and the s t a b i l i t y l i m i t when the Κ v a l u e s were i n i t i a t e d too near u n i t y (a below about 0.6). At temperatures g r e a t e r than 218 K, the t r i v i a l s o l u t i o n was avoided f o r α > 0.35. C e r t a i n l y , no trivial s o l u t i o n s were encountered when COSS was i n i t i a t e d a t the W i l s o n Κ v a l u e s (a = 1 ) . A l t h o u g h convergence t o the t r i v i a l s o l u t i o n i s i m p o s s i b l e i n the u n s t a b l e r e g i o n , the number of i t e r a t i o n s r e q u i r e d c o u l d become large. In F i g u r e 4, open boxes are used to i n d i c a t e p o i n t s where convergence was not a c h i e v e d . Convergence f a i l u r e i s d i s c u s s e d i n more d e t a i l l a t e r . The c a l c u l a t i o n s f o r m i x t u r e I a t 65 bar were repeated u s i n g the f i r s t a c c e l e r a t e d s u c c e s s i v e s u b s t i t u t i o n p r o c e d u r e (ACSS1). The convergence b e h a v i o r i s shown i n F i g u r e 5. I t w i l l be seen t h a t ACSSl and COSS behave s i m i l a r l y as r e g a r d s the tendency t o converge to the t r i v i a l s o l u t i o n . S i n c e b o t h p r o c e d u r e s can be regarded as steep descent a l g o r i t h m s , n e i t h e r can reach the t r i v i a l s o l u t i o n i n the u n s t a b l e temperature i n t e r v a l . A s i d e from some i s o l a t e d p o i n t s , ACSSl and COSS converge t o the c o r r e c t s o l u t i o n a t the same v a l u e s of a. F i g u r e 6 shows the r e s u l t s of ACSSl f l a s h c a l c u l a t i o n s f o r m i x t u r e I at 70 b a r . The i s o b a r at t h i s p r e s s u r e does not i n t e r s e c t the u n s t a b l e r e g i o n . P o t e n t i a l l y , t h e r e f o r e , the t r i v i a l s o l u t i o n c o u l d be reached a t any temperature w i t h i n the two-phase r e g i o n between about 221 and 244 K. The f i g u r e shows, however, t h a t i f α i s l a r g e enough, ( i . e . , α > 0.6) the t r i v i a l s o l u t i o n i s never r e a c h e d . The r e s u l t s are of the same type as seen i n F i g u r e s 3-5 and demonstrate t h a t i n i t i a t i n g w i t h the W i l s o n Κ values i s sufficient to avoid t r i v i a l s o l u t i o n s even a l o n g a difficult retrograde isobar. Convergence r e s u l t s f o r the m i x t u r e of F i g u r e 2 a t 40 and 100 b a r , u s i n g COSS, are shown i n F i g u r e s 7 and 8. The i s o b a r at 40 bar extends from a low-temperature l i q u i d - l i q u i d r e g i o n , through a v a p o r - l i q u i d r e g i o n , t o the phase boundary a t about 287 K. The m i x t u r e i s t h e r m o d y n a m i c a l l y u n s t a b l e a t temperatures below 259 K, t h e r e f o r e i t i s i m p o s s i b l e f o r COSS to converge t o the trivial s o l u t i o n at any temperature below 259 K, r e g a r d l e s s of the i n i t i a l Κ v a l u e s used. The p l o t i n F i g u r e 7 c o n t a i n s o n l y a v e r y s m a l l r e g i o n (which i s not c r o s s - h a t c h e d ) bounded by 259 < Τ < 287 Κ and α < 0.3 i n w h i c h t r i v i a l s o l u t i o n s were found. As shown i n F i g u r e 7, two d i f f e r e n t two-phase s o l u t i o n s were p o s s i b l e i n the temperature i n t e r v a l 188 < Τ < 202 Κ. Depending on the i n i t i a l Κ v a l u e s , i t was p o s s i b l e t o r e a c h e i t h e r a v a p o r - l i q u i d solution or a l i q u i d - l i q u i d solution. There i s o b v i o u s l y a t h r e e - p h a s e r e g i o n i n t h i s v i c i n i t y and i t s e x i s t e n c e c o m p l i c a t e s f l a s h c a l c u l a t i o n s when the number of phases s p e c i f i e d i s l e s s than the number of phases w h i c h s h o u l d a c t u a l l y e x i s t a t e q u i l i b r i u m ( a c c o r d i n g t o the model). No attempt was made t o l o c a t e the three-phase r e g i o n . I t s h o u l d be n o t e d , however, t h a t convergence

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

RIJKERS A N D H E I D E M A N N

!

!

160.0

180.0

F i g u r e 3.

Single-Stage Flash Calculations

I

200.0

I

220.0

TEMPERATURE (K)

485

1

»

240.0

260.0

Region o f S o l u t i o n f o r M i x t u r e I a t 20 Bar

TEMPERATURE (K) F i g u r e 4.

Region o f S o l u t i o n f o r M i x t u r e I a t 65 Bar Using COSS

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

486

210.0

215.0

220.0

225.0

230.0

235.0

240.0

245.0

250.0

255.0

TEMPERATURE (K) F i g u r e 5.

Region o f S o l u t i o n f o r M i x t u r e I a t 65 Bar U s i n g

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ACSS1

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

23.

RIJKERS A N D H E I D E M A N N

Τ 180.0

F i g u r e 7.

ι 180.0

F i g u r e 8.

I 200.0

487

Single-Stage Flash Calculations

I 220.0

I I 240.0 260.0 TEMPERATURE (K)

I 280.0

Region o f S o l u t i o n f o r M i x t u r e II

ι

ι

200.0

220.0

ι

1

240.0 260.0 TEMPERATURE (K)

I 300.0

r 320.0

a t 40 Bar

1

I

»

280.0

300.0

320.0

Region o f S o l u t i o n f o r M i x t u r e II

a t 100 Bar

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

488

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

to a n o n - t r i v i a l s o l u t i o n was o b t a i n e d a t a l l temperatures and a t a l l α v a l u e s i n t h e v i c i n i t y o f t h e t h r e e phase r e g i o n . F i g u r e 8 shows convergence p r o p e r t i e s o f t h e m i x t u r e o f F i g u r e 2 a t 100 b a r . The p r e s s u r e i s j u s t below t h e d i p i n t h e phase boundary curve a t about 235 Κ so t h a t t h e m i x t u r e remains i n two phases from v e r y low temperatures t o about 309 K. There i s a temperature i n t e r v a l from about 228 t o 249 Κ where t h e m i x t u r e i s outside the l i m i t of d i f f u s i o n a l s t a b i l i t y . L i k e w i s e , from about 297 t o 309 Κ t h e m i x t u r e i s o u t s i d e t h e l i m i t o f s t a b i l i t y . Only i n these two s e p a r a t e i n t e r v a l s a r e t r i v i a l s o l u t i o n s p o s s i b l e when flash c a l c u l a t i o n s a r e performed using COSS. These trivial s o l u t i o n s a r e avoided i f α i s g r e a t e r than about 0.45, as shown i n F i g u r e 8. A l s o i n d i c a t e d i n F i g u r e 8 a r e a few p o i n t s where convergence was n o t a c h i e v e d . These p o i n t s a r e a l l i n t h e i n t e r m e d i a t e s t a b l e temperature i n t e r v a l w i t h v e r y l o w α v a l u e s . Slow Convergence Problems In some r e g i o n s o f t h e phase diagrams t h e s u c c e s s i v e s u b s t i t u t i o n a l g o r i t h m s converged v e r y s l o w l y . I n o r d e r t o reduce computation time a d u a l a l g o r i t h m was used. The computations were begun w i t h the s u c c e s s i v e s u b s t i t u t i o n p r o c e d u r e , b u t i f convergence was n o t a c h i e v e d w i t h i n a r e a s o n a b l e number o f i t e r a t i o n s t h e Newton-Raphson a l g o r i t h m f o r u p d a t i n g t h e Κ v a l u e s was begun. T h i s procedure d i d n o t change t h e c o n c l u s i o n s as t o whether o r not t h e t r i v i a l s o l u t i o n was u l t i m a t e l y t o be reached so l o n g as t h e f l a s h c a l c u l a t i o n was w e l l on i t s way t o convergence and so l o n g as t h e r e was t o be a s p l i t i n t o two phases o f f i n i t e amount. The second c o n d i t i o n i s n e c e s s a r y because t h e J a c o b i a n m a t r i x of e q u a t i o n (26) i s u n d e f i n e d when V = 0 o r V = 1. The problem a r i ses w i t h m a t r i x IJ * g i v e n by e q u a t i o n (22) . The d u a l a l g o r i t h m improved convergence b e h a v i o r c o n s i d e r a b l y . In p r e p a r i n g F i g u r e 4, 3283 independent f l a s h c a l c u l a t i o n s were required. W i t h COSS a l o n e , 238 o f these c a l c u l a t i o n s d i d n o t converge w i t h i n 100 i t e r a t i o n s . W i t h t h e d u a l a l g o r i t h m o n l y 27 poorly i n i t i a t e d c a l c u l a t i o n s were l e f t unconverged. Figure 7 r e q u i r e d 3179 f l a s h c a l c u l a t i o n s , 382 o f w h i c h d i d n o t converge w i t h COSS a l o n e . A l l c a l c u l a t i o n s converged w i t h t h e d u a l a l g o r i t h m . The reason f o r convergence f a i l u r e even o f t h e Newton-Raphson p r o c e d u r e r e q u i r e s some d i s c u s s i o n . What was a c h i e v e d i n these cases was an o s c i l l a t o r y b e h a v i o r between a s o l u t i o n l i k e t h e t r i v i a l s o l u t i o n and one l i k e t h e c o r r e c t s o l u t i o n . These f a i l u r e s were g e n e r a l l y a s s o c i a t e d w i t h i n i t i a l guesses f o r t h e Κ v a l u e s near u n i t y ( a near z e r o ) . Speed o f Convergence The speeds o f convergence o f f o u r d i f f e r e n t a l g o r i t h m s f o r u p d a t i n g the Κ v a l u e s a r e compared i n F i g u r e s 10 through 13. The m i x t u r e o f F i g u r e 1 i s t h e b a s i s f o r t h e comparison. F l a s h c a l c u l a t i o n s were performed a t 930 p o i n t s throughout t h e two-phase r e g i o n f o r t h i s m i x t u r e and a t each p o i n t t h e c a l c u l a t i o n was i n i t i a t e d w i t h t h e W i l s o n Κ v a l u e s (a = 1 ) .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

23.

RUKERS AND HEIDEMANN

Single-Stage

Flash

Calculations

489

F i g u r e 9 s h o w s t h e COSS r e s u l t s . A t e a c h o f t h e 930 p o i n t s t h e number o f iterations to convergence has been inserted in the diagram. Through most o f the two-phase region, convergence was o b t a i n e d i n 25 i t e r a t i o n s o r f e w e r a n d t h e a v e r a g e i t e r a t i o n c o u n t t o c o n v e r g e n c e was j u s t o v e r 1 5 . A t o n l y t h r e e p o i n t s o f t h e 930 was c o n v e r g e n c e u n a t t a i n a b l e i n 100 i t e r a t i o n s a n d t h e s e p o i n t s a r e very c l o s e to the c r i t i c a l p o i n t of the m i x t u r e . F i g u r e s 10 a n d 11 show i t e r a t i o n c o u n t s f o r t h e A C S S 1 a n d A C S S 2 algorithms, respectively. These two algorithms perform quite c o m p a r a b l y , w i t h a v e r a g e i t e r a t i o n s t o convergence o f 9.84 and 9 . 8 5 . ACSS1 is apparently superior to ACSS2 on the grounds that convergence was o b t a i n e d w i t h i n 100 i t e r a t i o n s a t a l l 9 3 0 p o i n t s w i t h ACSS1 and ACSS2 f a i l e d t o c o n v e r g e a t one p o i n t . T h e t o t a l e x e c u t i o n t i m e a t 9 3 0 p o i n t s was 1 8 1 . 8 , 1 4 1 . 6 , a n d 1 6 6 . 3 s e c o n d s f o r COSS, ACSS1 and A C S S 2 , r e s p e c t i v e l y . On t h i s a c c o u n t A C S S 1 a p p e a r s a l s o t o be s u p e r i o r t o A C S S 2 a n d t o s u c c e s s i v e s u b s t i t u t i o n w i t h o u t a c c e l e r a t i o n (COSS). F i g u r e 12 shows t h e r e s u l t s o b t a i n e d w h e n t h e Newton-Raphson a l g o r i t h m was employed to update Κ values. As b e f o r e , these calculations were a l l initiated with the Wilson Κ values. A s t r i k i n g r e s u l t i s t h a t t h e a v e r a g e number o f i t e r a t i o n s p e r c o r r e c t s o l u t i o n i s o n l y 4 . 7 1 , l e s s than h a l f the average of the other three procedures. However, the iteration count g i v e s only a partial p i c t u r e o f t h e e f f i c i e n c y o f t h e a l g o r i t h m a n d two n e g a t i v e factors must be c o n s i d e r e d . The f i r s t n e g a t i v e f e a t u r e i s t h a t t h e N-R u p d a t i n g p r o c e d u r e f o r t h e Κ v a l u e s f a i l e d t o r e a c h t h e c o r r e c t s o l u t i o n a t 40 o f 9 3 0 points. T h i s p o o r c o n v e r g e n c e b e h a v i o r was a r e s u l t o f a n u n d e f i n e d J a c o b i a n at the phase b o u n d a r i e s . I f e q u a t i o n s ( 6 ) a n d (7) d i d n o t y i e l d a two p h a s e s o l u t i o n , s m a l l q u a n t i t i e s o f L o r V w e r e i n t r o duced. C o n s e q u e n t l y the x^ and y^ were e v a l u a t e d from e q u a t i o n s (6) o r (7) a n d w e r e n o r m a l i z e d b y a p p l y i n g e q u a t i o n s ( 9 ) o r ( 1 0 ) . This approach c l e a r l y d i d not solve the problem: At 25 p o i n t s the t r i v i a l s o l u t i o n w a s r e a c h e d a n d a t t h e r e m a i n i n g 15 p o i n t s the procedure d i d not converge. The second point to consider is that each N-R iteration r e q u i r e s c o n s i d e r a b l e m a t r i x m a n i p u l a t i o n and i s much more c o n s u m i n g of computer time than a s u c c e s s i v e substitution iteration. The total time required for the 930 p o i n t s o f Figure 12 i s 424.3 s e c o n d s ; m o r e t h a n t w i c e t h e t i m e r e q u i r e d f o r t h e COSS p r o c e d u r e . The f a i l u r e t o c o n v e r g e i s no d o u b t a more s e r i o u s c o n c e r n t h a n t h e computer t i m e consumed i n t h e N-R p r o c e d u r e . Computing time c e r t a i n l y depends on t h e s k i l l o f t h e programmer and no d o u b t a N-R r o u t i n e c a n be w r i t t e n w h i c h w i l l be more e f f i c i e n t t h a n o u r s . Discussion All examples presented here support the conclusion that the successive s u b s t i t u t i o n a l g o r i t h m cannot converge to the trivial s o l u t i o n , h o w e v e r i n i t i a t e d , a t p o i n t s where t h e homogeneous phase is unstable. This property i s due t o t h e n a t u r e o f successive s u b s t i t u t i o n as a steep d e s c e n t p r o c e d u r e f o r m i n i m i z i n g the G i b b s free energy.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

490

80.0

70.0 4

Cl C2 C3 nC4 r»C5 nC6

-

0.8608 0.0247 0.0067 0.0045 0.0024 0.0009

ALGORITHM USED : COSS MO. Of CALC. EXEC: 930 NO. Of CALC CONV.: 927 N O . or com. soot: 9 2 7 TOT. NO. ITERS. : 14.705 m. ΙΤ/CORftSOLM. : 15.5394 EXEC TIC s : «1787 COEXISTENCE

50.0

152 4336 32 28252321 20«17 I «42363229262321 2 0 « 17 « 16 V «35322926242220» « 17 « IS V

CURVE

DET(B) = 0 (dP/âv) = o

4 Ο

CALCULATION

NOT

J 21 21 2 0 2 0 » « 1 7 » 19 β Η 13 13 12 Ο η H 1 JO» «««««17«««Μ131312121111111 1««««««17«««1413131212111111»11 «17T7171717«1S15H1313121211 H » » » 10» c

CONVERGED

si-!*

40.0

17171

*»»»*»»»»***»»

7 1 5 1 3 » » « « « 1 5 1 4 13 13121211 11 « 10 « r « M M « « « « « M 1 3 l 3 1 2 1 2 1 1 11 « « » B«13H«««HH13131211 H H « « »~ ---- - JH13««eHH13l312l11111»» ' ' f- à Ζ Ί ï * » « « « » » »

30.0

i

s

I

H

1 4

1 4

β

1 5

H

H

fl

, ï 5 î î > 2 ^ Η Η β β β ΐ 4 « β β ι ι η β β » 9 5 5 1 t ï « e i l 11 » » « 9 9 1 » À5 S 2 5 £A * « « « H 1 3 1 3 1 2 Î 1 Î 1 » » 999881 S 17 « « « 14 M « / » 17 W W « 1 3 l 3 l 2 t t » » » 999888 f~21 2 0 » « 517 14 « *2021£Ϊ* « 8 8 98 97 87 87 87 8 7 7 6 7 6 ^ 202\ « V T «71«3H11 2 2ΤΙ1 «2 1101 » 9 8 8 7 7 7 7 7 6 6 6 6 10222221 2 0 » « 17 « « 2 2 2 2 « * 14 1312 11 Ο 9 9 7 7 7 7 6 6 6 6 6 6 6 24232220» « « Τ7 » 2 3 2 3 2 2 · β 43Ί2 ΤΙ « 9 9 8 8 7 6 6 6 6 6 6 6 5 5 5 Ι 2 6 2 3 2 1 2 0 « « « « « 2 4 2 9 2 4 » « 1 3 1 % if » 9 9 8 8 7 7 6 6 6 6 5 5 5 5 5 2321 » « « « « » 21 32322217 14 12 H » Α / 9 8 8 7 7 7 7 6 g ^7 7 6 6 6 6 6 6 5 5 5 5 5 5 5 5 17 17 » » 2031 442317 13 12 « τ 240.0 180.0 200.0 220.0 160.0 ΰ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

20.0

1

M

e

e

e

e

M

U

1 3

1 7 1 8 1 3

10.0

4

9

8

8

7

1 7

#

260.0

TEMPERATURE (Κ) Figure Wilson

9. Iteration Κ Values

Counts

f o r Mixture

I.

COSS

Initiated

with

80.0

70.0 Η

cokrosmoN C02 - O.ttOO CI - 0.8606 C2 - 0.0247 C3 - 0.0067 nC4 - 0.0045 nC5 - 0.0024 nC8 - 0.0009

ALGORITHM USED : ACSS1 NO. Or CALC EXEC: 930 NO. Or CALC CONV.: 930 NO. ( Τ COfM. SOUL: 930 TOT. NO. ITERS. : 9.«2 Mf. IT/C0RR.SOLR : 9.8409 EXEC TIC · : 141596 . COEXISTENCE

50.0

4

• (dP/dv) 40.0

30.0

20.0

0.0

-I

CURVE

• DET(B)=0 =0

. n a n e n ctn n β » 9 β β β 9tt»121t131311»» 9 8 8 88 .. _ 8 1 1 1 3 1 2 1 3 1 3 1 1 1 1 » » 9 8 8 8 8 « 1 2 » 913H121211»»» 988888 *2»111112»1111»tt 99888887 » « 911 H » tt tt » 9 8 8 8 8 8 8 7 7 _ . 911 tf JO » 9 1 1 1 1 1 1 1 1 1 1 » » 9 8 8 8 8 8 7 7 7 99 9 H J D / 9 » » H H 1 2 1 I » » » 9 8 8 8 8 7 7 7 7 9 9»»/M0»»H1211tttttt 9 9 8 8 8 7 7 7 7 7 , ~ . . . » » » .V· 9 » 11 12 11 » « » 9 9 8 8 8 7 7 7 7 6 6 "9 9H tt » 9 9/fTi 1 t H 1 2 » 1 1 » t t 9 9 8 8 8 7 7 7 7 6 6 6 9 9 9«/l» 11 11 12 H 11 tt tt 9 9 8 8 7 7 7 7 6 6 6 6 6 I 9 911 W « H « « « β » 9 9 8 8 7 7 7 6 6 6 6 6 6 6 8 » 9 9 911 13 flAl 12 11 « « " 9 9 8 7 7 7 6 6 6 6 6 6 6 9*5" ^ ^811 » » 9 » 9 911 13 14/13/12 » 9 » 9 8 7 7 7 7 7 6 6 6 6 6 6 9 9 9 9 7 7 7 6 6 6 6 6 6 6 5 5 S S 5 S 6 l l » « « 1 1 1 2 » 912 14 14 I f f 11 9 9 9 9 7 6 6 6 6 6 6 6 9 9 9 9 9 9 " «1312121211 H » 1 2 « « M 1 2 21 H1 W I 1 7 7 7 7 6 6 6 6 6 6 9 9 9 9 9 9 9 9 6 6 6 6 9 9 9 9 9 9 9 9 ' " 1 2 M 1 3 1 3 1 3 » 1 4 12 WT7M1211 ~ 6 9 9 9 9 9 9 4 4 4 H 1 3 1 2 » H It II 1 2 H » « M 1 2 » 8 8 8 1/7 7 6 6 6 6 6 n n n n n « e i 3 i i 9 9 8 8 7 6 6 6/6/« 6 6 5 9 5 9 9 4 4 4 4 4 4 I 240.0 160.0 180.0 200.0 220.0

260.0

TEMPERATURE (K) Figure Wilson

10. I t e r a t i o n Κ Values

Counts

f o r Mixture

I.

ACSS1

Initiated

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

with

23.

RIJKERS A N D H E I D E M A N N

COMPOS

70.0

J

-

Flash

491

Calculations

ALCORfTHM USED : ACSS2 MO. OF CALC. EXEC: 930 NO OF CALC. CONV.. 929

mow

C 0 2 - o.ttoo C I - αββΟβ

C2 C3 nC4 nC3 nC6

Single-Stage

0 0247 C 0067 30045 0 0024 00009

NO. UT CORR. SOLK.: 929

TOT. NO. ITERS. : 9.255 M. IT/CORK.SOLK : 9.8547 EXEC. T I C > : «6.336

COEXISTENCE

CURVE

• DET(B)-0

{dP/dv)--o n 1012 9 1 0 » 12 10tttttt 12 9 9 9 9

40.OJ tu H « « jf « βΗ η « 9yn m ' 9 1 4 9«Î1 tt/« 9 ' 8 8tt η τιtt» » tt ^ 9 9 9 « T1tt« Λ » 8 99 9 T i n « 9 « 8 9 913 910 « 9 « « '910 9 9 913 12 910 9 « « /9A3 « « -^8 913 13 12 8W 9 9 9T1/iZtt 9 9 „ 8 913 1 3 Π 8 9 9 9 912 1 2 V 9 9 9 ^9 810 12 12 tt 8 9 910 13 13 12tt/ 9 9 8 8 913 U 10 10 9 9 910 913\ Hn 12 1210 tt«fett y « 99888 8 ilO 9tt 10 8 β 9 913 13 10 ' ' 9 8 8 7 I ΠH tttt tt'tt 9 9TC tt 8 8 8tt14Ul3T1T1tttt • 10 10 A / 8 7 7 6 6 5 9 9T0 1 3 « 1 4 T 1 1 2 « 1 0 9 9 88 7 7/β/β 6 6 5 5

20.0

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

tttt

r

0.0

160 Figure Wilson

11. I t e r a t i o n Κ Values

COMPOSITION

C02 - 0.1000 C1 - 0.8606 C2 - 0.0247 C3 - 0.0067 nC4 - 0.0045 nC5 - 0.0024 nC8 — 0.0009

180.0

200.0 TEMPERATURE (K)

Counts

f o r Mixture

220.0 I.

ACSS2

240.0 Initiated

260.0 with

ALGORITHM USED : NR NO. Or CALC. EXEC: 930 NO. Or CALC CONV.: 9B NO. Or CORK. SOLK: 890 TOT. NO. ITERS. : 4,661 m. IT/CORR.SOLN. : 4.7135

EXEC TME ·

COEXISTENCE

: 424.301

CURVE

DET(B)=0

(dp/dv) = o 1—Ι — l

1

CALCULATION CONVERGED

NOT

C>3 C A L C . G I V E S T R I V I A L

SOL'N

Λ 5 Χ 5>f

\x\w\fi\ 44 4 4 -4 4 4 4 4 4 4 55554455 44445557

\\\m\\ S 5 5 S 5fil5 5 5 5 5 5 5 5^5 5 3 5 6 6 5 5 f i 555 5 5 5 5 Afi5 5 4 5 5 5 4/4/• • • • 180.0 200.0 TEMPERATURE

Figure

12. I t e r a t i o n

with Wilson

Counts f o r M i x t u r e

220.0

260.0

240.0

(K) I.Newton-Raphson

Initiated

Κ Values

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

492

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Resorting to Newton-Raphson updating of Κ values, however, amounts to giving up the minimization character of the algorithm in favor of using an equation solving technique. The N-R scheme can (it has been demonstrated) converge to the trivial solution even at unstable points. In the examples it appears that convergence to trivial solutions is unlikely if the Κ values are initiated appropriately. For the mixture of Figure 1, initiating with the Wilson Κ values was adequate to avoid convergence to the trivial solution anywhere inside the two-phase region. This is true even though the region of instability is a very small part of the two-phase region. Initiating ^ith Κ values near unity can result in convergence to trivial solutions. This implies that there may be some risks involved in using Κ values at a converged solution to initiate a flash calculation at a nearby point, particularly near the critical. The results regarding speed of convergence indicate a definite advantage in accelerating the successive substitution algorithm. The first procedure, ACSS1, appears sufficient. The Newton-Raphson procedure used here certainly reduces the iteration count dramatically but has associated with it an increased risk that the trivial solution or non-convergence will be reached. Also, it is not certain that a reduced number of iterations implies a reduction in computer time. Other types of flash calculations than the constant Τ and Ρ flash examined in this manuscript are important. The occurrence of trivial solutions in isenthalpic flash routines, for instance, requires further study. Acknowledgments This work was supported by the National Science and Engineering Research Council of Canada. Literature Cited 1.

Poling, B.E.; Grens, E.A.; Prausnitz, J.M. Ind. Engg. Chem. Proc. Des. Dev. 1981, 20, 127. 2. Gundersen, T. Comp. Chem. Engg 1982, 6, 245. 3. Coward, I.; Gayle, S.E.; Webb, D.R. Trans. I. Chem. E. 1978, 56, 19. 4. Veeranna, D.; Rihani, D.N. Fluid Phase Equil 1984, 16, 41. 5. Asselineau, L . ; Bogdanic, G.; Vidal, J . Fluid Phase Equil 1979, 3, 273. 6. Michelsen, M.L. Fluid Phase Equil 1980, 4, 1. 7. Mehra, R.K.; Heidemann, R.A.; Aziz, K. Soc. Pet. Engrs J 1982, 22, 61. 8. Michelsen, M.L. Fluid Phase Equil 1982, 9, 1. 9. Michelsen, M.L. Fluid Phase Equil 1982, 9, 21. 10. Peng, D-Y.; Robinson, D.B. Ind. Engg. Chem. Fundamentals 1976, 15, 59. 11. Null, H.R. "Phase Equilibrium in Process Design"; Wiley Interscience Inc.: New York, 1970.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

23.

12. 13. 14. 15. 16. 17. 18. 19.

RIJKERS AND HEIDEMANN

Single-St age Flash Calculations

493

Nghiem, L.X.; Aziz, K.; Li, Y.K. Soc. Pet. Engrs J1 1983, 23, 521. Anderson, T . F . ; Prausnitz, J.M. Ind. Engg. Chem. Proc. Des. Dev. 1980, 19, 9. Mehra, R.K; Heidemann, R.A.; Aziz, K. Can. J . Chem. Eng. 1983, 61, 590. Nghiem, L.X.; Heidemann, R.A. In 2nd European Symposium on Enhanced Oil Recovery 1982, p.303 Ed. Techniqs., Paris, 1982. Zeleznik, F . J . J . Assoc. Comp. Mach. 1968, 15, 265. Rijkers, M.P.W. M.Sc. Thesis, The University of Calgary, Calgary, Alberta, Canada, 1985. Amundson, N.R. "Mathematical Methods in Chemical Engineering"; Prentice-Hall, Inc.: Englewood Cliffs, N . J . , 1966. Heidemann, R.A.; Khalil, A.M. AIChE J . 1980, 26, 769.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch023

RECEIVED November 8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

24 Four-Phase Flash Equilibrium Calculations for Multicomponent Systems Containing Water R. M . Enick, G. D. Holder, J . A. Grenko, and A. J . Brainard

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

Chemical and Petroleum Engineering Department, University of Pittsburgh, Pittsburgh, PA 15261

A technique for predicting one- to four-phase flash equilibrium is presented for multicomponent systems containing water, such as CO /crude o i l / water mixtures which characterize the carbon dioxide miscible flooding of petroleum reservoirs. The Peng-Robinson equation of state is used to describe the aqueous and hydrocarbon phases. An accelerated and stabilized successive substitution method is used to obtain convergence, even in the near c r i t i c a l region. Additional hydrocarbon phases are introduced by using a fugacity based testing scheme. Water-free flash calculations are first performed, yielding one, two or three phase equilibrium. A comprehensive search strategy is then used to consider eleven general classifications of systems which may result, including water-rich liquid/hydrocarbon-rich liquid/CO -rich liquid/vapor equilibrium. Improved methods for obtaining i n i t i a l estimates of additional phases are presented and a reliable scheme of searching for additional hydrocarbon-rich phases is introduced which considers all three possible phase identities. 2

2

General Objectives. Multiple phase behavior is often encountered in the gas miscible flooding of petroleum reservoirs. In water-free carbon dioxide/crude o i l systems, for example, three phases often exist in equilibrium at low temperatures. (1) However, water is almost universally present, either interstitially or because i t is injected for mobility control;(2) hence its presence should also be considered in phase behavior studies. Because of the relatively low miscibility of water with both carbon dioxide and hydrocarbons at reservoir conditions, an aqueous phase will almost always exist, resulting in the possible presence 0097-6156/86/0300-0494$07.50/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

24.

ENICK ET A L .

Four-Phase

Flash Equilibrium

Calculations

o f a s many a s f o u r p h a s e s . In a s p h a l t 1 c crude systems, a f i f t h p h a s e , a s o l i d p r e c i p i t a t e , may a l s o f o r m , ( 3 ) b u t p r e d i c t i n g i t s presence or c o m p o s i t i o n i s beyond t h e scope this study.

of

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

G e n e r a l l y , w a t e r - f r e e f l a s h c a l c u l a t i o n s a r e made when t h e s e systems a r e modeled,(_4) a n d a n e q u a t i o n o f s t a t e i s u s e d t o c a l c u l a t e component f u g a c i t i e s . Temperature, pressure and o v e r a l l composition a r e t y p i c a l l y s p e c i f i e d i n the f l a s h c a l c u l a t i o n technique i n order to determine the amounts and c o m p o s i t i o n s of each p o s s i b l e p h a s e . The o b j e c t i v e o f t h i s s t u d y i s t o d e v e l o p a n e f f i c i e n t a l g o r i t h m for CC^/hydrocarbon/water systems i n which the n u m b e r o f p h a s e s , one t o f o u r , i s d e t e r m i n e d a n d i n w h i c h t h e c o m p o s i t i o n a n d amount o f e a c h p h a s e , i n c l u d i n g t h e a q u e o u s phase, i s a c c u r a t e l y described. Emphasis i s p l a c e d on f o u r phase e q u i l i b r i u m and the e f f e c t s of the presence o f water on the phase behavior of a CC^/hydrocarbon m i x t u r e . Review of the L i t e r a t u r e : Multiphase Flash Calculations. The m o d e l f o r t h e two p h a s e f l a s h p r o b l e m was p r e s e n t e d i n 1952 by R a c h f o r d a n d R i c e ( 5 _ ) a n d many i m p r o v e m e n t s w e r e i n t r o d u c e d i n subsequent s t u d i e s , ( 6 - 8 ) but multiphase f l a s h c a l c u l a t i o n s w e r e n o t a d d r e s s e d u n t i l 1969 when Deam a n d Maddox(^) presented the three-phase f l a s h e q u i l i b r i u m problem. S i n c e t h a t t i m e many i m p r o v e m e n t s i n t h e a l g o r i t h m s u s e d have l e d t o more r a p i d c o n v e r g e n c e r a t e s a n d t o t h e c o r r e c t i d e n t i f i c a t i o n o f t h e number a n d c o m p o s i t i o n o f t h e s t a b l e equilibrium phases. These improvements have a d d r e s s e d the two d i s t i n c t p r o b l e m s w h i c h c h a r a c t e r i z e f l a s h calculations. The f i r s t i s d e f i n i n g a t h e r m o d y n a m i c m o d e l , such as an equation of s t a t e , which g i v e s r e s u l t s i n agreement w i t h e x p e r i m e n t a l measurements. The s e c o n d p r o b l e m i s f i n d i n g a numerical s o l u t i o n to the f l a s h c a l c u l a t i o n w i t h the g i v e n thermodynamic model. The m a j o r c o n t r i b u t i o n s t o these improvements i n c l u d e e f f i c i e n t n u m e r i c a l t e c h n i q u e s such as the over-relaxâtion method, a c c e l e r a t e d successive s u b s t i t u t i o n , and the m u l t i - v a r i a n t Newton-Raphson t e c h n i q u e , the development of e q u a t i o n s of s t a t e , a l g o r i t h m s f o r o b t a i n i n g i n i t i a l estimates f o r the composition of each phase, and the a p p l i c a t i o n of Gibbs free energy m i n i m i z a t i o n p r i n c i p l e s for e v a l u a t i n g the s t a b i l i t y of any e q u i l i b r i u m phase.(10-25) T h e s o l u t i o n scheme p r e s e n t e d b y R i s n e s a n d D a l e n ( 2 5 ) serves as a foundation f o r t h i s work. Improved a l g o r i t h m s h a v e b e e n d e v e l o p e d f o r d e t e r m i n a t i o n o f t h e number o f p h a s e s , the i n i t i a l e s t i m a t i o n o f phase c o m p o s i t i o n s , and the s e a r c h s t r a t e g y f o r d e t e r m i n i n g i f a n a d d i t i o n a l phase s h o u l d e x i s t o r i f a n e x i s t i n g p h a s e s h o u l d be e l i m i n a t e d . Emphasis i s p l a c e d o n q u a n t i t a t i v e l y d e s c r i b i n g how t h e a d d i t i o n o f

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS water to a hydrocarbon based m i x t u r e a f f e c t s compositions of e q u i l i b r i u m phases.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

Two

t h e number a n d

Phase And M u l t i p h a s e F l a s h C a l c u l a t i o n s

Governing Equations and Computational Algorithms. The e q u a t i o n s employed t o d e s c r i b e t h e p h a s e s , d e f i n e t h e r m o d y n a m i c e q u i l i b r i u m , f a c i l i t a t e t h e s o l u t i o n o f two phase a n d m u l t i p h a s e s o l u t i o n s , i n i t i a t e e s t i m a t e s o f a n y phase's composition, and e f f i c i e n t l y a d j u s t these i n i t i a l estimates a r e presented i n t h i s s e c t i o n . In a d d i t i o n t o the system temperature, pressure and o v e r a l l composition, the input data r e q u i r e d f o r the s o l u t i o n of these f l a s h e q u i l i b r i u m problems c o n s i s t of the c r i t i c a l temperature, c r i t i c a l p r e s s u r e a n d a c e n t r i c f a c t o r o f each component, a s w e l l a s a b i n a r y i n t e r a c t i o n parameter f o r each p a i r of components. A l l p h a s e s a r e d e s c r i b e d by a n e q u a t i o n o f s t a t e . The Peng-Robinson equation of s t a t e i s presented i n Table l a along with i t s corresponding expressions for compressibility f a c t o r , f u g a c i t y c o e f f i c i e n t and mixing r u l e s , i n Table l b , Equations 1-13. Table Ρ -

la.

Peng-Robinson

Equation of

RT _ a v-b v(v+b) + b(v-b)

m V

b = 0 . 0 7 7 8 0 RT / P c c a(T)

-

α ·

= 1 + in(l-Tr

0

5

m -

0.37464 +

Table

Z

3

0.45724

lb.

-

0 e 5

2

Ζ »

Β » k

-

)

1.54226ω -

(4)

0.26992ω

+ (A-3B -2B)Z 2

-

Pv/RT

A - aP/R T

Ψ

(3)

2

2

In

'

(5)

2

Compressibility Factor and Fugacity

(l-B)Z

where

A

(2)

R T <x/P c c 2

S t a t e ( 15)

2

3

-

0

(6)

(7) (8)

2

bP/RT

(Z-l)

(AB-B -B )

Coefficient

- ln(Z-B)

(9) -

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

24.

ENICK ET A L .

Table

lb.

Four-Phase

2Σχ a 1

2/2B"

i

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

1

t

b, - Ji)

k

a

b = |x b

Ζ +

(/2+1)Β )

Ζ -

(/2-l)B

in (

b

(10)

(11)

i

= f j ^ i j

a.. ij

497

Calculations

Continued

(

a

Flash Equilibrium

(12)

= (l-σ. . ) a , ij i

0

e

5

a. * J 0

(13)

5

T h e g o v e r n i n g e q u a t i o n s f o r two p h a s e w a t e r - f r e e e q u i l i b r i u m i n c l u d e o v e r a l l a n d component m a t e r i a l b a l a n c e s , m o l e - f r a c t i o n c o n s t r a i n t s , and the thermodynamic e q u i l i b r i u m c r i t e r i a , (Equations 14-17). T h e s e 2n+2 i n d e p e n d e n t e x p r e s s i o n s d e f i n e t h e two p h a s e e q u i l i b r i u m p r o b l e m o f 2n+2 unknowns, L , V, x^ a n d y ^ . The o v e r a l l m a t e r i a l b a l a n c e , L + V = Ν = 1

(14)

component

material

Lx

= Uz

+ v

t

y±

mole ±

(15)

±

fraction =

Σχ

balances,

contraints,

= 1

Σγ

±

(16)

and thermodynamic e q u i l i b r i u m c r i t e r i a (a) (b)

f i = f i (17) s y s t e m o f p r e d i c t e d p h a s e s must minimize Gibbs energy of system define the f l a s h c a l c u l a t i o n problem. Two p h a s e f l a s h c a l c u l a t i o n s may be p e r f o r m e d f o l l o w i n g the procedure. The e q u i l i b r i u m c o n s t a n t s a r e d e f i n e d a s K

i

L

y i

=

/

v

x

i

(

Eliminating Σχ

±

L from

+ V Σ(γ -χ ) ±

±

-

(15)

a n d summing o v e r a l l components -

Σζ

±

1 (N=l)

The g ( V ) f u n c t i o n , u s e d t o i s defined as

g(V)

-

Σ^-χρ

.

Σ

1

solve

(K.-l)z . +

(

R

where V i s the r o o t o f g(V) In the s i n g l e phase s t a t e ,

i

1

=

)

v

1

8

)

gives (19)

(14),

(15),

(16),

and

(18),

(20)

0.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

498 a.

if

V < 0 t h e n V = 0,

b. i f V > 1 t h e n V - 1, The b r i n g - b a c k p r o c e d u r e expressions, if K i

= z,

i

±

y is

i

new

if

=

γ κ

.

γ =

ί

1

1

1

(21)

1

(y /K )/Σ(γ /Κ )(22) following ±

±

±

±

(25) ί

(26)

= 0

(27)

= iq/Zz^iq

(28)

Σζ /Κ ±

Κ χ /ΣΚ χ

(24)

i Z i

= γκ

«

(23)

= 0

i

ί

new

±

.

1/EK

±

±

±

V = 1 then Σ(1-1/γΚ )ζ

K

y

= z, x = d e f i n e d by t h e

±

V = 0 t h e n l(yK -l)z

γ *

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

x

±

V a l u e s a r e assumed f o r the e q u i l i b r i u m c o n s t a n t s d e f i n e d by E q u a t i o n 1 8 . The η f u g a c i t y e q u a t i o n s a r e t h e n r e p l a c e d by the e q u i l i b r i u m c o n s t a n t e q u a t i o n s , e n a b l i n g t h e s t r a i g h t f o r w a r d s o l u t i o n f o r the unknowns. The m o l e s o f v a p o r , V , a r e f i r s t d e t e r m i n e d by u s i n g N e w t o n ' s method t o f i n d t h e r o o t o f g ( V ) = 0. This function i s a mathematical c o m b i n a t i o n o f t h e 2n+2 d e f i n i n g e q u a t i o n s a n d a t g ( V ) = 0 t h e amount o f f l a s h e d v a p o r a n d l i q u i d a r e known ( L = l - V f o r s y s t e m o f one t o t a l m o l e ) . Compositions are then c a l c u l a t e d b y s i m u l t a n e o u s l y s o l v i n g E q u a t i o n s 1 4 , 15 a n d 16. If the r o o t f a l l s o u t s i d e o f t h e t w o - p h a s e i n t e r v a l , V < 0 o r V > 1, the system i s i n a s i n g l e phase s t a t e . The c o m p o s i t i o n o f the n o n - e x i s t i n g phase i s then c a l c u l a t e d a s i f the system i s a t the s a t u r a t i o n p r e s s u r e . In order to h a s t e n the r e - e n t r y of a phase l o s t i n the c o r r e c t i o n process i n t o the s o l u t i o n a l g o r i t h m w h e r e i t may e v e n t u a l l y a p p e a r i n t h e f i n a l a n s w e r , t h e " b r i n g - b a c k " p r o c e d u r e may be u s e d ( E q u a t i o n s 23-28). S i n c e a common m u l t i p l e , γ , a p p l i e d t o e a c h o f t h e K ^ ' s d o e s not e f f e c t the c o m p o s i t i o n of the the c o r r e s p o n d i n g phases ( i t d o e s a f f e c t t h e a m o u n t s o f e a c h p h a s e ) , t h i s f a c t o r may be i n t r o d u c e d t o f o r c e t h e g ( V ) f u n c t i o n t o z e r o . This a d j u s t m e n t keeps t h e n o n - e x i s t i n g phase a t t h e edge o f t h e t w o - p h a s e b o u n d a r y , w h e r e i t s e x i s t e n c e w i l l s u b s e q u e n t l y be tested. Two p r o c e d u r e s f o r i n i t i a l i z i n g K - v a l u e e s t i m a t e s may be used. I d e a l l y , these estimates should assure that c a l c u l a t i o n s s t a r t i n t h e two p h a s e r e g i o n , a r e a s c l o s e a s possible to the a c t u a l values, and d i s p l a y s u f f i c i e n t c o n t r a s t to a v o i d a t r i v i a l s o l u t i o n where a l l K ^ ' s e q u a l unity. The W i l s o n f o r m u l a ( 1 4 ) , i s a n e m p i r i c a l e x p r e s s i o n w h i c h p r o v i d e s K - v a l u e e s t i m a t e s w h i c h meet t h e s e c r i t e r i a under most c o n d i t i o n s . K

±

-

Pri'

1

exp[5.3727(l+u>)(l-Tri"" )] 1

(29)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

24.

ENICK ET A L .

Four-Phase

Flash Equilibrium

Calculations

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

In the n e a r - c r i t i c a l r e g i o n , however, the e v a p o r a t i o n technique c o n s i s t e n t l y y i e l d s better estimates. T h i s may b e a t t r i b u t e d t o t h e t e c h n i q u e b e i n g b a s e d u p o n t h e same e q u a t i o n of s t a t e used t o d e f i n e two-phase e q u i l i b r i u m . The e v a p o r a t i o n t e c h n i q u e c o n s i s t s o f t h e f o l l o w i n g sequence of c a l c u l a t i o n s : 1.

Assume L =

1.0

2.

Calculate

f 's

3.

Evaporate

.01 mole;

4.

Continue u n t i l

5.

Perform m a t e r i a l balance,

i

composition proportional to

fj/s

L = 0.50 K| = y^/x^

I m p r o v e d K - v a l u e s may b e o b t a i n e d on f u g a c i t y e x p r e s s i o n s

by t h e methods

based

f

1 L

= x ^ P

(30)

f

i v

= y ^ v P

(3D

an I t e r a t i v e correction of K-values, substitution i s defined as K, i

t

+

-

1

l

/ l „ - K. R . * " iL iV i i T

where

t

t R

by

successive

* i t e r a t i o n number i "

f

iL

/

f

iV

(32) (33)

I n o r d e r t o r e d u c e t h e number o f i t e r a t i o n s i n t h e n e a r c r i t i c a l r e g i o n , w h e r e c o n v e r g e n c e may r e q u i r e h u n d r e d s o f i t e r a t i o n s , t h e b a s i c s u c c e s s i v e - s u b s t i t u t i o n method must be accélérated(25). This accelerated successive s u b s t i t u t i o n i s defined as (1-Ι^Γ K

t

t

+

1

1

-

where k

(34) = (R^-D/iR^' -!) 1

±

(35)

T h i s a c c e l e r a t i o n t e c h n i q u e may be i m p l e m e n t e d o n l y a f t e r t h e b a s i c s u c c e s s i v e - s u b s t i t u t i o n method i s raonotonically a p p r o a c h i n g a s o l u t i o n , a s i n d i c a t e d by a n e r r o r norm, p, o f a p p r o x i m a t e l y 10 . F u r t h e r m o r e , t h e a c c e l e r a t i o n must be performed a l t e r n a t e l y w i t h the c o n v e n t i o n a l technique, and any a c c e l e r a t e d K - v a l u e w h i c h cause t h e e r r o r norm t o i n c r e a s e m u s t be r e j e c t e d . The c r i t e r i o n f o r s o l u t i o n acceptance i s based on t h e thermodynamic e q u i l i b r i u m c r i t e r i o n , E q u a t i o n 17, which r e q u i r e s that the l i q u i d and v a p o r phase f u g a c i t i e s o f e a c h component be e q u a l . The s o l u t i o n acceptance i s expressed as

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

500

Ρ = Σ ^ - 1 )

< ε

2

(36)

A f u g a c i t y r e s i d u a l e r r o r norm, E q u a t i o n 36, o f 1 0 " " ^ i s a t t a i n a b l e f o r many s y s t e m s , i n c l u d i n g t h e e x a m p l e p r o b l e m s presented i n a subsequent s e c t i o n . The e q u a t i o n s w h i c h g o v e r n m u l t i p h a s e e q u i l i b r i u m a r e d i f f e r e n t from t h e two-phase e q u a t i o n s o n l y i n t h a t two a d d i t i o n a l phases a r e i n t r o d u c e d . For a four-phase system, t h e s e 4n+4 independent equations define t h e e q u i l i b r i u m p r o b l e m o f 4n+4 u n k n o w n s . These e q u a t i o n s i n c l u d e t h e o v e r a l l material balance, + L

h

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

x

+ L

2

component l

L

l i

x

+

L

+ V = Ν = 1

3

material 2 2i x

+

balances,

3 3i

L

(37)

x

+

η "

ν

mole

fraction constraints,

Σχ

= Zx

1;ί

and

»

2 i

Ζχ$ι

N

z

i

< > 38

Σy = 1

β

(39)

i

thermodynamic e q u i l i b r i u m c r i t e r i a

<>

f

(b)

system o f predicted

a

i

L

=

l

iL

f

=

2

f

iL

=

3

f

i V

< > 40

phases

must m i n i m i z e G i b b s

energy

The m u l t i p h a s e f l a s h c a l c u l a t i o n p r o c e d u r e i s a l s o q u i t e s i m i l a r t o t h a t o f t h e two phase s y s t e m . The e q u i l i b r i u m c o n s t a n t s a r e d e f i n e d i n r e f e r e n c e t o t h e vapor phase a s K

l i

- ?ΐ/ 11 χ

K

2 i- y i / 2 i x

K

The g a s p h a s e c o m p o s i t i o n ,

V

Z

i

/

[

1

+

l

L

"

1

}

+

L

3 i - *ΐ/*31

(

L Z(x -y ) 1

u

i

24 (

"

X

)

+

L

3 ^T (

+ L ^ x ^ - y ^ + L ^ x ^ - y . )

Defining t h e gj f u n c t i o n s , used t o determine distribution, g

j

( L

r

L

2

, L ) = « x 3

1

)

f o r Ν = 1, i s g i v e n b y

'

E l i m i n a t i n g V f r o m E q u a t i o n 38 a n d s u m m i n g o v e r components, g i v e s Ey.+

4

- y ^

(

4

2

)

a l l

= Σζ. = 1

(43)

t h e phase

(44)

Combining E q u a t i o n s 4 1 , 4 2 , 43 and 44

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

24.

Four-Phase

ENICK ET A L .

(

1)ζ

-b"

s

g

Flash Equilibrium

Calculations

ι

i l

J

1

+

l
L

(45)

VlÇ"

+

" »

+

L

3 K^- -

1)

(

The g j f u n c t i o n s a r e u s e d t o s o l v e E q u a t i o n s 3 7 , 3 8 , 3 9 , 4 1 , w h e r e L j , L a n d L3 a r e t h e r o o t s o f g j ( L j , L , L3) - 0 . T h i s i s v a l i d o n l y i f the gas phase i s p r e s e n t and the y±'s are normalized. I f the gas phase i s not p r e s e n t , assume the o i l - r i c h Lj phase e x i s t s . The o i l r i c h p h a s e c o m p o s i t i o n i s t h e n g i v e n by 2

n=

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

x

2

-

+

Eliminating components, 2

2

-x

i

)

u

+ L Z(x 3

One may now d e f i n e functions hj

(L ,

L ,

2

-

+

V)

3

hj

3

i

-x )

v

(

K

i r

1

)

i

(

-

+

to

(

4

^

1

"

}

)

Σζ.=1

the

gj

(47)

(48)

ÏC7 " J i _ 1

+

6

a l l

«

l±

functions, analagous

]

= Σ(χ-μ-*η)

= Σ

1

+

+ VZ(y -x )

u

C o m b i n i n g E q u a t i o n s 4 1 , 4 6 , 47 a n d

h.

»

f r o m E q u a t i o n 38 a n d s u m m i n g o v e r yields

Σχ + L E ( x u

»

)

48

i

z

(49)

^

X

"

)

+

V

(

K

l

i

"

1

}

The h j f u n c t i o n s a r e u s e d t o s o l v e E q u a t i o n s 3 7 , 3 8 , 3 9 , 4 1 , w h e r e L , L3 a n d V a r e t h e r o o t s o f h j ( L , L 3 , V) = 0. This i s v a l i d o n l y i f the hydrocarbon r i c h l i q u i d phase i s p r e s e n t and the X j / s a r e n o r m a l i z e d . 2

2

The e m p i r i c a l e q u a t i o n s u s e d t o p r o v i d e i n i t i a l e s t i m a t e s f o r a n a d d i t i o n a l h y d r o c a r b o n phase a r e l i s t e d below.

X

2i

y

i

X

l i

"

=

=

[

(

x

l i

T

i

r

2

/

P

r

i

)

/

i l l

(

x

l i

T

r

i

2

/

P

i

r

)

]

L

2

p h â S Q

[(x^Tr^/Pr^/gix^Tr^/Pr.)]

V phase

[(

L

x

2

i

P

r

i

/

T

r

i

2

)/ilJ(x

2

1

Pr /Tr i

i

2

)]

l P

hase

(

5

0

)

(51)

(52)

T h e s e e s t i m a t e s n o r m a l l y meet t h e t h r e e p r e v i o u s l y described c r i t e r i a for i n i t i a l K-value estimates. These

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

502

e x p r e s s i o n s i n d i c a t e t h a t t h e c o n c e n t r a t i o n o f a component i n a g i v e n phase i s p r o p o r t i o n a l t o t h e amount o f t h a t component p r e s e n t i n t h e m i x t u r e , a s measured by i t s c o n c e n t r a t i o n i n a n o t h e r p h a s e , a n d by i t s t e n d e n c y t o be i n a l e s s o r more dense phase, a s e s t i m a t e d by i t s reduced p r o p e r t i e s . For e x a m p l e , i f a two phase f l a s h y i e l d s l » j - L I "tlibria> components w i t h a h i g h r e d u c e d temperature a n d a l o w reduced p r e s s u r e a r e l i k e l y t o a p p e a r i n a v a p o r phase whose existence i s being evaluated. T h e s e i n i t i a l e s t i m a t e s may b e a d j u s t e d by u s i n g t h e f o l l o w i n g p r o c e d u r e . If, f o r example, t h e 1*2 p h a s e i s b e i n g t e s t e d f o r , n e g l e c t v a r i a t i o n o f fugacity coefficient with composition. Then c o r r e c t t h e composition of the L phase 2

e i

u

2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

X

< l L /

X

2i

« 2Ï

f

and normalize x

2i

t + 1

f

I I /

(

5

3

)

concentrations.

Χ

/ Σ χ

" 21

( 5 4

21

>

Because o f the low m i s c i b i l i t y o f hydrocarbons a n d water, the i n i t i a l estimate of K j values and ^ values (n r e f e r s t o w a t e r ) may b e o b t a i n e d b y i m p l e m e n t i n g ^ t w o s t e p procedure. Trace amounts o f (.001 mole) water a r e added t o L j , L , and V phases, and the f values adjusted u n t i l n

2

f

_ nL^

n

f _ nL

= f 1

- f

T

nL

2

(55)

w

nV

T h e n , a s m a l l amount o f C 0 a n d e a c h 2

x

is

l

hydrocarbon

- ^ / [ ( T r t / T r / i P r . / P r / ]

S

introduced into

the aqueous

phase,

(56) L3, u n t i l

Use .|^ l e q u a t i o n 56 i f L j n o t p r e s e n t ; a n d u s e y i i f both L|, and L not present. I n i t i a l K-values a r e then given by K j i y /x χ

2

n

2

t

i±

S i n c e many h y d r o c a r b o n s h a v e a n e x t r e m e l y s m a l l s o l u b i l i t y i n w a t e r , c o m p u t a t i o n a l d i f f i c u l t i e s may o c c u r a s values of x ^ 3 i approach zero and as values of K 3 ^as approach i n f i n i t y . To p r e v e n t a n y d i v e r g e n c e p r o b l e m s , a n a r b i t r a r y b u t r e a s o n a b l e maximum v a l u e o f 1 0 i s set for Kvalues of hydrocarbons i n water. Aqueous-phase concentrations a r e automatically set equal to zero i f the Kj-^i-value exceeds t h i s l i m i t . L

1 2

The a c c e l e r a t e d s u c c e s s i v e s u b s t i t u t i o n t e c h n i q u e f o r m u l i t p h a s e p r o b l e m s may b e d e r i v e d i n a s i m i l a r m a n e u v e r a s t h e two p h a s e p r o b l e m , E q u a t i o n s 3 0 - 3 6 .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ENICK E T A L .

Four-Phase

Flash Equilibrium

The i t e r a t i v e c o r r e c t i o n substitution is K.. J

t

+

= K./f..

1

l

J

i

i L

/£.

j

of

«

K-values

K . ^ R , . ;t J

1

Î J

by

»

l

1 V

Calculations

R

successive

i t e r a t i o n number -

f

1 L

J

Bring-back

Y

=

g

and

y

i

the

\

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

E

^

=

=

/

factors

for

l i ^

K

^ ^ l i

j i ^

x

l i

x

l i

for f

hj o

g

l

h

f

t

v

J

i

o r

h

2

u J

+

1

i

iL

f

J

Ρ «

by

v

ι

- ν ~

one

i L (65)

iV successive

s u b s t i t u t i o n may

be

ι

(66)

\ *

R.Z-l ^ R. . - I s o l u t i o n acceptance c r i t e r i o n for

where k . . The is

±

i s given

(τ—^-τ^

κ

(

*

and fugacity

/ £ 0

iV

-

(62)

s

(63)

Therefore, accelerated expressed as

±

(61)

c o r r e c t i o n and b r i n g back procedures,

1

K.

)

i L . J

f

μ*

=

0

f

f

t

phase

f iV = -— i L . J

iv

By c o m b i n i n g t h e obtains

K

liquid

a

γ factor

6

are

^ disappearing S phase, w h i c h must be m u l t i p l i e d t o the other sets of K . . c o n s t a n t s

iL. J

f

functions disappearing

r

The r e l a t i o n b e t w e e n t h e

= Σχ

are

(

2

Ύ

functions

(58) (59)

i v

j i

K

bring-back K

gj

/f i

=

(67)

1

^(Rj.-l)

2

< ε

the

multiphase

system

(68)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

504

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

R e l a t i o n s h i p s for i t e r a t i v e c o r r e c t i o n u s i n g the c o n v e n t i o n a l and accelerated s u c c e s s i v e - s u b s t i t u t i o n techniques, Equations 53 a n d 6 4 , a r e s i m i l a r t o t h e two p h a s e s y s t e m e x p r e s s i o n s , E q u a t i o n s 32 a n d 3 4 . The b r i n g - b a c k c o r r e c t i o n f a c t o r s a r e p r e s e n t e d f o r b o t h t h e gj a n d h j f u n c t i o n s , E q u a t i o n s 5 8 - 6 2 , s e r v e t h e same p u r p o s e a s t h e b r i n g b a c k f a c t o r f o r two-phase c a l c u l a t i o n s , f a c i l i t a t i n g the r e - e n t r y of a phase t h a t i s l o s t i n the c o r r e c t i o n process even though i t appears i n the final solution. Search Strategy for Multiphase S o l u t i o n s . The b a s i c successive-substitution algorithm for solving multiphase problems i n v o l v e s a sequence of c o m p u t a t i o n s . The e x i s t e n c e of any a d d i t i o n a l hydrocarbon phase i s e v a l u a t e d after performing a w a t e r - f r e e two-phase f l a s h c a l c u l a t i o n . Subsequently, water i s i n t r o d u c e d i n t o the system and the r e s u l t a n t e f f e c t on the phase d i s t r i b u t i o n i s d e t e r m i n e d . This stepwise a d d i t i o n of phases and s e a r c h for a d d i t i o n a l or d i s a p p e a r i n g p h a s e s i s b e s t d e s c r i b e d by F i g u r e I. This flow chart represents a search strategy for multiphase s o l u t i o n s , w h i c h s y s t e m a t i c a l l y e v a l u a t e s t h e number o f p h a s e s p r e s e n t a t the s p e c i f i e d c o n d i t i o n s . Water-Free F l a s h E q u i l i b r i u m . The i n i t i a l s t e p i n t h e m u l t i p h a s e f l a s h e q u i l i b r i u m c a l c u l a t i o n Is the p r e d i c t i o n of the two-phase w a t e r - f r e e e q u i l i b r i u m . The r e s u l t s o f t h i s c a l c u l a t i o n f e l l into three categories: divergence, onephase e q u i l i b r i u m and two-phase e q u i l i b r i u m . I f d i v e r g e n c e o c c u r s , t h e o t h e r method o f o b t a i n i n g i n i t i a l e s t i m a t e s o f K - v a l u e s s h o u l d be e m p l o y e d . In g e n e r a l , both the W i l s o n formula and the e v a p o r a t i o n t e c h n i q u e y i e l d c o n v e r g e n t s o l u t i o n s t h r o u g h o u t t h e two p h a s e region. The u s e o f t h e W i l s o n f o r m u l a , h o w e v e r , p r o v e s a d v a n t a g e o u s i n many s y s t e m s n e a r t h e s a t u r t i o n c u r v e s , whereas the e v a p o r a t i o n t e c h n i q u e c o n s i s t e n t l y y i e l d s better i n i t i a l estimates i n the near c r i t i c a l r e g i o n . Since the W i l s o n f o r m u l a i s much s i m p l e r i n f o r m , i t i s u s e d f i r s t i n a l l calculations. If divergence occurs or i f f a l s e s o l u t i o n s a p p e a r , t h e e v a p o r a t i o n t e c h n i q u e may a l s o be e m p l o y e d . T h e f l a s h c a l c u l a t i o n may a l s o y i e l d a s i n g l e p h a s e h y d r o c a r b o n - r i c h l i q u i d , L j , or vapor, V. T h i s s i n g l e phase s o l u t i o n may a l s o be t h e c a r b o n d i o x i d e - r i c h u p p e r l i q u i d phase, L » In any case, a search for a n a d d i t i o n a l h y d r o c a r b o n phase i s r e q u i r e d i n o r d e r to e s t a b l i s h whether the Gibbs energy of the system i s a t i t lowest v a l u e . In t h i s s t u d y , t h e a l t e r n a t i v e method o f i n i t i a l i z i n g K - v a l u e s i s employed i f a s i n g l e phase s o l u t i o n a r i s e s , and the twophase r e s u l t i s c o n s i s t e n t l y o b t a i n e d i f i t does e x i s t . If no a d d i t i o n a l p h a s e i s p r e d i c t e d o r i f a l e s s s t a b l e two p h a s e s o l u t i o n i s o b t a i n e d , t h e s y s t e m may s a f e l y be categorized as s i n g l e phase. 2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2Φ 1HC + L

CHECK FOR HC

4φ

Figure

I YJ 1.

#3,

3>

CHECK FOR HC

2HC + L

3Φ

ι γι

IF 1HC + L

3

1Φ 1HC

#2

2Φ 1HC + L OR 2HC

1Φ

I

DIVERGE

WATER FREE 2φ FLASH

in

#5

PU

3

solution

4φ

2Φ 1HC + L OR 2HC

2

ADD H 0

I

2Φ

In

2

L ASSUME L , V CHECK FOR Li

x

ASSUME L , L CHECK FOR V

Multiphase

3Φ 2HC + L3

CHECK FOR HC

3Φ 2HC + L.

L

CHECK FOR L2

ASSUME L , V

γ

Y

v//7

1Φ 1HC

search

#6

2

Y

2Φ

2

3Φ

f

#9

ADD H 0

strategy.

#8

r

2φ

CHANGE INITIAL! K METHOD

IF MULTIPLE SOLUTIONS OCCUR, PICK THE ONE WHICH MINIMIZED GIBBS ENERGY

CHANGE INITIAL K METHOD

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

2 1HC + L.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

506

Two p h a s e e q u i l i b r i u m may a l s o be p r e d i c t e d b y t h e i n i t i a l flash calculation. Although referred to as l i q u i d a n d v a p o r , t h e s e s y s t e m s may be L ^ - L , 1> "~V l ~ A l t h o u g h two p h a s e e q u i l i b r i u m may b e e s t a b l i s h e d a t t h i s p o i n t , t h e s y s t e m c a n n o t be c l a s s i f i e d a s s u c h u n t i l i t h a s been a s c e r t a i n e d t h a t a t h i r d h y d r o c a r b o n phase cannot e x i s t a t the s p e c i f i e d c o n d i t i o n s . 2

2

o

r

L

V

e

I t s h o u l d be e m p h a s i z e d t h a t a l t h o u g h t h e p r e s e n c e o f a d d i t i o n a l phases u s u a l l y minimizes the Gibbs energy of the s y s t e m , t h e r e a r e i n s t a n c e s w h e r e t h e s y s t e m i s more s t a b l e i n a s i n g l e p h a s e s o l u t i o n a s o p p o s e d t o t h e two p h a s e solution.(19) S p e c i f i c a l l y , t h i s may o c c u r w i t h a s i n g l e p h a s e L s y s t e m w h i c h may s a t i s f y a l l o f t h e t w o p h a s e L j - V w a t e r - f r e e e q u i l i b r i u m c o n d i t i o n s except for the Gibbs energy minimization. F u r t h e r m o r e , near the t h r e e phase r e g i o n , f a l s e two p h a s e s o l u t i o n s may a l s o a p p e a r . For example, a t p r e s s u r e s j u s t below the t h r e e phase r e g i o n i n multicomponent s y s t e m s , L j - V s y s t e m s may a l s o s a t i s f y a l l e q u i l i b r i u m c o n d i t i o n s ( w i t h the e x c e p t i o n of Gibbs energy m i n i m i z a t i o n ) for L - V or l q - L systems. Two d i f f e r e n t s o l u t i o n s may o f t e n be o b t a i n e d n e a r t h e m u l t i p h a s e p h a s e r e g i o n u s i n g t h e two Riva l u e i n i t a t i o n t e c h n i q u e s . The s y s t e m w h i c h m i n i m i z e s t h e G i b b s e n e r g y w i l l be t h e s t a b l e s o l u t i o n .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

2

2

2

An a d d i t i o n a l h y d r o c a r b o n p h a s e may be t e s t e d f o r by f o r m i n g a s m a l l amount (.01 m o l ) o f f l u i d w i t h a c o m p o s i t i o n c h a r a c t e r i s t i c of the phase b e i n g searched f o r , yet d i s p l a y i n g s u f f i c i e n t c o n t r a s t w i t h the e x i s t i n g phases to prevent t r i v a l s o l u t i o n s . C o m p o n e n t f u g a c i t i e s may t h e n be c a l c u l a t e d and adjusted u s i n g the p r e v i o u s l y described procedures. This composition c o r r e c t i o n for the a d d i t i o n a l phases y i e l d s three p o s s i b l e r e s u l t s : (a) t h e c o m p o s i t i o n o f t h e a d d i t i o n a l p h a s e b e i n g i d e n t i c a l t o one o f t h e two o r i g i n a l phases (b) a d i s t i n c t phase w i t h a n a v e r a g e f u g a c i t y r a t i o ( c o m p o n e n t f u g a c i t y i n a n e x i s t i n g p h a s e d i v i d e d by component f u g a c i t y i n t h e a d d i t i o n a l p h a s e ) l e s s t h a n u n i t y , o r (c) a d i s t i n c t phase whose a v e r a g e f u g a c i t y r a t i o i s greater than u n i t y . O n l y i n t h e t h i r d c a s e w i l l t h e new p h a s e g r o w a n d m u l t i p h a s e e q u i l i b r i u m be e s t a b l i s h e d , r e f l e c t i n g a decrease i n the Gibbs energy of the system. T h e s e a r c h f o r a n a d d i t i o n a l h y d r o c a r b o n p h a s e may be s i m p l i f i e d i f the i d e n t i t y of the phase, i . e . L p L or V, c a n be a s c e r t a i n e d . T h i s may be a c c o m p l i s h e d i f t h e i d e n t i t i e s o f t h e two i n i t i a l p h a s e s c a n b e e s t a b l i s h e d . A l t h o u g h c a l c u l a t i o n s of the m i x t u r e c r i t i c a l p o i n t or phase c l a s s i f i c a t i o n schemes b a s e d on t h e m i x t u r e c r i t i c a l volume and cricondentherra are u s e f u l for t h i s purpose, they are somewhat l e n g t h y i n n a t u r e . Therefore, i n t h i s study a l l t h r e e p o s s i b l e i d e n t i t i e s , L , V, and are considered s e q u e n t i a l l y b y a s s u m i n g t h e t w o p h a s e s y s t e m t o be L j - V , L | L , and L - V , r e s p e c t i v e l y . I f the a d d i t i o n a l phase b e i n g t e s t e d f o r does not e x i s t a t the g i v e n c o n d i t i o n s , the 2

2

2

2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

24.

ENICK ET A L .

Four-Phase Flash Equilibrium Calculations

i n i t i a l e s t i m a t e o f i t s c o m p o s i t i o n w i l l be q u i c k l y a d j u s t e d to t h e c o m p o s i t i o n o f one o f t h e o r i g i n a l phases. Should a l l t h r e e a s s u m p t i o n s y i e l d no a d d i t i o n a l hydrocarbon phase, twophase e q u i l i b r i u m i s e n s u r e d . I f a n a d d i t i o n a l h y d r o c a r b o n phase i s found t o e x i s t i n a n y o f t h e t h r e e t e s t s , t h r e e phase e q u i l i b r i a i s e s t a b l i s h e d a n d t h e s e a r c h ended.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

At t h i s p o i n t , t h e w a t e r - f r e e m u l t i p h a s e f l a s h c a l c u l a t i o n s a r e complete a n d t h e system may be i n one- , two- , o r t h r e e - p h a s e e q u i l i b r i u m . I f water i s p r e s e n t , i t i s now added t o t h e system. I n t r o d u c t i o n o f Water t o System. The i n t r o d u c t i o n o f water i n t o t h e h y d r o c a r b o n system w i l l n o t s i m p l y r e s u l t i n t h e f o r m a t i o n o f a n aqueous phase accompanied by i n s i g n i f i c a n t changes i n t h e h y d r o c a r b o n phase b e h a v i o r . W a t e r - f r e e e q u i l i b r i u m may be s h i f t e d t o t h e e x t e n t t h a t h y d r o c a r b o n phases may d i s a p p e a r o r a p p e a r , and i n some cases t h e aqueous phase i t s e l f may not form. These e f f e c t s a r e d e t a i l e d i n t h i s s e c t i o n a n d o u t l i n e d by t h e s e a r c h s t r a t e g y i l l u s t r a t e d i n F i g u r e 1 and t h e g e n e r a l i z e d f l o w diagram o f c o m p u t a t i o n s , F i g u r e 2. The e f f e c t s o f water w i l l be d i s c u s s e d a s they r e l a t e t o the t h r e e g e n e r a l c l a s s i f i c a t i o n s o f w a t e r - f r e e e q u i l i b r i a : one- , two- and t h r e e - p h a s e systems. I n each c a s e , t h e moles of water i n t r o d u c e d i n t o t h e system may be used a s t h e i n i t i a l e s t i m a t e o f L 3 . The i n i t i a l i z a t i o n o f L , L , and V s i m p l y r e q u i r e s t h e n o r m a l i z a t i o n o f t h e w a t e r - f r e e phase d i s t r i b u t i o n such t h a t t h e sum o f the mole f r a c t i o n s i s unity. 1

2

The a d d i t i o n o f water t o a s i n g l e phase system u s u a l l y r e s u l t s i n t h e f o r m a t i o n o f a n aqueous phase, L 3 , i n e q u i l i b r i u m w i t h t h e h y d r o c a r b o n phase. However, s e v e r a l o t h e r l e s s o b v i o u s p o s s i b i l i t i e s a l s o e x i s t . When a s m a l l amount o f water i s i n t r o d u c e d i n t o a h i g h - t e m p e r a t u r e h y d r o c a r b o n phase, t h e aqueous phase may n o t appear due t o the r e l a t i v e l y h i g h s o l u b i l i t y o f water i n t h e h y d r o c a r b o n phase under such c o n d i t i o n s . F u r t h e r m o r e , a second h y d r o c a r b o n phase may appear i f t h e s i n g l e phase m i x t u r e i s near t h e two phase r e g i o n , y i e l d i n g two h y d r o c a r b o n phases i n e q u i l i b r i u m . Another p o s s i b l e t r a n s i t i o n i n phase d i s t r i b u t i o n a r i s e s when water i s added t o a s i n g l e phase m i x t u r e near i t s s a t u r a t i o n p o i n t . For example, t h e i n t r o d u c t i o n o f water t o a s i n g l e phase L j m i x t u r e may s h i f t the w a t e r - f r e e e q u i l i b r i u m t o t h e e x t e n t t h a t t h e L j phase becomes s a t u r a t e d and a V phase a p p e a r s . T h e r e f o r e , when L j L 3 e q u i l i b r i u m e x i s t s , a bubble o f t h e V phase must be formed and i t s e x i s t e n c e t e s t e d . There i s even a p o s s i b i l i t y o f two a d d i t i o n a l hydrocarbon phases a p p e a r i n g , i n w h i c h case f o u r phases would c o e x i s t .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

508

Input: Nocomp. T. P.cs. moles of feed, displaced fluid ratio, flags, comp properties Calculate compositions and reduced properties Echo print input and start-up calculations

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

Calculate pure component parameters for EOS Estimate K-values for new phase

V

K-values found? Find roots of function by Newton's method

lter«lter*l|-

Calculate compositions and correction factors *

—

1

[

1

Repeat for all phases

<*> Γ

1 Apply 1 Calculate vofume%s phase densities ,s

Add another phase

7

I I( i/Vi) 3 f

βπΰ

conventional used last iteration? yes

Stop

F i g u r e 2.

Ύ c -orec o itnT—aj

h o t Single phase region?}**-' Apply conventional

Ï

J Output 1

K-values found

7

correction

I Iter-Iter-1| Did Accel \γά improve K-values 7

Apply acceleration c o r r e c t i o n

G e n e r a l i z e d f l o w diagram o f c o m p u t a t i o n a l

algorithm.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

ENICK ET A L .

Four-Phase Flash Equilibrium Calculations

When water i s added t o a two phase system, t h e most o b v i o u s r e s u l t i s t h a t a n aqueous phase w i l l be i n e q u i l i b r i u m w i t h two hydrocarbon phases. However, i f t h e two-phase system i s near t h e t h r e e phase boundary, t h e i n t r o d u c t i o n o f a w a t e r - r i c h l i q u i d phase may cause a n a d d i t i o n a l h y d r o c a r b o n phase t o a p p e a r , r e s u l t i n g i n f o u r phase e q u i l i b r i u m . Therefore, the existence of an a d d i t i o n a l hydrocarbon phase must be e v a l u a t e d whenever two h y d r o c a r b o n phases a r e i n e q u i l i b r i u m w i t h a n aqueous phase. At e l e v a t e d temperatures t h e aqueous phase may n o t form s i n c e t h e water may be c o m p l e t e l y d i s t r i b u t e d between t h e two h y d r o c a r b o n phases. Two phase e q u i l i b r i u m may a l s o r e s u l t when water i s added t o a two phase m i x t u r e near t h e dew p o i n t , c a u s i n g a h y d r o c a r b o n phase t o d i s a p p e a r . L a s t l y , i n systems near s a t u r a t i o n c o n d i t i o n s t h e two phase e q u i l i b r i u m may be s h i f t e d such t h a t one o f t h e h y d r o c a r b o n phases d i s a p p e a r s , l e a v i n g a s i n g l e hydrocarbon phase. Three c l a s s i f i c a t i o n s o f r e s u l t s o c c u r when water i s i n t r o d u c e d t o a t h r e e phase system, t h e f i r s t b e i n g t h e a d d i t i o n o f a n aqueous phase, y i e l d i n g f o u r phase equilibrium. The second i s c h a r a c t e r i z e d by a d i s a p p e a r i n g h y d r o c a r b o n phase, l e a v i n g two hydrocarbon phases i n e q u i l i b r i u m w i t h the aqueous phase. T h i r d l y , near t h e edge of t h e m u l t i p h a s e r e g i o n , two h y d r o c a r b o n phases may d i s a p p e a r , l e a v i n g one h y d r o c a r b o n a n d one aqueous phase i n equilibrium. R e s u l t s o f Example C a l c u l a t i o n s A s e r i e s o f c a l c u l a t i o n s f o r two systems w i l l i l l u s t r a t e t h e n i n e most common c a s e s w h i c h may r e s u l t upon t h e i n t r o d u c t i o n o f water t o one- , two- o r t h r e e - p h a s e systems, F i g u r e 1. Table I I d e s c r i b e s t h e c o n d i t i o n s f o r which each o f these cases o c c u r s . T a b l e I I . C l a s s i f i c a t i o n o f Example Computations

Figure

3 3 3 3 3 3 4 4 4 4 4

Range(K)

400.0-404.5 404.5-416.2 416.2-440.0 440.0-450.0 450.0-469.0 469.0-480.0 298.0-299.2 299.2-299.5 299.5-300.9 300.9-301.3 301.3-303.0

Case

1 3 5 7 8 2 5 10 9 6 5

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

510

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

The f i r s t system i s a multicoraponent hydrocarbon/water system d e s c r i b e d by Peng and Roblnson(14) and Heidemann( 13). The second i s a C C ^ / C H ^ / n C j ^ ^ / H ^ system which s e r v e s a s a model f o r raulticomponent crude o i l systems. The Peng-Robinson e q u a t i o n o f s t a t e was used t o d e s c r i b e a l l phases. Any c a r b o n d i o x i d e / w a t e r c h e m i c a l r e a c t i o n was i g n o r e d . The f u g a c i t y r a t i o e r r o r norm was s e t a t 10""^ f o r a l l c o m p u t a t i o n s . A p p r o x i m a t e l y 20 i n t e r a t i o n s ( F i g u r e 2) were r e q u i r e d f o r low t o moderate p r e s s u r e t h r e e phase (2 h y d r o c a r b o n and 1 aqueous) systems. Four phase computations r e q u t e d 25-35 i t e r a t i o n s . H i g h p r e s s u r e c a l c u l a t i o n s , e s p e c i a l l y near t h e c r i t i c a l p o i n t a n d s a t u r a t i o n c u r v e s , converged i n 35-50 i t e r a t i o n s . I t has been e s t a b l i s h e d t h a t when t h e Peng-Robinson e q u a t i o n o f s t a t e i s used i n c o n j u n c t i o n w i t h t h e m i x i n g r o l e g i v e n i n E q u a t i o n 13 o f T a b l e I , i t i s n o t p o s s i b l e t o p r e d i c t both t h e s o l u b i l i t y o f hydrocarbons o r C 0 i n t h e aqueous phase and t h e s o l u b i l i t y o f water i n t h e h y d r o c a r b o n or carbon d i o x i d e - r i c h phase. T h e r e f o r e , " o p t i m a l " i n t e r a c t i o n parameters a v a i l a b l e i n l i t e r a t u r e f o r such systems a r e u s u a l l y o b t a i n e d by matching aqueous phase c o m p o s i t i o n s , s i n c e such data a r e more r e l i a b l e t h a n vapor phase measurements. No a t t e m p t was made, t h e r e f o r e , t o o p t i m i z e t h e s e v a l u e s , w h i c h l i s t e d i n Table I I I . 2

Table I I I .

Non-Zero I n t e r a c t i o n Parameters

Figure 3 Water/Hydrocarbons

0.48

F i g u r e s 4,5 C 0

C0

C

-

2

CH

2

C H

4

0.100 -

4

H

16 34

C

H

16 34

H0

0.125

0.100

0.040

0.300

2

0.500

H0 2

Recent p r o g r e s s has been made, however, i n t h e development o f new, s i m p l e m i x i n g r u l e f o r asymétrie systems (27). When i t i s used i n c o n j u n c t i o n w i t h the m o d i f i e d " a " and "b" e q u a t i o n o f s t a t e p a r a m e t e r s , a c c u r a t e p r e d i c t i o n s o f mutual s o l u b i l i t i e s and aqueous phase d e n s i t i e s may be a t t a i n e d . Such improvements may e a s i l y be i n c o r p o r a t e d i n t o the f l a s h c a l c u l a t i o n t e c h n i q u e p r e s e n t e d i n t h i s s t u d y , g r e a t l y enhancing i t s c a p a b i l i t y t o a c c u r a t e l y d e s c r i b e t h e phase d e n s i t i e s and c o m p o s i t i o n s o f multicoraponent, m u l t i p h a s e asymétrie systems.(28)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ENICK

ETAL.

Four-Phase Flash Equilibrium Calculations

Phase d i s t r i b u t i o n r e s u l t s a r e p r e s e n t e d i n F i g u r e s 3, 4 and 5. W a t e r - f r e e r e s u l t s a r e i n d i c a t e d by the dashed l i n e s and e q u i l i b r i u m e s t a b l i s h e d a f t e r the i n t r o d u c t i o n of water i s r e p r e s e n t e d by the s o l i d l i n e s . P l o t s o f d e n s i t y and c o m p o s i t i o n , a l t h o u g h not p r e s e n t e d , a r e a l s o c o n t i n u o u s over phase b o u n d a r i e s , v e r i f y i n g the c o n s i s t e n c y o f t h i s multicoraponent, m u l t i p h a s e , f l a s h e q u i l i b r i u m t e c h n i q u e .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

Multlcomponent Hydrocarbon/Water System. Although t h i s system i s not c h a r a c t e r i s t i c o f r e s e r v o i r c o n d i t i o n s , i t c l e a r l y i l l u s t r a t e s s e v e r a l o f the e f f e c t s o f water on p h i se e q u i l i b r i u m and a l s o demonstrates t h a t t h e c o m p u t a t i o n scheme i s g e n e r a l i n n a t u r e ; i t i s not l i m i t e d t o carbon d i o x i d e / o i l / w a t e r systems. The r e s u l t s , F i g u r e 3, not o n l y d u p l i c a t e t h o s e p r e s e n t e d by Peng and Robinson(14) between 404.5 Κ and 450 K, but a l s o extend t o lower and h i g h e r t e m p e r a t u r e s . At 400 Κ the w a t e r - f r e e m i x t u r e i s a s i n g l e phase, h y d r o c a r b o n r i c h m i x t u r e ( L j ) . The system remains a t s i n g l e phase L j a s temperature i s i n c r e a s e d up t o 416.2 K, a t which p o i n t a vapor phase, V, forms. As temperature i s e l e v a t e d above t h i s bubble p o i n t t e m p e r a t u r e , the amount o f t h e vapor phase i n c r e a s e s s t e a d i l y as the L j phase d i m i n i s h e s u n t i l , a t 469 K, the dew p o i n t i s r e a c h e d . Above t h i s t e m p e r a t u r e , o n l y a s i n g l e V phase remains. When water i s i n t r o d u c e d i n t o the system, s i g n i f i c a n t changes i n phase b e h a v i o r o c c u r . At 400 K, a h y d r o c a r b o n r i c h l i q u i d , L j , i s i n e q u i l i b r i u m w i t h an aqueous phase, hy These two phases remain i n e q u i l i b r i u m w i t h o n l y s l i g h t changes i n t h e i r d i s t r i b u t i o n a s temperature i s i n c r e a s e d . At 404.5 K, however a vapor phase a p p e a r s . As t h e temperature i s e l e v a t e d above t h i s bubble p o i n t t e m p e r a t u r e , the vapor phase s t e a d i l y e n l a r g e s a s b o t h t h e L| and L 3 phase become s m a l l e r . At 440 K, the aqueous phase, L 3 , d i s a p p e a r s , l e a v i n g the two h y d r o c a r b o n phases i n e q u i l i b r i u m . The dew p o i n t i s reached a t 450 Κ and o n l y a s i n g l e phase vapor remains w i t h f u r t h e r i n c r e a s e s i n t e m p e r a t u r e . In t h i s example, the a d d i t i o n of water t o the system causes the f o r m a t i o n o f a n aqueous phase which d i s a p p e a r s a t 440 K, a 12 Κ r e d u c t i o n o f the bubble p o i n t temperature and a 19 Κ r e d u c t i o n o f the dew p o i n t t e m p e r a t u r e . ( ^ / C H ^ / n C ^ B ^ / I ^ O System. The r e s u l t s of the m u l t i p h a s e f l a s h c a l c u l a t i o n s f o r t h i s system a r e p r e s e n t e d i n F i g u r e 4. At 298 K, the w a t e r - f r e e system c o n s i s t s of a h y d r o c a r b o n - r i c h l i q u i d phase, L j , and a c a r b o n d i o x i d e - r i c h l i q u i d phase, l ^ . Only minor changes i n the l ^ - i ^ phase d i s t r i b u t i o n a r e e v i d e n c e d a s temperature i n c r e a s e s . At 299.2 K, however, a vapor phase f o r a s and t h r e e phase L ^ - l ^ - V e q u i l i b r i u m i s e s t a b l i s h e d . Further i n c r e a s e s i n temperature cause the vapor phase t o grow and the L 2 phase t o d i m i n i s h by

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

512

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

Pressure: 2.413 M Pa Dashed l i n e s : water-free system Phase Symbols

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

0 Δ +

L1-hydrocarbon r i c h L3-water r i c h V

ζ -Composition solid dashed Propane 0.1667 0.2273 n-Butane 0.1667 0.2273 0.2000 0.2728 n-Pentane 0.0666 0.0908 n-Hexane η-Octane 0.1818 0.1333 Water 0.0000 0.2667

480

F i g u r e 3. prediction.

E f f e c t o f water on phase d i s t r i b u t i o n ; Peng-Robinson

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

24.

ENICK ET A L .

Four-Phase Flash Equilibrium

Calculations

513

P r e s s u r e : 7.239 MPa Dashed l i n e s : w a t e r - f r e e system Phase

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

0 X Δ +

Symbols

L1-hydrocarbon r i c h L2-carbon d i o x i d e r i c h L3-water r i c h V

z-Composition solid Carbon d i o x i d e 0.750 Methane 0.025 n-Hexadecane 0.025 Water 0.200

dashed 0.93750 0.03125 0.03125 0.00000

F i g u r e 4. E f f e c t o f water on phase d i s t r i b u t i o n ; Peng-Robinson p r e d i c t i o n ; i s o b a r i c system, m u l t i p l e phase r e g i o n .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

514

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

Temperature: 299.7 Κ Dashed l i n e s : w a t e r - f r e e system Phase Symbols

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

0 X Δ +

L1-hydrocarbon r i c h L2-carbon d i o x i d e r i c h L3-water r i c h V

z-Composition solid Carbon d i o x i d e 0.750 Methane 0.025 n-Hexadecane 0.025 Water 0.200

dashed 0.93750 0.03125 0.03125 0.00000

co

ω ο CD

ω __> ο c ο -_J -Ρ D

CO

α ω CO Ο

_c ci-

F i g u r e 5. E f f e c t o f water on phase d i s t r i b u t i o n ; Peng-Robinson p r e d i c t i o n ; i s o t h e r m a l system, m u l t i p l e phase r e g i o n .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

ENICK E T A L .

Four-Phase Flash Equilibrium

Calculations

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

a p p r o K i u v i r e l y the same amount s i n c e o n l y i n s i g n i f i c a n t changes o c c u r i n t h e phase. The t h r e e phase r e g i o n ends a t 300.9 Κ as t h e \>2 phase d i s a p p e a r s and o n l y t h e L| and V phases r e m a i n . F u r t h e r i n c r e a s e s i n t e m p e r a t u r e caase o n l y i n s i g n i f i c a n t changes i n the L p V phase d i s t r i b u t i o n . The a d d i t i o n o f water t o t h i s model system causes d i s t i n c t changes i n the phase d i s t r i b u t i o n . From 298 Κ t o 299.5 K, t h e Lj_ and L£ phases a r e i n e q u i l i b r i u i r a w i t h a n aqueous phase, L 3 . Only minor changes i n t h e phase d i s t r i b u t i o n o c c u r o v e r t h i s range. At 299.5 K, a vapor phase forms and f o u r - p h a s e l ^ - I ^ - l ^ - V e q u i l i b r i u m i s e s t a b l i s h e d . As t e m p e r a t u r e i n c r e a s e s , t h e V phase becomes l a r g e r and the L 2 phase reduces a t about t h e same r a t e s i n c e o n l y v e r y s m a l l changes o c c u r i n t h e and L 3 phase d i s t r i b u t i o n . The f o u r - p h a s e r e g i o n ends a s the l»2 phase e v e n t u a l l y d i s a p p e a r s a t 301.3 K. Three-phase L 1 - L 3 - V e q u i l i b r i u m o c c u r s a t h i g h e r temperatures w i t h o n l y s l i g h t changes a p p a r e n t i n the phase d i s t r i b u t i o n . The i n t r o d u c t i o n o f water t o t h i s p a r t i c u l a r system causes the f o r m a t i o n o f a n aqueous phase, a .3 Κ i n c r e a s e i n t h e temperature a t which the V phase a p p e a r s and a .4 Κ i n c r e a s e i n t h e t e m p e r a t u r e a t which t h e L 2 phase d i s a p p e a r s . I n e f f e c t , the m u l t i p l e h y d r o c a r b o n phase r e g i o n becomes s l i g h t l y w i d e r and i s s h i f t e d toward e l e v a t e d t e m p e r a t u r e s i n t h e presence o f an aqueous phase. I n F i g u r e 5 t h e same system i s e v a l u a t e d a t a c o n s t a n t t e m p e r a t u r e over a range o f p r e s s u r e . I n t h i s f i g u r e , the a d d i t i o n o f water not o n l y r e s u l t s i n t h e f o r m a t i o n o f a n aqueous phase, but a l s o i n c r e a s e s the w i d t h of t h e m u l t i p l e hydrocarbon-phase r e g i o n and s h i f t s i t toward l o w e r pressures. Conclusions A multiphase equation of s t a t e f l a s h e q u i l i b r i u m c a l c u l a t i o n t e c h n i q u e has been d e v e l o p e d i n which: 1. The number of phases (one, two, t h r e e , o r f o u r ) i s d e t e r m i n e d , not assumed. 2. An improved method o f s e a r c h i n g f o r a t h i r d h y d r o c a r b o n phase i s i n t r o d u c e d w h i c h c o n s i d e r s a l l t h r e e p o s s i b l e i d e n t i t i e s o f the a d d i t i o n a l phase. 3. An e f f i c i e n t means o f e s t i m a t i n g the c o m p o s i t i o n o f t h e a d d i t i o n a l phase i s p r e s e n t e d w h i c h c o n s i d e r s t h e r e l a t i v e amount o f t h e component p r e s e n t i n t h e m i x t u r e a s w e l l a s i t s reduced p r o p e r t i e s . 4. A comprehensive s e a r c h s t r a t e g y w h i c h checks f o r a d d i t i o n a l o r d i s a p p e a r i n g h y d r o c a r b o n and aqueous phases i s used t o c o n s i d e r e l e v e n g e n e r a l c l a s s i f i c a t i o n s o f systems w h i c h may r e s u l t from t h e i n t r o d u c t i o n o f water i n t o one, two and t h r e e phase w a t e r - f r e e systems.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

516

5.

Example c a l c u l a t i o n s not o n l y i l l u s t r a t e t h e n u m e r i c a l e f f i c i e n c y o f the s o l u t i o n t e c h n i q u e , but a l s o d i s p l a y c o n t i n u i t y over phase r e g i o n b o u n d a r i e s .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

Legend o f Symbols a A b Β f g G h HC k Κ L m η Ν Ρ R Τ ν V y ζ Ζ α Ύ ε Ρ Φ Ψ ω

equation of state c o e f f i c i e n t equation of state c o e f f i c i e n t equation of state c o e f f i c i e n t equation of state c o e f f i c i e n t fugacity function of l i q u i d f r a c t i o n s Gibbs f r e e energy f u n c t i o n o f vapor f r a c t i o n and e x t r a l i q u i d f r a c t i o n s h y d r o c a r b o n phase, L^, L o r V a c c e l e r a t i o n parameter e q u i l i b r i u m constant l i q u i d moles equation of s t a t e c o e f f i c i e n t number o f components t o t a l number of moles pressure gas c o n s t a n t or (when s u b s c r i p t e d ) f u g a c i t y r a t i o temperature molar volume vapor moles mole f r a c t i o n i n vapor phase mole f r a c t i o n i n f e e d compressibility factor equation-of-state c o e f f i c i e n t bring-back f a c t o r tolerance f u g a c i t y r e s i d u a l norm phase fugacity coefficient accentric factor 2

Subscripts c g h i j k L η r V 1 2 3

c r i t i c a l state g-function h-function component number component number o r phase number component number l i q u i d phase component n, water ( i f p r e s e n t ) reduced vapor phase l i q u i d phase number, h y d r o c a r b o n - r i c h l i q u i d phase number, C 0 - r i c h l i q u i d phase number, aqueous 2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

24.

ENICK ET A L .

Four-Phase Flash Equilibrium Calculations

Superscripts t

i t e r a t i o n number i n i t i a l estimate

Literature Cited. 1. 2. 3.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch024

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Shelton, J.; Yarborough, L. Journal of Petroleum Technology September 1977, 1171-8. Stalkup, F. Miscible Displacement; SPE of AIME: New York, 1983, pp. 6-30. Stalkup, F." Journal of Petroleum Technology August 1978, pp. 1102-1112. Stalkup, F. Op. Cit., pp. 71-96. Rachford, H.; Rice, J." Journal of Petroleum Technology October 1952, p. 19. Organik, E.; Meyer, H. Journal of Petroleum Technology May 1955, pp. 9-13. Bennett, C.; Brasket, C.; Tierney, J. American Institute of Chemical Engineer's Journal 1960, Vol. 6, No. 1, pp. 67-70. Shelton, R.; Wood, R. The Chemical Engineer April 1965, pp. CE68-75. Deam, J.; Maddox, R. Hydrocarbon Processing July 1969, pp. 163-4. Erbar, J.H. Proceedings of the Vapor-Liquid Symposium, 1972, (Poland: Polish Academy of Science). Dluziewski, J.; Alder, S.; Ozardesh, H. Chemical Engineering Progress, November 1973, Vol. 69, No. 11, pp. 79-80. Lu, B.C.-Y.; Yu, P., Sugie, A.H. Chemical Engineering Science, 1974, Vol. 29, pp. 321-326. Heidemann, R. American Institute of Chemical Engineers Journal, 1974, Vol. 20, pp. 847-55. Peng, D.; Robinson, D. Canadian Journal of Chemical Engineering, December 1976, Vol. 54, pp. 595-98. Peng, D.; Robinson, D. Industrial and Engineering Chemistry Fundamentals, 1976, Vol. 15, No. 1, pp. 59-64. Wilson, G. Proceedings of the 65th National American Institute of Chemical Engineers Meeting, 1969, Paper No. 15C. Fussell, L. Society of Petroleum Engineers Journal August 1979, pp. 203-10. Fussell, D.; Yanosik, J. Society of Petroleum Engineers Journal June 1978, pp. 173-82. Baker, L.; Pierce, Α.; Luks, K. Society of Petroleum Engineers Journal October 1982, pp. 731-42. Mehra, R.; Heidemann, R.; Aziz, K. Society of Petroleum Engineers Journal February 1982, pp. 61-8. Fayers, F.J., ed., Enhanced Oil Recovery, "Phase Equilibrium Calculations in the Near Critical Region, by R. Risnes and V. Dalen," (New York: Elsevier Scientific Publishing Company, 1981), pp. 329-349.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

518

22. 23. 24.

25. 26.

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27.

28.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Carnahan, B.; Luther, H.; Wilkes, J. Applied Numerical Methods; John Wiley & Sons: New York; p. 61. Soave, G. Chemical Engineering Series, 1972, Vol. 27, pp. 1197-1203. Rabinowitz, P., Numerical Methods for Non-Linear Algebraic Equations, "A Fortran Subroutine for Non -Linear Algebraic Systems, by M.J. Powell" (Longon: Gordon and Breach Science Publishers, 1970), pp. 115161. Risnes, R.; Dalea, V. Society of Petroleum Engineers Journal February 1984, pp. 87-95. Abbott, M. American Institute of Chemical Engineers Journal, 1973, Vol. 19, No. 3, pp. 596-601. Panagiotopolous, Α.; Reid, R., "A New Mixing Rule for Cubic Equations of State for Highly Polar, Asymetric Systems," presented at the ACS Symposium on Equations of State-Theories and Applications (April 20-May 3, 1985) Miami, Florida. Enick, R.M.; Holder, G.D.; Mohamed, R.S. 60th Annual SPE Technical Conference and Exhibition, 1985, paper SPE 14148.

RECEIVED November

5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

25 Interpretations of Trouton's Law in Relation to Equation of State Properties Grant M . Wilson

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

Wiltec Research Co., Inc., 488 South 500 West, Provo, UT 84601

When corrections are made in Trouton's law for differences in molar volume of both liquid and vapor phases, then the resulting entropy deviation correlates with restricted motion in the liquid phase due to molecular size, shape, flexibility, and polarity. These entropy deviations also correlate with hard sphere equation of state properties which only depend on density without significant effect of temperature. Based on these results a new hard sphere equation of state is proposed which better fits the high density region of hard spheres based on molecular dynamics calculations than does the Carnahan - Starling equation. E n t r o p y d e v i a t i o n s o b t a i n e d by c o r r e c t i n g t h e e n t r o p y o f v a p o r i z a t i o n f o r d i f f e r e n c e s i n m o l a r volume i n b o t h the l i q u i d and v a p o r phases p l o t a l m o s t as a s i n g l e l i n e v e r s u s b o i l i n g p o i n t f o r a w i d e v a r i e t y o f n o n - p o l a r and p o l a r compounds. T h i s p r o v i d e s a new means f o r e s t i m a t i n g h e a t s o f v a p o r i z a t i o n and v a p o r p r e s s u r e s f o r a w i d e range o f compounds based on a measured b o i l i n g p o i n t and l i q u i d density. An e x a m i n a t i o n o f t h e e n t r o p y d e v i a t i o n t e r m shows t h a t i t c o r r e l a t e s w i t h r e s t r i c t e d m o t i o n i n the l i q u i d phase r e s u l t i n g f r o m m o l e c u l a r s i z e , s h a p e , f l e x i b i l i t y , and p o l a r i t y . F o r p a r a f f i n h y d r o c a r b o n s , the main e f f e c t appears t o be due t o r e s t r i c t e d m o t i o n as a r e s u l t o f t h e l e n g t h and f l e x i b i l i t y o f c h a i n m o l e c u l e s ; w h i l e f o r hydrogen bonded compounds, r e s t r i c t e d m o t i o n o c c u r s as a r e s u l t of hydrogen bonding. Compounds w i t h a d i p o l e moment a l s o e x h i b i t r e s t r i c t e d m o t i o n as a r e s u l t o f i n t e r a c t i o n s between the d i p o l e s . When d e v i a t i o n e n t r o p i e s a r e c a l c u l a t e d a t o t h e r t e m p e r a t u r e s b e s i d e s the b o i l i n g p o i n t , t h e y appear t o c o r r e l a t e w i t h h a r d sphere e q u a t i o n o f s t a t e p r o p e r t i e s w h i c h o n l y depend on d e n s i t y w i t h o u t any s i g n i f i c a n t e f f e c t o f t e m p e r a t u r e . As a r e s u l t a m o d i f i e d h a r d sphere equation o f s t a t e i s proposed which b e t t e r f i t s the h i g h d e n s i t y r e g i o n o f h a r d s p h e r e s b a s e d on m o l e c u l a r d y n a m i c s 0097-6156/86/0300-0520$06.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

25.

Interpretations of Trouton's Law

WILSON

521

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

c a l c u l a t i o n s than does t h e C a r n a h a n - S t a r l i n g e q u a t i o n . By a l l o w i n g one o f t h e p a r a m e t e r s t o v a r y , t h e same e q u a t i o n c a n be u s e d t o a c c o u n t f o r r e s t r i c t e d m o t i o n i n t h e l i q u i d phase r e s u l t i n g f r o m molecular size, f l e x i b i l i t y , p o l a r i t y , etc. The hard sphere e q u a t i o n o f s t a t e has n o t been w i d e l y used even t h o u g h i t a p p e a r s t o be m o r e a c c u r a t e t h a n t h e V a n d e r W a a l s r e p u l s i v e t e r m . The r e a s o n f o r t h i s i s p r o b a b l y a c o m p e n s a t i n g e f f e c t f o r e r r o r i n t h e a t t r a c t i o n t e r m when t h e Van d e r Waals t e r m i s u s e d . An e x a m i n a t i o n o f d a t a on a s i m p l e f l u i d shows t h a t t h e a t t r a c t i v e terms o f the R e d l i c h - K w o n g a n d P e n g - R o b i n s o n e q u a t i o n s f i t the low d e n s i t y r e g i o n q u i t e w e l l , but they s i g n i f i c a n t l y diverge at high densities. This explanation of a compensating e f f e c t b y t h e u s e o f t h e V a n d e r W a a l s r e p u l s i o n t e r m seems reasonable i n view of these d i f f e r e n c e s a t h i g h d e n s i t i e s .

Discussion T r o u t o n ' s l a w s t a t e s t h a t f o r many n o n - p o l a r f l u i d s t h e r a t i o o f Δ H/T i s about 21 cal/mole-°K. H i l d e b r a n d (4) has shown t h a t t h i s r a t i o i s more n e a r l y c o n s t a n t a t t e m p e r a t u r e s w h e r e t h e m o l a r volumes i n t h e gas phase a r e t h e same r a t h e r than when t h e p r e s s u r e s a r e t h e same. K i s t i a k o w s k y (5.) p r o p o s e d an e q u a t i o n w h i c h e s s e n t i a l l y c o r r e c t s f o r d i f f e r e n c e s i n v a p o r m o l a r volumes a t t h e b o i l i n g point to give the f o l l o w i n g equation: b

4£

- 8.75 + R l n T

v

(1)

T h i s e q u a t i o n does w e l l f o r l o w m o l e c u l a r w e i g h t n o n - p o l a r compounds, and thus i s r a t h e r r e s t r i c t e d i n i t s use. I n c o n t r a s t t o these e a r l y o b s e r v a t i o n s w h i c h f r e q u e n t l y s e r v e as t h e b a s i s f o r e s t i m a t i n g h e a t s o f v a p o r i z a t i o n , i t p r o v e s i n t e r e s t i n g t o c a l c u l a t e an e n t r o p y d e v i a t i o n a t t h e b o i l i n g p o i n t based on t h e d i f f e r e n c e between t h e a c t u a l e n t r o p y o f v a p o r i z a t i o n and an i d e a l e n t r o p y f o r t h e c h a n g e i n m o l a r v o l u m e i n g o i n g f r o m the l i q u i d phase t o t h e vapor phase as f o l l o w s :

or

A t _ t h e _ b o i l i n g p o i n t , t h e v a p o r phase i s n e a r l y i d e a l ; so (S°- S )/R - (S°- S ) / R i s v i r t u a l l y z e r o . T h u s , E q u a t i o n 2 r e d u c e s t o t h e following: V

I

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

522

V

L

V

L

I

The d i f f e r e n c e , ( S - S ) / R , i s g i v e n b y ΔΗ/RT^; a n d ( S - S ) / R i s g i v e n by t h e f o l l o w i n g e q u a t i o n :

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

A t one atmosphere V • RT/P where Ρ equals one atmosphere. 3 c a n t h e r e f o r e be w r i t t e n as f o l l o w s :

D E

A S ! ΔΗ

Equation

RT

m

DEV

A p l o t o f A S / R f o r hydrocabons and r a r e gases i s g i v e n i n F i g u r e 1. Data i n t h i s and subsequent p l o t s a r e based on A n t o i n e c o n s t a n t s f o r h e a t s o f v a p o r i z a t i o n ; and t h e R a c k e t t e q u a t i o n f o r l i q u i d volumes u s i n g d a t a and c o n s t a n t s g i v e n i n R e i d e t a l (7.). F i g u r e 1 shows t h a t Δ S /R varies linearly with T for paraffin hydrocarbons. Branched p a r a f f i n s such as neo-pentane appear t o p l o t on t h e same l i n e , w h i l e c y c l i c p a r a f f i n s p l o t about o n e - h a l f t o one u n i t below t h e l i n e . A c l u e t o t h e r e a s o n f o r v a r i a t i o n o f Δ S / R v e r s u s T^ o f t h e p a r a f f i n h y d r o c a r b o n s i s g i v e n i n F i g u r e 1 b y t h e fact that Δ S /R o f t h e r a r e gases i s a c o n s t a n t independent o f T^. The r a r e gases a r e s p h e r i c a l atoms w h i l e t h e p a r a f f i n hydrocarbons are l o n g f l e x i b l e m o l e c u l e s . Also, the deviation entropies of the a r o m a t i c hydrocarbons a r e c l o s e r t o t h e r a r e gases compared t o t h e p a r a f f i n hydrocarbons a t t h e same T^. T h i s i s c o n s i s t e n t w i t h l e s s f l e x i b i l i t y o f the a r o m a t i c hydrocarbons. This suggests t h a t t h e main r e a s o n f o r t h e v a r i a t i o n o f Δ S / R v e r s u s T i s t h e i n c r e a s e i n m o l e c u l a r f l e x i b i l i t y as the c h a i n l e n g t h i n c r e a s e s . F i g u r e 2 shows AS / R o f o t h e r compounds i n c l u d i n g w a t e r and a m m o n i a . S u r p r i s i n g l y , w a t e r and ammonia p l o t on a l m o s t t h e same l i n e as t h e p a r a f f i n hydrocarbons. I n t h i s case, the i n c r e a s e i n entropy c o m p a r e d t o t h e r a r e g a s e s p r e s u m a b l y i s n o t due t o m o l e c u l a r f l e x i b i l i t y , b u t i n s t e a d i s due t o r e s t r i c t e d m o t i o n f r o m o r i e n t a t i o n o r o t h e r s i m i l a r e f f e c t s i n t h e l i q u i d p h a s e due t o hydrogen bonding. C u r i o u s l y these e f f e c t s appear t o be about e q u a l i n m a g n i t u d e t o f l e x i b i l i t y e f f e c t s f o r n o n - p o l a r compounds w i t h equal b o i l i n g p o i n t s . Thus w h e t h e r Δ S /R i s a r e s u l t o f f l e x i b i l i t y o r p o l a r e f f e c t s , t h e n e t r e s u l t i s about t h e same f o r compounds w i t h t h e same b o i l i n g p o i n t . F i g u r e 2 a l s o shows a dashed l i n e f o r A s / R o f t h e d i a t o m i c h a l o g e n compounds. A g a i n , t h i s p l o t d e v i a t e s s i g n i f i c a n t l y from the n - p a r a f f i n p l o t because t h e f l e x i b i l i t y does n o t i n c r e a s e a s t h e b o i l i n g p o i n t i n c r e a s e s . However, t h e r e does appear t o be some r e s t r i c t i o n o f m o t i o n as t h e m o l e c u l a r w e i g h t s and s i z e s o f t h e s e d i a t o m i c m o l e c u l e s i n c r e a s e b e c a u s e Δ S 7 R d o e s i n c r e a s e s l i g h t l y v e r s u s T^. F i g u r e 3 shows D E V

fe

V

D E V

b

V

D

D E V

D E

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

25.

WILSON

0

Interpretations of Trouions Law

100

200

300 T

b'°

523

400

500

600

K

F i g u r e 1 . E n t r o p y D e v i a t i o n from I d e a l Gas o f Hydrocarbons Rare Gases a t t h e i r B o i l i n g P o i n t s .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

and

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

524

F i g u r e 3. E n t r o p y Various Functional

Deviation Groups,

from I d e a l Gas o f

Compounds

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

with

25.

WILSON

Interpretations of Trouions Law

525

DEV

a d d i t i o n a l Δ S / R data. This p l o t shows that the alcohols, g l y c o l , and g l y c e r o l e x h i b i t A S /R v a l u e s which a r e h i g h e r than the p a r a f f i n hydrocarbons by one to two u n i t s ; thus suggesting that i n these cases the o r i e n t i n g e f f e c t s of hydrogen bonding of these compounds are g r e a t e r than the f l e x i b i l i t y effects which r e s t r i c t the motion of non-polar compounds. However, even w i t h these differences, the error i n predicting the heat of vaporization would be 20% or less because ΔS/R i s about 10 or more. The r e s u l t s i n Figures 1, 2, and 3 provide i n t e r e s t i n g features p e r t a i n i n g to entropy e f f e c t s i n the l i q u i d phase. I n a d d i t i o n , these f i g u r e s can be used d i r e c t l y f o r e s t i m a t i n g heats of vaporization. The estimated heats of vaporization combined w i t h the A n t o i n e equation w i t h a g e n e r a l i z e d v a l u e f o r "C" p r o v i d e s an e x c e l l e n t means f o r e s t i m a t i n g vapor p r e s s u r e s based o n l y on a measured b o i l i n g p o i n t and l i q u i d d e n s i t y . For t h i s purpose the f o l l o w i n g generalized c o r r e l a t i o n f o r "C" i s recommended. Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

D

C°K *= - . 0 0 2 8 9 6 T

1,672

(6)

b

When t h i s v a l u e f o r "C" i s used, the other parameters are as follows : ( T

Β

b

+

0

)

2

Λ

H

-M

-

Τ,

RT,

D

D

(7)

A = 6.633 +

(8)

ln(P,mmHg)= A -

(9)

A l s o from Τ^,ΔΗ/RT^, and the c r i t i c a l c o n s t a n t s of a compound can be e s t i m a t e d . T h i s i s almost analogous to the e s t i m a t i o n of hydrocarbon c r i t i c a l constants based on density and b o i l i n g point, but w i t h hydrocarbons the molecular weight can also be estimated, unfortunately, t h i s i s not possible i n general without knowing the class of compound involved. Besides the immediate u t i l i t y of F i g u r e s 1, 2, and 3 they provide insight into entropy e f f e c t s i n the l i q u i d phase that prove worth p u r s u i n g . I f the d e v i a t i o n entropy of a compound i s determined along the saturation curve w i t h corrections made f o r the vapor phase e f f e c t s i n Equation 2 at high reduced temperatures, then curves a r e obtained such as the curve f o r argon shown i n F i g u r e 4. Non-random entropy e f f e c t s i n non-polar l i q u i d s are very small as i s evidenced by the f a c t t h a t the product UV i s a c o n s t a n t ; or O U / a v ^ - U / V (2.). Thus the main e f f e c t on A s / R w i l l be the v a r i a t i o n of l i q u i d volume w i t h temperature along the s a t u r a t i o n curve. I n f a c t , the curve i n Figure 4 f o r argon bears considerable resemblance to the curve p r e d i c t e d f o r hard spheres based on the equation of Carnahan and S t a r l i n g (3.). An a l t e r n a t e curve i s L

D E V

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

526

p l o t t e d i n F i g u r e 4 based on the f o l l o w i n g e q u a t i o n :

- - ν In ( 1 - |)

-|Ë.

(10)

W h i c h i n v o l v e s two p a r a m e t e r s i n c o n t r a s t t o one Carnahan-Starling equation. E q u a t i o n 10 g i v e s a c o m p r e s s i b i l i t y f a c t o r as f o l l o w s : Ζ = 1 + V

=

used hard

i n the sphere

(11)

-^—

V - b

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

C o m p r e s s i b i l i t y f a c t o r s from t h i s e q u a t i o n w i t h ν - 3.5 a r e p l o t t e d i n F i g u r e 5 w h e r e c o m p a r i s o n i s made w i t h t h e C a r n a h a n - S t a r l i n g equation: ζ , * + y * y (1 - y )

y

2

y

4

)

3 α 2

J

, b = § Ν

πσ

3

4V C o m p a r i s o n i s a l s o made w i t h r e s u l t s o f m o l e c u l a r d y n a m i c s c a l c u l a t i o n s w i t h h a r d spheres by A l d e r and W a i n w r i g h t (1_). F i g u r e 5 shows t h a t E q u a t i o n 11 f o l l o w s the r e s u l t s of A l d e r and W a i n w r i g h t at h i g h d e n s i t i e s much more c l o s e l y than does the C a r n a h a n - S t a r l i n g equation. One c o u l d i n t r o d u c e a second p a r a m e t e r t o the CarnahanS t a r l i n g e q u a t i o n i n o r d e r t o improve agreement, but the i n c e n t i v e seems r a t h e r low i n v i e w of the s i m p l i c i t y of E q u a t i o n 11. Without a c o r r e c t i o n , the C a r n a h a n - S t a r l i n g e q u a t i o n c o r r e s p o n d s t o a much s o f t e r f l u i d at h i g h d e n s i t i e s than i s a c t u a l l y p r e d i c t e d from m o l e c u l a r d y n a m i c s . The r e a s o n f o r t h i s i s n o t c l e a r , and i t a p p e a r s t o be i n e r r o r b e c a u s e t h e m i n i m u m v o l u m e t h a t t h e h a r d spheres can occupy i s the c l o s e packed c r y s t a l volume shown a t v / v = 1.0 i n F i g u r e 5. The f o r m a t i o n o f the c r y s t a l r e q u i r e s a t r a n s i t i o n f r o m a f l u i d s t a t e t o the c r y s t a l l i n e s t a t e , and the minimum volume i n the f l u i d s t a t e i s about 14% l a r g e r than the volume o f the c l o s e packed c r y s t a l . T h e r e f o r e , E q u a t i o n 11 w h i c h r e p r e s e n t s a m o d i f i e d hard sphere e q u a t i o n i s proposed f o r the c a l c u l a t i o n o f hard sphere properties at high densities. E q u a t i o n 11 was f i t t e d t o t h e d a t a p l o t t e d i n F i g u r e 4 by f i t t i n g two p o i n t s a l o n g t h e c u r v e . T h i s r e q u i r e s t h e a s s u m p t i o n that temperature e f f e c t s are n e g l i g i b l e . A l t e r n a t i v e l y , both p a r a m e t e r s c a n be f i t t e d a t e a c h t e m p e r a t u r e by a s s u m i n g t h a t ( 3 U p V ) i s e q u a l t o -Δϋ/V. When t h i s i s done, two s i m u l t a n e o u s equations i n v o l v i n g A S /R and T(8S/3v) can be s o l v e d f o r b and V as f o l l o w s : 0

T

D

T

A Q

Δ|_

d

e

κ

v

m

_

v

l n

( 1

_ b

}

(

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1

0

-

a

)

25.

WILSON

τ

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

527

Interpretations of Trouton's Law

1

ι

1

Γ

b = 21.8

cc/mole

ι h

0

2.0

2.5

3.0

v /v c

F i g u r e 4. Curve·

D e v i a t i o n E n t r o p y o f Argon A l o n g t h e L i q u i d

Saturation

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

528

F i g u r e 5. C o m p r e s s i b i l i t y F a c t o r o f Hard Spheres from CarnahanS t a r l i n g E q u a t i o n Versus M o l e c u l a r Dynamics.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

25.

WILSON

Interpretations of Trouions Law

. _Δϋ P V ^ +

K

J

R dV

RT

RT

529

eχ 1

+

V

(

^L

}

( 1 3 )

Vb/y

In Equation 13, Ρ i s the s a t u r a t i o n p r e s s u r e ; so except at h i g h reduced temperature, PV /RT i s very s m a l l . Values of i n t e r n a l energy were determined from ΔΗ where a correction was made f o r Δ Ζ both i n c a l c u l a t i n g ΔΗ and i n c o n v e r t i n g AH to Αϋ and f o r the energy content of the vapor. When t h i s was done, v a l u e s of b were obtained as are plotted i n Figure 6. This figure shows that derived v a l u e s of b i n the m o d i f i e d hard sphere e q u a t i o n correspond q u i t e c l o s e l y to the minimum l i q u i d volume p r e d i c t e d from the Rackett equation at zero reduced temperature. A l s o the volume of the molecules can be estimated from the Rackett equation by taking the r a t i o of 0.74/1.14 from F i g u r e 5 times the minimum l i q u i d volume from the Rackett equation. T h i s r a t i o p r o v i d e s the lower dashed l i n e i n F i g u r e 6 which i s a c t u a l l y h i g h e r than d e r i v e d v a l u e s of b in the Carnahan-Starling equation. These r e s u l t s strongly suggest that the d e r i v e d v a l u e s of b f o r argon i n the two equations are consistent with the i n t e r p r e t a t i o n given i n Figure 5. Figure 6 suggests that the extrapolated l i q u i d volume based on the Rackett equation at zero reduced temperature corresponds c l o s e l y to the v a l u e of b to be used i n Equations 10 and 11. I f t h i s i s assumed, then values of Δ S / R versus b/V can be plotted f o r other materials as are shown i n Figure 7. These curves appear to conform r a t h e r c l o s e l y except t h a t they d i f f e r by n e a r l y a constant m u l t i p l y i n g factor depending on the compound. Curiously, even polar compounds such as ammonia seem to f i t the p a t t e r n . These r e s u l t s suggest that the only difference between these compounds and argon i s the v a l u e of ν used i n Equation 10. For non-polar compounds, t h i s seems reasonable because the thermal pressure i s probably more c l o s e l y related to the free volume per molecular segment than i t i s to the free volume per molecule. Equation 11 gives the f o l l o w i n g :

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

L

DEV

Ζ - 1 +

V

1

- - ν

(14)

V

At h i g h d e n s i t i e s , the o n l y s i g n i f i c a n t term i s ( l - b / V ) / v which corresponds to the f r e e volume per segment where ν r e p r e s e n t s a parameter proportional to the number of segments i n the molecule. Thus i t appears t h a t E q u a t i o n 11 can not o n l y be used f o r hard spheres but also f o r long or bulky molecules such as those p l o t t e d i n Figure 15. For polar compounds V would not have the s i g n i f i c a n c e of number of segments; but i f ν i s also interpreted as a parameter proportional to r e s t r i c t e d degrees of freedom i n the l i q u i d phase; then i t s t i l l has meaning f o r polar compounds. Figure 8 shows the r e s u l t of simultaneously f i t t i n g b and V f o r various compounds at T 0.6. Results at other temperatures d i f f e r only s l i g h t l y . This plot shows that ν i s a l i n e a r function of b f o r the p a r a f f i n hydrocarbons. Also, the aromatic hydrocarbons plot on v i r t u a l l y the same l i n e ; thus showing that any reduction i n degrees of freedom of the a r o m a t i c hydrocarbons i s a l s o r e f l e c t e d i n a s m a l l e r b. The r a r e gases d e v i a t e s l i g h t l y from the l i n e but the s

r

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

530

25

Ί

1

Γ

R a c k e t t E q n . a t Z e r o °K "V

20

Minimum L i q u i d V o l

:

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

^ M o d i f i e d Hard Sphere Eqn.

15

R a c k e t t Eqn.

. ^ A c t u a l Vol. of Molecules

C a r n a h a n - S t a r l i n g Eqn. 10

0.5

0.6

0.7

0.8

1.0

0.9

b/V F i g u r e 6. M o l e c u l a r Volume o f S t a r l i n g and R a c k e t t E q u a t i o n .

A r g o n Compared

with

Carnahan-

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

WILSON

Interpretations of Trouton's Law

531

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

25.

0.5

0.6

0.7

0.8

0.9

1.0

b/v

F i g u r e 7. E n t r o p y D e v i a t i o n from I d e a l Gas V e r s u s b/V o f V a r i o u s Compounds·

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

532

10

n-Paraffins^5

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

EtOH • JO

Q'

Phenol •

Diphenyl H0 2

Ο

Acetone Q-Q • Ο Naphthalene Benzene ~ Cyclohexane P'

HC1

o

V

Ar

c

c

l

4

ν = 2 +

.04b

Key: Ο n-Paraffins Smoothed n-Paraffins • O t h e r Compounds

p' Xe

200

100 b,

cc/mole

F i g u r e 8. Degree o f Freedom Parameter V e r s u s b of and Other Compounds, M o d i f i e d Hard Sphere E q u a t i o n .

n-Paraffins

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

25.

WILSON

Interpretations of Trouions Law

533

difference i s rather small. Greater differences are shown by polar molecules such as water, e t h a n o l , phenol, acetone, and HC1 which d e v i a t e s i g n i f i c a n t l y ; thus showing a rather large r e s t r i c t i o n i n molecular motion of the compounds i n the l i q u i d phase compared to the sizes of the molecules.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

Why Van der Waals RT/(V-b)î Many equations of state have been proposed over the years. Some of these c o n t a i n the RT/(V-b) term from the Van der Waals e q u a t i o n while others replace i t w i t h another term including the hard sphere e q u a t i o n of Carnahan and S t a r l i n g . I t seems s i g n i f i c a n t t h a t the two equations commonly i n use today; the Soave m o d i f i c a t i o n of the Redlich-Kwong equation and the Peng-Robinson equation of state use the o r i g i n a l term of Van der Waals. Why should t h i s be so when a r e p u l s i o n term based on hard spheres seems much more reasonable? A l s o , from the r e s u l t s above i t appears t h a t r e a l f l u i d s conform f a i r l y c l o s e l y to a m o d i f i e d hard sphere e q u a t i o n of s t a t e . M o l e c u l a r d y n a m i c i s t s a l s o c h i d e us f o r p e r s i s t i n g i n the use of what i s considered to be an obsolete form. The answer i s probably not because of r e s i s t a n c e to new i d e a s . I n s t e a d i t appears t h a t both equations of s t a t e g i v e b e t t e r r e s u l t s f o r l i q u i d p r o p e r t i e s when the o r i g i n a l Van der Waals term i s used compared with the use of the Carnahan-Starling equation. I f t h i s i s so, i t suggests that there i s a compensation of e r r o r s between the a t t r a c t i o n and r e p u l s i o n terms i n these equations of s t a t e . An examination of the i n t e r n a l energy of a simple f l u i d according to Lee-Kesler (7) i n F i g u r e 9 shows t h a t t h e r e are s i g n i f i c a n t d i f f e r e n c e s at h i g h d e n s i t i e s between a l l m o d i f i c a t i o n s of the Redlich-Kwong equation and the a c t u a l curve. At lower d e n s i t i e s , the Redlich-Kwong equation f i t s the curve q u i t e a c c u r a t e l y ; but i t doesn't bend back as the data do at h i g h d e n s i t i e s . By c o n t r a s t , a Lennard-Jones f l u i d (6) appears to conform w i t h the simple f l u i d at high densities even though the a c e n t r i c f a c t o r of a Lennard-Jones f l u i d i s about -0.058; which i s not a simple f l u i d . A t h i r d curve corresponding to the Peng Robinson equation of state could be drawn i n Figure 9. At low d e n s i t i e s i t conforms as w e l l as does the Redlich-Kwong equation, and at high densities i t deviates less severely than the Redlich-Kwong equation of state. These deviations at high densities i n d i c a t e t h a t both equations d e v i a t e s i g n i f i c a n t l y from the t r u e energy curve at high densities. This problem i s also evident from anjexamination of data i n Figure 10 where 0 U / 3 V ) i s plotted versus U/V f o r v a r i o u s compounds. This f i g u r e shows t h a t non-polar compounds appear to conform c l o s e l y to a slope of unity while polar compounds d e v i a t e s i g n i f i c a n t l y from t h i s . A s l o p e of u n i t y i s c o n s i s t e n t w i t h the upper end of the curve f o r a s i m p l e f l u i d at high densities shown i n Figure 9. The Redlich-Kwong equation gives a s l o p e of about 0.65 f o r the range of d e n s i t i e s i n the l i q u i d region while the Peng-Robinson equation gives a slope of about 0.75; thus showing t h a t both equations d e v i a t e s i g n i f i c a n t l y at h i g h densities. T

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

534

EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

Τ

b/v F i g u r e 9. I n t e r n a l Energy o f a Simple F l u i d (ω=0) a t T = (Lee-Kesler). r

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1.0

25. WILSON

Interpretations of Trouions Law

535

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

1000

Nonpolar ^ Compounds

500

Ο

•

MP a iFO)

J

OiBuOH Ο Ο EtOH Ο BuOH PrOH MeOH 0

500 Δυ/V,

1000

MPa

F i g u r e 10. V a r i a t i o n o f (aU/9V) Versus Δϋ/V a t 20°C ( f r o m A. F. B a r t o n , CRC Handbook o f S o l y . Parameters, 1983). T

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

M.

536

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Notation Upper Case L e t t e r s : B>

88

Antoine Constants

cj 88

H enthalpy Ν Avogadro's number Ρ pressure R gas constant S entropy Τ absolute temperature U i n t e r n a l energy V volume Ζ Compressibility factor Lower Case L e t t e r s : b volume analogous Van der Waals b y b/4V i n Carnahan-S tar l i n g hard sphere equation Superscripts: DEV = deviation from i d e a l I ideal L l i q u i d phase V vapor phase molar property or average property ° property at a low enough pressure to behave as an i d e a l gas Subscripts: b at atmospheric b o i l i n g point c value at c r i t i c a l point r reduced property Greek L e t t e r s : Δ • incremental change hard sphere diameter ν = degree of freedom parameter i n proposed modified hard sphere equation of state 88

88

88

88

88 88

88

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch025

88

88

88

88

88

88

88

88

88

88

88

σ

88

References 1. Alder, B. J., and Wainwright, Τ. Ε., J. Chem. Phys. 1960, 33, 1439. 2. Barton, A. F. Μ., CRC Handbook of Solubility Parameters and other Cohesion Parameters, CRC Press, Inc.; Boca Raton, Florida, (1983). 3. Carnahan, N. F., and Starling, Κ. Ε., J. Chem. Phys. 1969, 51, 635. 4. Hildebrand, J. H., J. Am. Chem. Soc. 1915, 37, 970. 5. Kistiakowsky, W., Z. Phys. Chem. 1923, 107, 65. 6. Nicolas, J. J. Gubbins, Κ. E., Street, W. B. and Tildesley, D. J., Molecular Physics 1979, 37, 1429. 7. Reid, R. C., Praunitz, J. M., and Sherwood, T. K.,The Properties of Gases and Liquids McGraw-Hill Book Company, Third Edition, (1977). RECEIVED November 8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

26 Selection and Design of Cubic Equations of State J.-M. Yu, Y. Adachi, and B.C.-Y. Lu Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario K1N 9B4, Canada

Fourteen known cubic equations of the van der Waals type, Ρ = RT/(V-b) - a(T)/(V + ubV + wb ), were evaluated through the calculations of eight properties of normal alkanes. The roles of u and w in these calculations were demonstrated. A new design proce dure was developed and a new relationship between u and w was suggested.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

2

2

R e l i a b l e methods f o r p r e d i c t i n g p h y s i c a l p r o p e r t i e s o f pure compo nents and t h e i r m i x t u r e s (such as s i n g l e and two-phase p r o p e r t i e s , and v a p o r - l i q u i d e q u i l i b r i a ) a r e f r e q u e n t l y r e q u i r e d i n process d e s i g n and m a t e r i a l h a n d l i n g . In view of t h e wide range of s t a t e c o n d i t i o n s found i n p r a c t i c a l a p p l i c a t i o n s , and t h e frequent l a c k o f e x p e r i m e n t a l d a t a , c o n s i d e r a b l e a t t e n t i o n has been p a i d t o t h e development of these methods. In p a r t i c u l a r , a number of e q u a t i o n s of s t a t e have been proposed i n t h e l i t e r a t u r e t o meet t h i s demand. The most popular equations of s t a t e a r e c u b i c i n volume ( o r d e n s i t y ) . A g e n e r i c e x p r e s s i o n f o r t h e c u r r e n t l y popular c u b i c equations of s t a t e may be represented i n t h e form of an extended van der Waal s (VDW) e q u a t i o n , Ρ =KL V-b V +ubV+wb a

2

(1) 2

The q u a d r a t i c e x p r e s s i o n i n volume (V +ubV+wb ) r e p l a c e s t h e V term i n t h e denominator of the a t t r a c t i v e term of t h e o r i g i n a l VDW e q u a t i o n . When t h e parameters u and w are a s s i g n e d c e r t a i n p a r t i c u l a r v a l u e s , Equation 1 can be reduced t o t h e o r i g i n a l VDW equation (u=w=0)U), t h e Redlich-Kwong (RK) equation (u=l, w=0)(2) and i t s v a r i o u s m o d i f i e d forms, t h e Peng-Robinson (PR) equation (u=2, w=-l)(_3), t h e Heyen (H) equation (u+w=l)(4), t h e Schmidt-Wenzel (SW) e q u a t i o n (u+w=l)(5), and a number of other e q u a t i o n s . Some general f e a t u r e s of Equation 1 or i t s e q u i v a l e n t e x p r e s s i o n have been d i s cussed by Abbott {6). There i s evidence i n t h e l i t e r a t u r e (7-9) t h a t p r a c t i c a l l y i d e n t i c a l v a p o r - l i q u i d e q u i l i b r i u m (VLE) values ( T - P - c o m p o s i t i o n ) 2

2

0097-6156/ 86/ 0300-0537$06.75/ 0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

2

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

538

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

can be o b t a i n e d from c u b i c e q u a t i o n s of s t a t e c o n t a i n i n g two t o f o u r parameters, and these r e s u l t s are f r e q u e n t l y comparable t o t h o s e o b t a i n e d from more complex e q u a t i o n s of s t a t e . I t has been f u r t h e r demonstrated (10) t h a t by t r e a t i n g the parameter " a " of E q u a t i o n 1 temperature dependent, the VDW e q u a t i o n i s as capable as o t h e r c u b i c e q u a t i o n s f o r c a l c u l a t i n g VLE v a l u e s . On the o t h e r hand, the c a p a b i l i t i e s of the a v a i l a b l e c u b i c e q u a t i o n s f o r r e p r e s e n t i n g v o l u m e t r i c p r o p e r t i e s vary from e q u a t i o n t o e q u a t i o n , e s p e c i a l l y i n the c a l c u l a t i o n of l i q u i d volumes. Kumar and S t a r l i n g (11) suggested t h a t " a t a p a r t i c u l a r t e m p e r a t u r e , a h i g h e r density"~3ependence l e a d s t o a more a c c u r a t e equation of s t a t e " . They i n c r e a s e d the c o m p l e x i t y of Equation 1, and proposed a f i v e - p a r a m e t e r c u b i c e q u a t i o n of s t a t e f o r p r e d i c t i n g l i q u i d d e n s i t i e s and low temperature vapor p r e s s u r e s . One approach i s t h e r e f o r e t o s e p a r a t e the i s s u e s of VLE c a l c u l a t i o n s and v o l u m e t r i c p r e d i c t i o n s i n the a p p l i c a t i o n of e q u a t i o n s of s t a t e . In o t h e r words, d i f f e r e n t equations are used f o r d i f f e r ent p u r p o s e s . Another approach i s t o f u r t h e r improve the p e r f o r m a n ce of the a v a i l a b l e e q u a t i o n s or develop new e q u a t i o n s t o s a t i s f y the requirements of both VLE and v o l u m e t r i c c a l c u l a t i o n s . The t a s k i s a c t u a l l y reduced t o d e v e l o p i n g a s u i t a b l e e q u a t i o n , w i t h a compromise between s i m p l i c i t y and accuracy f o r r e p r e s e n t a t i o n of v o l u m e t r i c d a t a . In both approaches, g u i d e l i n e s f o r s e l e c t i n g the a p p r o p r i a t e e q u a t i o n s are r e q u i r e d . There i s s t i l l a need t o f u r t h e r e v a l u a t e the a v a i l a b l e c u b i c e q u a t i o n s i n a s y s t e m a t i c manner, so t h a t the u t i l i z a t i o n of a c u b i c e q u a t i o n can reach i t s f u l l p o t e n t i a l . The purpose of t h i s study i s t h e r e f o r e t o e v a l u a t e a v a i l a b l e c u b i c equations of s t a t e of the VDW t y p e as r e p r e s e n t e d by Equation 1 t o i d e n t i f y the m e r i t s and l i m i t a t i o n s of these e q u a t i o n s f o r the purpose of s e l e c t i o n , and t o suggest a s u i t a b l e procedure f o r d e s i g n i n g new c u b i c e q u a t i o n s of t h e same t y p e but t a i l o r e d t o s p e c i f i c p u r p o s e s . E v a l u a t i o n o f Cubic Equations

of

State

The t e c h n i q u e s used i n the improvement of the o r i g i n a l VDW e q u a t i o n f o r p h y s i c a l p r o p e r t y p r e d i c t i o n s , w i t h o u t changing the e x p r e s s i o n of Equation 1, may be grouped i n t o the f o l l o w i n g t h r e e c a t e g o r i e s : 1. M o d i f i c a t i o n of the e x p r e s s i o n used i n the denominator of the a t t r a c t i v e term. 2 . A p p l i c a t i o n of a v o l u m e - t r a n s l a t i o n t e c h n i q u e t o the o r i g i n a l VDW e q u a t i o n and the e q u a t i o n s of the f i r s t c a t e g o r y . 3. I n t r o d u c t i o n of temperature dependence t o one or more p a r a m e t e r s . The e q u a t i o n s of RK, PR and SW are examples of the f i r s t c a t e g o r y . The C l a u s i u s (C) e q u a t i o n (12) i s an example of the second c a t e g o r y . The Soave form of the RK e q u a t i o n (SRK)(13) and t h e m o d i f i c a t i o n of the RK e q u a t i o n by Hamam et a l . ( H C L ) Q T ) are examples of the t h i r d c a t e g o r y . In a d d i t i o n t o the above mentioned SRK, PR, SW, HCL and H equat i o n s , we have i n c l u d e d i n our c o n s i d e r a t i o n the Harmens-Knapp (HK) e q u a t i o n ( 1 5 ) , the P a t e l - T e j a (PT) e q u a t i o n ( 1 6 ) , the r e s u l t i n g e q u a t i o n from volume t r a n s l a t i o n of the SRK e q u a t i o n (TSRK) proposed by Peneloux et a l . ( 1 7 ) , the f o u r - p a r a m e t e r e q u a t i o n of Adachi et a l . ( A L S ) ( 9 ) , and the CI e q u a t i o n proposed by Peneloux et a l . ( 1 8 ) .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

26. YU ET AL.

539

Design of Cubic Equations of State

The o r i g i n a l VDW, RK and C equations are not e v a l u a t e d because t h e i r poor performance on v o l u m e t r i c p r e d i c t i o n s i s known. On the o t h e r hand, the t h r e e - p a r a m e t e r RK e q u a t i o n s suggested by Adachi et a l . (3RK)(19) and by F u l l e r (F) ( 2 0 ) , and the M a r t i n Equation (21) (a v e r s i o n of the C l a u s i u s e q u a t i o n ) i n two m o d i f i e d v e r s i o n s W C (Adachi et a l . (19)) and KMC (Kubic (22)) are c o n s i d e r e d . Thus, a t o t a l of f o u r t e e n c u b i c equations are e v a l u a t e d i n t h i s work. Among these e q u a t i o n s , the RK e q u a t i o n i s one of the most s u c c e s s f u l two-parameter c u b i c e q u a t i o n s . In the SRK e q u a t i o n , Soave made an e f f o r t i n 1972 t o c l o s e l y reproduce vapor p r e s s u r e s of pure compounds by assuming the parameter " a " of the o r i g i n a l e q u a t i o n t o be temperature dependent. This m o d i f i c a t i o n enhanced t h e a p p l i c a b i l i t y of the RK e q u a t i o n f o r c a l c u l a t i n g VLE v a l u e s . Many of the equations which have subsequently appeared i n t h e l i t e r a t u r e have adopted the same or s i m i l a r m o d i f i c a t i o n s . As f a r as the r e p r e s e n t a t i o n of v o l u m e t r i c data f o r pure f l u i d s i s c o n c e r n e d , M a r t i n (21) concludes t h a t the C l a u s i u s - t y p e e q u a t i o n i s the best of the s i m p l e r c u b i c e q u a t i o n s . However, the c a l c u l a t i o n of v o l u m e t r i c p r o p e r t i e s at s a t u r a t i o n c o n d i t i o n s w i t h o u t c o n s i d e r i n g the e q u a l i t y of f u g a c i t y at the same time as a p p l i e d by M a r t i n c o u l d i n t r o d u c e i n t e r n a l i n c o n s i s t e n c y i n the c a l c u l a t e d v a l u e s . As mentioned above, the C l a u s i u s e q u a t i o n can be o b t a i n e d from a volume t r a n s l a t i o n of the VDW e q u a t i o n . I t i s d i f f i c u l t t o envisage t h a t the volume t r a n s l a t e d VDW e q u a t i o n i s s u p e r i o r t o the v o l u m e - t r a n s l a t e d RK e q u a t i o n (TSRK). A comparison of the c a l c u l a t e d r e s u l t s i s d e f i n i t e l y of i n t e r e s t . It should be mentioned t h a t i n the two m o d i f i e d v e r s i o n s of the M a r t i n e q u a t i o n , MMC (19) and KMC ( 2 2 ) , the parameter " a " of Equation 1 was t r e a t e d as temperature dependent. Parameters o t h e r than " a " of Equation 1 have been assumed t o be temperature dependent i n some c u b i c e q u a t i o n s . For example, the H e q u a t i o n c o n t a i n s t h r e e parameters, two of which are assumed t o be temperature dependent. It i s d e s i g n a t e d i n t h i s work as a 3P2T e q u a t i o n . The o t h e r equations are s i m i l a r l y d e s i g n a t e d . The i n v e s t i g a t e d equations of s t a t e are thus grouped i n t o s i x types as shown i n Table I. The r e l a t i o n s h i p s between u and w f o r these e q u a t i o n s , the parameters which are t r e a t e d temperature dependent as w e l l as t h e p r o p e r t i e s (other than T and P ) used t o determine the paramet e r s of these e q u a t i o n s are a l s o presented i n Table I. In the t a b l e , the two-parameter e q u a t i o n proposed by Harmens (HA) (23) was a l s o i n c l u d e d , because i t s u and w r e l a t i o n s h i p formed p a r t o f the b a s i s i n the development of the SW e q u a t i o n . c

R e p r e s e n t a t i o n of P h y s i c a l

c

Properties

The p r o p e r t i e s of ten normal alkanes from methane t o n-decane, o b t a i n e d from g e n e r a l i z e d c o r r e l a t i o n s and t a b u l a t i o n s a v a i l a b l e i n the l i t e r a t u r e were used as the b a s i s f o r comparing the performance of the f o u r t e e n e q u a t i o n s . A t o t a l of e i g h t p r o p e r t i e s were c o n s i d e r e d . The vapor p r e s s u r e s , p , were o b t a i n e d from the g e n e r a l i z e d c o r r e l a t i o n proposed by Gomez-Nieto and Thodos (24) f o r nonpolar substances. The s a t u r a t e d l i q u i d volumes, V * , were o b t a i n e d from a m o d i f i e d R a c k e t t e q u a t i o n using the i n p u t parameters suggested by Spencer and A l d e r ( 2 5 ) . The s a t u r a t e d vapor volumes, V , were o b t a i n e d from the c o r r e l a t i o n of B a r i l e and Thodos ( 2 6 ) . The second v

v

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

540

Table I

Features of Some Cubic Equations Ρ = RT/(Y-b) - a ( T ) / ( V

TYPE 2P1T

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

3P1T

4P1T 2P2T 3P2T 3P3T

EOS

u

w

VDW RK SRK PR HA HK SW PT 3RK MMC TSRK CI ALS HCL Η KMC F

0 1 1 2 3 1-w 1-w 1-w

0 0 0 -1 -2

fU) fU)

f(co) f(ω) f(
fU) f(T)

a

. y

of S t a t e

+ ubV + w b )

2

2

b

Fitted

b.

none none v v v P , C r i t i c a l Isotherm P ,V% =0.7) P , V * ( T . = 0.6 t o 1.0) v A pv ρν λ

b

ah)

c

Properties

P

a(T) a(T) a(T) a(T) a(T) f(co) 0 a(T) u /4 a(T) (2u -u-l)/9 a(T) - ( u - 4 u - 4 ) / 8 a(T) a(T) f(u>) 0 a(T) fU,b) a(T) u /4 a(T) 0 a(T)

p

b^ bj b b

p

v

v

c c

r

V

r

b^ bj

2

2

b

2

p

c

> ν

PV*

bj b

P , Critical P ,V* P\V* P\B, p ,V*,V v

c

bU)

Isotherm

V

b(T) b(T) b(T)

2

j V

v

v

v i r i a l c o e f f i c i e n t s , B, were o b t a i n e d from the c o r r e l a t i o n o f Tsonopoulos (27)« For these f o u r p r o p e r t i e s , p o i n t s were taken at 0.02 i n t e r v a l s i n the T range of 0.5 t o 0.80 and at T equals 0 . 8 5 , 0.90, 0.95 and 0.98 f o r a t o t a l of 20 Τ v a l u e s . The c o r r e l a t i o n o f Lee and K e s l e r (28) was used t o o b t a i n the values f o r l i q u i d c o m p r e s s i b i l i t y T F c t o r ( Z , 0.30 < T < 0 . 9 9 , 0.01 < P < 1 0 . 0 , 315 p o i n t s ) , c o m p r e s s i b i l i t y f a c t o r of vapor ( Z , 0.55 < T < 0 . 9 9 , 0.01 < P < 0 . 8 , 56 p o i n t s ) , c o m p r e s s i b i l i t y f a c t o r of gas above t h e c r i t i c a l temperature ( Z , 1.01 < T < 4 . 0 0 , 0.01 < P < 1 0 . 0 , 240 p o i n t s ) and c o m p r e s s i b i l i t y f a c t o r along the c r i t i c a l isotherm ( Z , T = 1 . 0 , 0.01 < Ρ < 10.0, 15 p o i n t s ) . A summary of the c a l c u l a t i o n r e s u l t s , i n terms of o v e r a l l average a b s o l u t e percent d e v i a t i o n s , i s presented i n Table I I . The r e p o r t e d values may be s l i g h t l y d i f f e r e n t from s i m i l a r c a l c u l a t i o n s a v a i l a b l e i n the l i t e r a t u r e due t o the d i f f e r e n c e i n t h e covered ranges of Tp and P and i n the number of data p o i n t s s e l e c t e d f o r the c a l c u l a t i o n . I t i s w e l l known t h a t a c c u r a t e r e p r e s e n t a t i o n of vapor p r e s s u res i s e s s e n t i a l f o r v a p o r - l i q u i d e q u i l i b r i u m c a l c u l a t i o n s . For t h i s r e a s o n , vapor p r e s s u r e values have been used t o determine t h e values of parameter " a " , or Q (= a Pc/R Τ ) . The g e n e r a l i z e d e x p r e s s i o n s of t h e o r i g i n a l authors f o r " a were used t o c a l c u l a t e p and p r a c t i c a l l y a l l e q u a t i o n s t e s t e d y i e l d e d a c c e p t a b l e r e s u l t s . As f a r as V* i s c o n c e r n e d , the m o d i f i e d M a r t i n (MMC) and the SRK equations y i e l d the l a r g e s t d e v i a t i o n s . The d e v i a t i o n s i n t h e c a l c u l a t e d V v a l u e s f o l l o w c l o s e l y t o those f o r p , and t h e d e v i a t i o n s i n Z v a l u e s f o l l o w c l o s e l y t o those f o r V^. p

r

x

p

p

v

p

r

s u p

f

r

c

r

p

2

2

a

v

v

v

A

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Table II

Summary of O v e r a l l Average A b s o l u t e Percent D e v i a t i o n s the C a l c u l a t e d P h y s i c a l P r o p e r t i e s f o r Ten Normal

Property p

v

Z* Z sup Z B v

z

c

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

Property p

v

V* V

v

Z* Z sup Z Β v

z

c

541

Design of Cubic Equations of State

26. YU ET AL.

NT

SRK

200 200 200 3150 560 2400 150 200

1.51 13.4 1.23 11.2 0.95 2.37 8.50 17.2

TSRK 1.51 3.77 1.20 3.67 0.64 1.35 5.42 16.1

PR 2.55 5.47 2.54 4.96 0.58 1.54 4.68 15.5

SW

HK 1.02 4.37 1.45 4.14 0.55 1.47 4.05 L5.1

1.44 2.78 1.22 2.96 0.38 1.41 4.08 15.6

PT 1.95 2.56 2.01 3.03 0.40 1.28 4.13 15.4

in

3RK

MMC

1.29 4.00 1.08 3.37 0.56 1.36 4.92 15.9

2.33 12.4 2.51 12.6 0.44 1.89 5.24 15.4

CI

ALS

HCL

Η

KMC

F

0.96 3.65 1.26 3.93 0.49 1.30 6.24 15.6

1.42 2.61 1.74 3.13 0.45 1.30 4.00 15.4

1.88 0.65 4.10 2.61 2.24 2.98 8.37 23.2

10.5 2.26 10.3 5.53 2.24 3.72 8.90 16.9

3.56 5.41 4.01 13.5 0.19 2.09 5.32 1.71

1.83 2.01 1.97 5.28 0.99 6.16 12.6 17.4

The d i f f e r e n c e s i n the c a l c u l a t e d Z , Z and Z v a l u e s among t h e 14 e q u a t i o n s are not too s i g n i f i c a n t . The KMC e q u a t i o n g i v e s t h e lowest d e v i a t i o n s i n the c a l c u l a t e d Β v a l u e s , which were used i n the d e t e r m i n a t i o n of the parameters of the e q u a t i o n . A l t h o u g h some of the f i n d i n g s mentioned above c o u l d be e n v i saged from the f e a t u r e s l i s t e d i n Table I, t h e r e are s e v e r a l i n t e r e s t i n g p o i n t s r e v e a l e d by the r e s u l t s of Table I I . 1. The c u r r e n t l y popular c u b i c equations of s t a t e (SRK and PR) do not y i e l d the best r e s u l t s . 2 . The performance of the 3P2T Η e q u a t i o n i s i n f e r i o r t o t h a t of the 3P1T PT e q u a t i o n . Indeed, the o v e r a l l performance of e q u a t i o n s c o n t a i n i n g more than one temperature-dependent parame t e r i s g e n e r a l l y i n f e r i o r , i n d i c a t i n g t h a t these equations are m a i n l y s u i t a b l e f o r r e p r e s e n t i n g the p h y s i c a l p r o p e r t i e s used i n the f o r c e d f i t t i n g p r o c e d u r e . 3. The M a r t i n e q u a t i o n i s not s u i t a b l e f o r VLE and v o l u m e t r i c calculations simultaneously. 4 . A t h r e e - p a r a m e t e r c u b i c e q u a t i o n w i t h o n l y the parameter " a " t r e a t e d temperature dependent i s adequate f o r the purpose of t h i s s t u d y . The performance of the s i x equations of t h i s type (HK, SW, PT, 3RK, TSRK, CI) seems t o be adequate. F u r t h e r examination of the d e v i a t i o n s o b t a i n e d f o r i n d i v i d u a l a l k a n e s (as shown i n Table I I I ) from the 2P1T and 3P1T e q u a t i o n s reveals that: 1. The PR e q u a t i o n y i e l d s l a r g e r d e v i a t i o n s i n p v a l u e s f o r l a r g e r m o l e c u l e s , and the d e v i a t i o n s i n V* i n c r e a s e w i t h i n c r e a s e i n molecular weight. v

s u p

c

v

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

542

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

2.

For t h e SRK e q u a t i o n , t h e d e v i a t i o n s i n V * a l s o i n c r e a s e w i t h increase in molecular weight, 3 . A l l 2P1T and 3P1T equations y i e l d l a r g e d e v i a t i o n s i n B, and t h e deviations increase with increase in molecular weight. 4 . The TSRK e q u a t i o n y i e l d s d e v i a t i o n s i n p i d e n t i c a l t o those of the SRK e q u a t i o n , c o n f i r m i n g the f a c t t h a t volume t r a n s l a t i o n does not a f f e c t the p c a l c u l a t i o n s . 5. As f a r as t h e d e v i a t i o n s i n V * and are c o n c e r n e d , the PT e q u a t i o n appears t o be s l i g h t l y s u p e r i o r i n t h e f a m i l y o f e q u a t i o n s (HK SW and PT) which can be r e p r e s e n t e d by t h e u + w = 1 r e l a t i o n s h i p . Among t h e equations r e s u l t i n g from volume t r a n s l a t i o n (MMC, TSRK, and C I ) , t h e performance of the TSRK e q u a t i o n i s b e t t e r because t h e i n d i v i d u a l d e v i a t i o n s are about t h e same f o r t h e t e n a l k a n e s . The d e v i a t i o n s obtained from t h e CI e q u a t i o n tend t o i n c r e a s e w i t h i n c r e a s e i n m o l e c u l a r w e i g h t . A comparison of the SW, PT, TSRK and CI e q u a t i o n s r e v e a l s t h a t e q u a t i o n s w i t h substance-dependent Q values y i e l d b e t t e r r e p r e s e n t a t i o n of these two p r o p e r t i e s . A l l equations appear t o r e p r e s e n t V * and Ζ t o t h e same degree of a c c u r a c y . 6. Although n e a r l y a l l t h e equations y i e l d low d e v i a t i o n s i n t h e c a l c u l a t e d Z and Z v a l u e s , t h e r e i s a tendency toward l a r g e r d e v i a t i o n s at l a r g e r m o l e c u l a r w e i g h t s . 7. The f a m i l y of equations r e p r e s e n t e d by the u + w = 1 r e l a t i o n s h i p y i e l d s lower d e v i a t i o n s i n the c a l c u l a t i o n of Z v a l u e s than t h e e q u a t i o n s o b t a i n e d from t h e v o l u m e - t r a n s l a t i o n method, and the i n d i v i d u a l d e v i a t i o n s are about the same. Thus, t h e r e s u l t s r e p o r t e d i n Tables II and I I I p r o v i d e some guidance i n t h e s e l e c t i o n of e q u a t i o n s among t h e 14 c u b i c e q u a t i o n s f o r r e p r e s e n t i n g p h y s i c a l p r o p e r t i e s of pure normal f l u i d s . The c h o i c e depends on the p r o p e r t i e s t o be emphasized and the m o l e c u l a r weight of t h e substances t o be c o n s i d e r e d . v

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

v

a c

λ

v

s u p

c

Design o f Cubic E q u a t i o n s o f S t a t e In E q u a t i o n 1, i f u and w are c o n s i d e r e d as c o n s t a n t s f o r a l l substances (such as t h e VDW, RK, SRK, PR and HA e q u a t i o n s ) , t h e r e s u l t a n t e q u a t i o n would be a two-parameter e q u a t i o n . I f u and w are r e l a t e d by an exact mathematic r e l a t i o n s h i p (such as the HK, SW, PT, 3RK, MMC and TSRK e q u a t i o n s ) , Equation 1 would become a t h r e e parameter c u b i c e q u a t i o n . I f u and w are not r e l a t e d t o each o t h e r through an exact mathematic r e l a t i o n s h i p (such as t h e ALS e q u a t i o n ) , E q u a t i o n 1 would y i e l d a f o u r - p a r a m e t e r e q u a t i o n . Although t h e CI e q u a t i o n i n i t s o r i g i n a l form (18) appears t o be a f o u r - p a r a m e t e r e q u a t i o n , t h e u and w r e l a t i o n s h i p of t h i s e q u a t i o n can be a p p r o ximated by the e x p r e s s i o n shown i n Table I. In o t h e r words, i t may be approximates as a v o l u m e - t r a n s l a t e d PR e q u a t i o n . I f we adopt t h e c l a s s i c a l c o n d i t i o n s at the c r i t i c a l p o i n t as two of our c o n s t r a i n t s , (ôP/ôV) = 0 (2) T c

(ô P/ôV2) 2

T c

=0

%

(3)

a two-parameter e q u a t i o n y i e l d s a constant c r i t i c a l c o m p r e s s i b i l i t y f a c t o r , ζ ; and a t h r e e - , or f o u r - p a r a m e t e r e q u a t i o n may y i e l d substance-dependent ζ v a l u e s . 0

Γ

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

6

1.86 0.87 1.03 0.90 1.43 0.56 1.57 0.97 1.09 2.04

10

C C C C^ C C C C C C

1.64 1.12 1.42 1.31 1.49 1.08 1.63 0.85 1.71 2.81

SRK

3.99 7.08 8.92 10.2 12.4 15.1 16.2 18.5 19.5 21.8

10

9

8

7

6

5

3

2

x

9

8

7

6

5

4

3

2

C, C C C C C C C C C

10

9

8

7

d C, C, C* Cc C C C C C

Property

2.11 3.09 2.97 2.53 1.08 2.05 1.42 2.11 3.35 4.70

9.25 7.04 5.83 4.97 3.55 2.28 2.95 4.99 5.85 7.96

1.15 2.51 2.61 2.42 0.94 2.35 1.28 2.19 4.16 5.94

PR

1.36 1.10 1.19 1.01 1.40 1.11 1.96 1.93 1.60 1.87

5.11 4.97 4.96 5.13 4.89 4.00 4.26 3.57 3.74 3.05

1.00 0.67 0.85 0.75 1.00 0.68 1.41 1.37 1.10 1.42

HK

3RK

1.32 2.73 2.72 2.35 0.64 1.64 1.08 1.58 2.10 3.36

1.60 1.01 1.21 1.04 1.54 0.88 1.73 0.95 1.11 1.80

1.64 1.12 1.42 1.31 1.49 1.08 1.63 0.85 1.71 2.81

T

TSRK (0.50 < r

2.06 1.60 1.55 1.55 2.80 1.90 2.86 2.38 2.89 3.66

v

MMC

Vapor Pressure, p

PT

3.50 3.10 2.88 2.83 2.56 2.36 2.26 2.32 2.30 2.42

4.13 4.60 4.49 3.93 3.62 4.22 3.32 3.33 4.00 4.35

5.39 5.22 6.24 9.49 16.0 13.2 13.4 15.1 17.1 23.0

0.49 1.60 1.73 1.52 0.39 1.22 0.38 1.25 1.78 1.87

1.51 2.85 2.77 2.26 0.77 1.53 1.47 2.03 2.12 2.80

1.80 0.86 0.98 0.67 1.54 0.52 1.72 1.15 0.57 0.95

2.42 1.86 1.76 1.74 3.11 1.92 3.09 2.58 2.88 3.77

Saturated Vapor Volume, V

3.56 3.28 2.97 2.50 2.37 3.21 2.34 2.83 2.25 2.46 v

1.91 1.04 1.24 0.96 1.33 0.50 1.42 0.68 1.00 1.93

p

p

(0.50 < T

3.69 3.69 3.68 3.67 3.69 3.74 3.75 3.74 4.04 4.05

CI

0.56 2.20 1.85 1.27 0.90 1.12 1.26 1.61 1.62 1.89

ALS

2.82 3.09 2.92 2.41 2.32 3.17 2.26 2.48 2.30 2.36

1.47 1.07 1.18 1.13 0.89 1.19 1.30 1.63 1.22 1.52

1.02 2.42 1.98 1.44 1.23 1.44 1.69 2.12 1.91 2.19

< 0.98)

2.33 2.14 2.47 2.55 3.45 4.66 4.46 5.12 4.34 4.98

< 0.98)

0.97 0.66 0.78 0.79 0.69 0.96 1.13 1.18 1.02 1.41

< 0.98)

Saturated Liquid Volume, V* (0.50 < T

0.37 1.73 1.88 1.69 Ό.39 J O.39 1.57 0.54 1.48 2.19 2.59

SW

page

0.58 0.76 1.20 1.27 1.76 3.02 2.49 3.23 2.63 3.18

1.55 1.04 1.42 1.44 1.10 1.60 0.90 1.38 3.19 4.72

F

C o n t i n u e d on n e x t

3.92 3.12 2.70 3.14 3.92 4.29 5.98 6.94 9.26 10.8

1.85 3.61 4.30 4.55 5.28 4.04 3.46 1.50 1.85 5.15

KMC

2.46 1.52 1.87 1.71 1.30 1.60 1.11 1.16 2.92 4.02

4.14 3.95 5.83 8.16 10.8 11.5 11.0 12.9 15.6 19.4

1.42 1.15 1.49 0.84 2.18 1.53 2.71 3.74 4.17 3.38

3.44 2.32 4.02 5.93 6.70 10.3 11.5 15.0 20.0 25.3

H

2.17 4.46 5.36 5.57 6.31 4.56 3.72 1.62 1.80 4.55

2.41 2.01 2.32 2.83 4.93 3.37 4.22 4.47 5.89 8.58

0.97 0.32 0.16 0.71 0.75 0.31 0.43 0.60 0.85 1.36

0.89 0.64 0.70 0.76 1.78 0.76 0.99 1.08 3.82 7.42

HCL

Table III Comparison of Average Absolute Percentage Deviations Obtained from 14 Cubic Equations of State for Ten Normal Alkanes

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

10

10

9

8

7

6

5

4

3

2

C, C C C C C C C C C

10

9

8

7

Cg C C C C

C5

C, CI C, C*

υ

7

8

c c

IÏ

2

1

c c

Property

0.88 0.71 0.60 0.52 0.45 0.43 0.45 0.49 0.57 0.67

8.35 6.64 5.46 4.52 3.52 3.03 2.96 3.52 4.97 6.65

PR

SW

PT

3RK l

MMC

3.16 3.11 3.09 3.06 3.02 2.99 2.92 2.94 2.98 3.09

3.78 3.34 3.14 3.01 2.91 2.88 2.95 3.25 3.88 4.54

0.44 0.48 0.51 0.53 0.55 0.56 0.58 0.59 0.61 0.62

0.17 0.23 0.28 0.32 0.37 0.40 0.45 0.49 0.52 0.56

0.20 0.27 0.31 0.35 0.39 0.42 0.46 0.49 0.52 0.55

0.20 0.31 0.39 0.46 0.55 0.61 0.68 0.74 0.80 0.84

0.18 0.25 0.31 0.36 0.42 0.47 0.53 0.58 0.62 0.66

v

3.72 5.82 7.69 9.40 11.6 13.3 15.6 17.5 19.7 21.9

1.14 1.55 1.83 2.05 2.32 2.51 2.75 2.94 3.17 3.39

2.13 1.66 1.48 1.37 1.31 1.31 1.36 1.44 1.57 1.73

ALS

HCL

0.18 0.29 0.38 0.47 0.60 0.68 0.80 0.89 1.00 1.10

p

3.58 3.43 3.40 3.38 3.43 3.46 3.81 3.85 4.08 4.28

3.25 3.17 3.20 3.34 3.75 3.96 4.47 4.59 4.69 4.90

3.64 3.29 3.17 3.11 3.06 3.03 3.00 2.99 2.99 2.99

P

p

3.60 2.26 2.23 2.76 3.09 3.09 2.77 2.26 1.83 2.26

< 0.99, 0.01 < r < 10.0)

CI

1.15 1.26 1.33 1.39 1.46 1.51 1.57 1.62 1.68 1.73

1.71 1.36 1.18 1.09 1.08 1.16 1.35 1.52 1.73 1.92

0.83 0.87 0.92 1.01 1.14 1.27 1.44 1.59 1.78 1.96

1.16 1.20 1.20 1.20 1.23 1.27 1.36 1.48 1.67 1.89

0.98 1.08 1.18 1.32 1.57 1.84 2.21 2.52 2.89 3.25

0.98 1.06 1.11 1.17 1.24 1.32 1.44 1.55 1.71 1.89

sup

0.88 0.89 0.97 1.07 1.22 1.33 1.48 1.60 1.73 1.88

0.84 0.86 0.93 1.01 1.15 1.27 1.45 1.62 1.82 2.03

r

0.39 0.38 0.39 0.40 0.43 0.45 0.48 0.50 0.53 0.56

0.76 1.20 1.50 1.78 2.12 2.39 2.72 3.00 3.32 3.63

1.94 2.00 2.05 2.11 2.18 2.24 2.33 2.41 2.50 2.60

5.09 5.01 5.00 5.02 5.08 5.21 5.44 5.78 6.42 7.25

H

p

0.10 0.13 0.16 0.18 0.20 0.21 0.22 0.23 0.23 0.24

5.85 7.57 9.07 10.5 12.4 13.9 16.0 17.8 20.1 22.3

KMC

1.05 1.52 1.92 2.30 2.79 3.17 3.65 4.03 4.45 4.88

3.17 3.21 3.29 3.37 3.53 3.68 3.89 4.09 4.35 4.62

1.22 1.23 1.29 1.49 1.79 2.08 2.45 2.76 3.15 3.54

(1.01 < T < 4.00, 0.01 < P < 10.0)

0.58 0.49 0.43 0.39 0.40 0.42 0.47 0.51 0.56 0.63

Compressibility Factor of Vapour, Z (0.55 < T < 0.99, 0.01 < P < 0.8)

4.21 4.53 4.65 4.66 4.55 4.37 4.12 3.74 3.41 3.18

3.44 3.02 2.93 2.90 2.86 2.86 2.85 2.87 2.91 3.00

Γ

TSRK

Liquid Compressibility Factor, l (0.30 < τ

HK

Compressibility Factor of Gas above the Critical Temperature, Z

0.19 0.39 0.56 0.71 0.89 1.04 1.21 1.35 1.52 1.67

3.67 5.30 6.83 8.27 10.2 11.7 13.7 15.5 17.6 19.6

SRK

Table III Continued

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

4.80 5.15 5.41 5.67 6.00 6.27 6.62 6.91 7.24 7.57

0.95 0.95 0.95 0.95 0.96 0.98 1.01 1.03 1.06 1.08

5.12 5.19 5.13 5.09 5.12 5.17 5.36 5.40 5.57 5.74

F

Ο

£

"O

g ? ζ

>

00

=

S

χ m

^ § m

^1

§ Ο

^ >

3

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

0

C*

Cq

c'

7

C* C* C* C

Co

C, C,

9

8

7

6

5

3

c c c" c c c c c

2

Property

7.73 11.8 14.0 15.7 17.5 18.8 20.2 21.2 22.2 23.1

4.33 5.30 6.19 7.02 8.07 8.89 9.93 10.8 11.8 12.7

SRK

HK

SW

PT

3RK

MMC

TSRK c

T

CI

ALS

r

HCL

H

7.83 10.9 12.6 14.0 15.6 16.7 18.0 18.9 19.9 20.7

4.44 4.21 4.18 4.21 4.26 4.46 4.72 4.94 5.43 5.99

7.16 10.6 12.4 13.8 15.5 16.4 17.5 18.4 19.2 19.9

3.31 3.50 3.64 3.77 3.94 4.12 4.36 4.47 4.59 4.70 4.12 3.70 3.79 4.04 3.93 4.03 4.34 4.24 4.61 4.49

3.99 4.35 4.46 4.68 4.81 4.94 5.34 5.49 5.47 5.68

3.84 4.11 4.34 4.65 5.01 5.29 5.66 6.06 6.50 6.98 4.12 4.48 4.74 4.98 5.28 5.53 5.84 6.10 6.40 6.69

7.81 11.2 13.0 14.3 15.8 16.8 18.0 18.8 19.6 20.3

7.63 11.1 12.9 14.2 15.7 16.7 17.8 18.6 19.5 20.2

7.77 11.4 13.3 14.7 16.2 17.3 18.4 19.1 20.0 20.6

7.49 11.1 12.9 14.2 15.5 16.7 18.0 18.7 19.5 19.9

7.59 11.4 13.3 14.8 16.4 17.5 18.6 19.5 20.4 21.2

3.30 3.46 3.74 3.71 4.00 3.98 4.31 4.45 4.37 4.67

7.26 10.8 12.7 14.2 15.7 16.7 17.9 18.8 19.7 20.4

< 0.98)

7.59 11.1 12.9 14.3 15.7 16.9 18.0 19.0 20.0 20.7

p

3.15 3.92 5.06 5.77 6.50 6.96 7.40 7.67 7.89 8.05

Second Virial Coefficient, Β (0.50 < T

3.59 3.72 Ï.12 4.23 4.32 4.21 4.09 4.12 4.19 4.26

8.54 14.7 18.1 20.7 23.5 25.5 27.8 29.4 31.1 32.7

2.92 5.48 6.45 7.19 6.19 8.96 9.91 10.7 11.5 12.4

13.4 13.7 14.6 15.5 16.5 17.4 18.3 19.0 19.8 20.5

7.27 7.77 8.19 8.49 8.85 9.12 9.45 9.68 9.98 10.2

Compressibility Factor along the Critical Isotherm, Z ( r = 1.0, 0.01 < P < 10.0)

PR

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

1.09 0.77 0.87 1.00 1.20 1.43 1.88 2.34 2.93 3.56

3.81 4.10 4.33 4.61 4.99 5.30 5.70 6.20 6.78 7.34

KMC

8.81 12.6 14.6 16.0 17.7 18.8 20.0 20.8 21.7 22.5

9.80 10.7 11.3 11.8 12.4 12.9 13.5 14.0 14.5 15.0

F

546

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Should the parameters of a two-parameter e q u a t i o n be t r e a t e d temperature dependent up t o the c r i t i c a l p o i n t by a f o r c e d f i t t i n g p r o c e d u r e , the s e l e c t i o n of p and V would be d e s i r a b l e f o r t h i s purpose. However, such an equation can no l o n g e r s a t i s f y both Equations 2 and 3. In o t h e r words, the r e p r e s e n t a t i o n of the c r i t i c a l i s o t h e r m i n the c r i t i c a l region cannot be s a t i s f a c t o r y . Although a 3P2T e q u a t i o n (having the f l e x i b i l i t y of o b t a i n i n g substance-dependent ζ v a l u e s ) might overcome t h i s d i f f i c u l t y , two such e q u a t i o n s were e v a l u a t e d i n t h i s work, Η and KMC, and both e q u a t i o n s y i e l d e d poorer o v e r a l l performance than the e v a l u a t e d 3P1T equations. Hence, a f u r t h e r examination of the l i m i t a t i o n s and b e h a v i o r of 3P1T c u b i c e q u a t i o n s of s t a t e would be u s e f u l i n t h e d e s i g n of new c u b i c e q u a t i o n s . A l l c u b i c equations of s t a t e s u f f e r c e r t a i n s h o r t c o m i n g s . As d i s c u s s e d above, no e q u a t i o n e v a l u a t e d c o u l d s a t i s f a c t o r i l y r e p r e sent a l l the e i g h t p h y s i c a l p r o p e r t i e s s i m u l t a n e o u s l y . I f one f o r c e s an e q u a t i o n t o p r o v i d e a c c e p t a b l e Β v a l u e s , such as the KMC e q u a t i o n , l a r g e r d e v i a t i o n s would occur i n o t h e r p r e d i c t i o n s , such as the V* and Z* v a l u e s . On the o t h e r hand, i f one i g n o r e s the r e p r e s e n t a t i o n of B, t h e r e i s a chance of o b t a i n i n g an e q u a t i o n which can s i m u l t a n e o u s l y p r o v i d e p r e d i c t i o n s of the o t h e r seven p h y s i c a l p r o p e r t i e s w i t h reasonable r e s u l t s . The parameter v a l u e s of " a " capable of p r e d i c t i n g vapor p r e s s u r e s of pure substances would i n general be s u i t a b l e f o r v a p o r - l i q u i d e q u i l i b r i u m c a l c u l a t i o n s . Because some a p p r o x i m a t i o n i s i n v o l v e d i n c o r r e l a t i n g Q by means of a temperature f u n c t i o n , some d e v i a t i o n s occur i n t h e calculated p values. T h i s shortcoming i s not due t o the form of the c u b i c e q u a t i o n of s t a t e , but due t o the temperature f u n c t i o n itself. The d e v i a t i o n s i n p due t o t h i s inadequacy are shown i n Table III. T h i s inadequacy i s a l s o r e f l e c t e d i n the c a l c u l a t e d V v a l u e s . A p l o t of p and V d e v i a t i o n s as a f u n c t i o n of T would r e v e a l t h a t one d e v i a t i o n curve i s m i r r o r image of the o t h e r a l o n g t h e z e r o - d e v i a t i o n l i n e f o r the 2P1T and 3P1T e q u a t i o n s . S i n c e the d e v i a t i o n s i n the c a l c u l a t e d Z and Z v a l u e s are p r a c t i c a l l y the same among the 2P1T and 3P1T equations and these d e v i a t i o n s are of t h e l e a s t concern of t h i s s t u d y , the f o l l o w i n g d i s c u s s i o n i s m a i n l y c e n t e r e d on the d e v i a t i o n s of V * , Z* and Z P . To i l l u s t r a t e the drawbacks of e q u a t i o n s c o n t a i n i n g two temp e r a t u r e - d e p e n d e n t parameters, the d e v i a t i o n curves o b t a i n e d f o r the HCL and Η e q u a t i o n s are p l o t t e d as a f u n c t i o n of T i n F i g u r e 1. The c o r r e l a t i o n equations o r i g i n a l l y proposed ( 4 , 14) were used t o generate the curves f o r n-hexane. As both equations used p and V f o r d e t e r m i n i n g t h e i r parameters, they y i e l d low d e v i a t i o n s i n t h e s e two p r o p e r t i e s , even though the accuracy of the r e p r e s e n t a t i o n was l o s t somewhat i n the c o r r e l a t i o n s . However, the d e v i a t i o n s o b t a i n e d i n the V v a l u e s are too l a r g e , due t o the s h i f t of e r r o r s through the l i m i t a t i o n s of a c u b i c e q u a t i o n . The r e s u l t s o b t a i n e d from a m o d i f i e d PR e q u a t i o n w i t h both parameters a and b s i m u l t a n e o u s l y f i t t e d t o p and V* of n-hexane are a l s o i n c l u d e d i n F i g u r e 1 f o r c o m p a r i s o n . The s h i f t of e r r o r s t o V i s a l s o very e v i d e n t . The d i f f i c u l t y encountered by equations c o n t a i n i n g two t e m p e r a t u r e dependent parameters f o r r e p r e s e n t i n g the c r i t i c a l region i s d e p i c t e d v

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In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

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Design of Cubic Equations of State

Y U ET AL.

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Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

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American Chemical Society Library 1155 16th Si» N.W. Washington, D.C. 20038 In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

E Q U A T I O N S O F STATE: T H E O R I E S A N D

548

APPLICATIONS

i n F i g u r e 2 , u s i n g the e x p r e s s i o n i n terms of a(=a/a ) and p ( = b / b ) , and p l o t t e d i n terms of T . The shape of the two c u r v e s , p a r t i c u l a r l y near the c r i t i c a l p o i n t , prevents simple c o r r e l a t i o n of the parameters i n terms of T . The i n f l u e n c e of values of u and w on the r e p r e s e n t a t i o n of V v a l u e s by means of the 2P1T e q u a t i o n s (SRK, PR and HA) i s i l l u s t r a t e d i n F i g u r e 3. The t h r e e e q u a t i o n s belong t o the u + w = 1 f a m i l y and t h e i r " a " parameters were determined by f i t t i n g vapor p r e s s u r e s . In F i g u r e 3, the c a l c u l a t e d d e v i a t i o n s i n V and V f o r methane, n-hexane and n-decane are p l o t t e d . None of the t h r e e e q u a t i o n s y i e l d e d r e a l l y s a t i s f a c t o r y r e s u l t s . I t i s of i n t e r e s t t o note t h a t the general p a t t e r n s of e r r o r d i s t r i b u t i o n are about the same. It i s i m p o s s i b l e t o o b t a i n low d e v i a t i o n s f o r substances w i t h d i f f e r e n t m o l e c u l a r weights from two-parameter e q u a t i o n s by s i m p l y a d j u s t i n g the u and w v a l u e s . T h i s c o n f i r m s the c o n c l u s i o n reached above t h a t t h r e e - p a r a m e t e r e q u a t i o n s are d e s i r a b l e . The d e v i a t i o n contours of V o b t a i n e d from E q u a t i o n 1 are p l o t t e d on u-w diagrams i n F i g u r e s 4 and 5 f o r methane, n-hexane and n-decane. The numbers i n d i c a t e d on the diagrams r e f e r t o the number of carbons of the a l k a n e s , and the dots r e p r e s e n t the minimum d e v i a t i o n s . The temperature range covered i n F i g u r e 4 (0.5 < Tr < 0.98) i s l a r g e r than t h a t i n F i g u r e 5 (0.5 < Tr < 0 . 8 5 ) . On the o t h e r hand, the c o n t o u r s i n F i g u r e 4 r e p r e s e n t average a b s o l u t e d e v i a t i o n s of 2.5% i n V , w h i l e i n F i g u r e 5, 1%. As p o i n t e d out by Schmidt and Wenzel (J5), the f o l l o w i n g c o n s t r a i n t s must be s a t i s f i e d t o a v o i d the c o n d i t i o n t h a t V + ubV + w b = 0 f o r V > b, c

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Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

A

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The n o n e x i s t i n g area i n d i c a t e d i n F i g u r e s 4 and 5 r e p r e s e n t s t h e area excluded by Equation 4. A comparison of the two diagrams i n d i c a t e s t h a t by narrowing the temperature range, the r e l a t i v e p o s i t i o n s of the two groups of d e v i a t i o n contours s h i f t towards l a r g e r v a l u e s of w and s m a l l e r v a l u e of u. In phase e q u i l i b r i u m c a l c u l a t i o n s , i t would be d e s i r a b l e t o have more v o l a t i l e components w e l l r e p r e s e n t e d at h i g h e r T v a l u e s , and l e s s v o l a t i l e components w e l l r e p r e s e n t e d at lower T values. The d e v i a t i o n contours f o r Z i n the temperature and p r e s s u r e ranges of 0.30 < T < 0.99 and 0.01 < P < 10.0 are p l o t t e d on t h e u-w diagram i n F i g u r e 6. The contours r e p r e s e n t 4% average a b s o l u t e d e v i a t i o n s . S i m i l a r l y , the d e v i a t i o n contours f o r Z P i n t h e temperature and p r e s s u r e ranges of 1.01 < T < 4.00 and 0.01 < P < 10.0 are p l o t t e d i n F i g u r e 7. The contours r e p r e s e n t C and C , and the average a b s o l u t e d e v i a t i o n s vary from 1.5 t o 2.5%. When the d e v i a t i o n contours of F i g u r e s 4-7 are superimposed, t h e o v e r l a p p i n g areas f o r the alkanes are p l o t t e d on the u - w diagram i n F i g u r e 8. The d e v i a t i o n contour of V f o r C is also i n c l u d e d i n the f i g u r e f o r r e f e r e n c e . Any l i n e or curve p a s s i n g through these contours w i l l y i e l d the f o l l o w i n g average a b s o l u t e d e v i a t i o n s f o r the alkanes i n the t h r e e p h y s i c a l p r o p e r t i e s : V : 1% (0.5 < T < 0 . 8 5 ) ; 2.5% (0.5 < T < 0.98) p

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In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

Figure 2

V a r i a t i o n of α and β w i t h T PR e q u a t i o n

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In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

550

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

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Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

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In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

26.

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Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

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In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

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Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

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In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Design of Cubic Equations of State

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Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

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In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

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Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

• OPTIMUM POINT

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D e v i a t i o n c o n t o u r s of V * i n the temperature range of 0.5 < T < 0 . 8 5 . Curves r e p r e s e n t d e v i a t i o n s of 1% in V*. r

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In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Design of Cubic Equations of State

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

YU ET AL.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

556

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Figure 7

D e v i a t i o n contours of Z P . i n d i c a t e d on each c u r v e . S U

Deviation l e v e l s are

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

Figure 8

O v e r l a p p i n g areas r e s u l t i n g from s u p e r i m p o s i n g d e v i a t i o n c o n t o u r s of F i g u r e s 4 - 7 ·

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

558

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Z : 4% (0.3 < Tr < 0.99) Z P < 2.5% (1.01 < T < 4 . 0 0 , 0.01 < P < 10.0) The s o l i d curves i n F i g u r e 8 r e p r e s e n t t h e l o c i of t h e minimum d e v i a t i o n p o i n t s f o r t h e t e n a l k a n e s , w i t h t h e e x c e p t i o n of t h e V * c u r v e , which extends t o C . The f a m i l y of e q u a t i o n s r e p r e s e n t e d by t h e u + w = 1 r e l a t i o n s h i p on t h e u-w diagram cannot y i e l d good p r e d i c t i o n of V* v a l u e s , e s p e c i a l l y f o r alkanes of l a r g e r m o l e c u l a r w e i g h t s . The d e s i r e d r e l a t i o n s h i p should have a p o s i t i v e s l o p e on t h e u-w d i a g r a m . A

S U

r

p

2 0

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

Concluding

Remarks

For t h e purpose of o b t a i n i n g s a t i s f a c t o r y r e s u l t s s i m u l t a n e o u s l y f o r v a p o r - l i q u i d e q u i l i b r i u m values and v o l u m e t r i c p r o p e r t i e s , a t h r e e parameter van der Waals type e q u a t i o n w i t h the parameter " a " f i t t e d t o vapor p r e s s u r e values i s d e s i r a b l e . Using d e v i a t i o n contours on u-w diagrams p r o v i d e s a means f o r designing a d e s i r a b l e equation f o r representing s p e c i f i c physical p r o p e r t i e s of t h e substances c o n c e r n e d . For t h e r e p r e s e n t a t i o n of V , Z * and Z P f o r normal a l k a n e s , t h e r e l a t i o n s h i p between u and w should y i e l d a p o s i t i v e s l o p e on the u-w diagram, c o n t r a r y t o t h e s u g g e s t i o n of Schmidt and Wenzel. A p o s s i b l e e q u a t i o n i s t h a t r e p r e s e n t e d by t h e u - w = 3 r e l a t i o n s h i p as shown i n F i g u r e 8 . A

S U

Acknowledgments The authors are indebted t o t h e Natural Science and E n g i n e e r i n g Research C o u n c i l of Canada f o r f i n a n c i a l s u p p o r t .

Literature Cited 1.

van der Waals, J.D. doctoral dissertation, Leiden, Holland, 1873. 2. Redlich, O., Kwong, J.N.S. Chem. Rev. 1949, 44, 239. 3. Peng, D.Y., Robinson, D.B. Ind. Eng. Chem. Fundam. 1976, 15, 59. 4. Heyen, G. Proc. 2nd Int. Conf. on Phase Equilibria and Fluid Properties in the Chemical Industry, 1980, p. 9; In "Chemical Engineering Thermodynamics"; Newman, S.A., Ed.; Ann Arbor Science Publishers, Ann Arbor, 1983; Chap. 15. 5. Schmidt, G., Wenzel, H. Chem. Eng. Sci. 1980, 35, 1503. 6. Abbott, M.M. AIChEJ 1973, 19, 596; in "Equations of State in Engineering and Research", Chao, K.C., Robinson, R.L. Jr., Eds.; Advances in Chemistry Series No. 182, American Chemical Society: Washington, D.C. 1979; Chap. 3. 7. Döring, R., Knapp, H. Proc. 2nd Int. Conf. on Phase Equilibria and Fluid Properties in the Chemical Industry, 1980, p. 34. 8. Lira, R., Malo, J.M., Leiva, M.A., Chapela, G. In "Chemical Engineering Thermodynamics"; Newman, S.A., Ed.; Ann Arbor Science Publishers, Ann Arbor, 1983; Chap. 14. 9. Adachi, Y., Lu, B.C.-Y., Sugie, H. Fluid Phase Equilibria 1983, 11, 29. 10. Adachi, Y., Lu, B.C.-Y. AIChEJ 1984, 30, 991.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

26. YU ET AL.

Design of Cubic Equations of State

11. Kumar, K.H., Starling, K.E. Ind. Eng. Chem. Fundam. 1982, 21, 255. 12. Clausius, R. Ann Phys. Chem. 1880, IX, 337. 13. Soave, G. Chem. Eng. Sci. 1972, 27, 1197. 14. Hamam, S.E.M., Chung, W.K., Elshayal, I.M., Lu, B.C.-Y. Ind. Eng. Chem. Process Des. Dev. 1977, 16, 51. 15. Harmens, Α., Knapp, H. Ind. Eng. Chem. Fundam. 1980, 19, 291. 16. Patel, N.C., Teja, A.S. Chem. Eng. Sci. 1982, 37, 463. 17. Peneloux, Α., Rauzy, E., Freze, R. Fluid Phase Equilibria 1982, 8, 7. 18. Freze, R., Chevalier, J.-L., Peneloux, Α., Rouzy, E. Fluid Phase Equilibria, 1983, 15, 33. 19. Adachi, Y., Lu, B.C.-Y., Sugie, H. Fluid Phase Equilibria 1983, 13, 133. 20. Fuller, G.G. Ind. Eng. Chem. Fundam. 1976, 15, 254. 21. Martin, J.J. Ind. Eng. Chem. Fundam. 1979, 18, 81. 22. Kubic, W.L. Fluid Phase Equilibria 1982, 9, 79. 23. Harmens, A. Cryogenics 1977, 17, 519. 24. Gomez-Nieto, M., Thodos, G. Ind. Eng. Chem. Fundam. 1978, 17, 45. 25. Spencer, C.F., Alder, S.B. J. Chem. Eng. Data 1978, 43, 137. 26. Barile, R.G., Thodos, G. Can. J. Chem. Eng. 1965, 43, 137. 27. Tsonopoulos, C. AIChEJ 1974, 20, 263. 28. Lee, B.I., Kesler, M.G. AIChEJ 1975, 21, 510. RECEIVED November 8, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

559

27 A n Improved Cubic Equation of State 1

R. Stryjek and J. H. Vera

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 2A7, Canada

The Peng-Robinson equation of state has been modified to extend its use to low reduced temperatures. The modified form, called the PRSV equation, is able to reproduce pure compound vapor pressures down to 1.5 kPa with accuracy comparable to Antoine's equation for hydrocarbons, polar and associated compounds. Hydro carbon/hydrocarbon and aromatic hydrocarbon/polar com pound vapor-liquid equilibria have been correlated using a single binary parameter with an accuracy simi lar to the γ-ψ approach. The binary parameter is slightly temperature dependent. Two parameters are required to correlate vapor-liquid equilibria of satu rated hydrocarbon/polar compound systems. Water/ methanol system is well represented with one binary parameter model. Representation of the systems water/ ethanol and water/propanols requires the use of two binary parameters. The use of c u b i c e q u a t i o n s o f s t a t e t o c o r r e l a t e v a p o r - l i q u i d e q u i l i b r i u m d a t a has r e c e i v e d i n c r e a s e d a t t e n t i o n i n the l a s t few y e a r s . For the purposes of t h i s s t u d y , i m p o r t a n t c o n t r i b u t i o n s have been presented by M a t h i a s ( 1 ) , M a t h i a s and Copeman ( 2 ) , Soave (3) and Gibbons and Laughton ( 4 ) . M a t h i a s (1) proposed an improved temperature dependence f o r the a t t r a c t i v e term of the Redlich-Kwong e q u a t i o n of s t a t e . M a t h i a s and Copeman (2) suggested a form f o r the temperature dependence of the a t t r a c t i v e term of the Peng-Robinson e q u a t i o n o f state. Soave (3) showed t h a t even the s i m p l e van d e r W a a l s equa t i o n of s t a t e may be used t o c o r r e l a t e v a p o r - l i q u i d e q u i l i b r i a w i t h an a p p r o p r i a t e temperature dependence of the a t t r a c t i v e term f o r pure compounds and the use of the m i x i n g r u l e s proposed by Huron and V i d a l ( 5 ) . Gibbons and Laughton (4) presented a modi f i e d form o f the Redlich-Kwong e q u a t i o n o f s t a t e i n c l u d i n g two a d j u s t a b l e parameters per pure compound. T

1

Current address: Institute of Physical Chemistry, PAN, Warsaw, Poland. 0097-6156/ 86/ 0300-O560$06.00/ 0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

27.

Improved Cubic Equation of State

STRYJEK AND VERA

561

Pure Compounds In t h i s work we have s e l e c t e d the c u b i c e q u a t i o n of s t a t e proposed by Peng and Robinson ( 6 ) , namely,

2

Ρ = Η _ v

"

b

v

2

(ί)

+ 2bv - b

2

with 2

2

a = (0.45724 R T / P ) a

(2)

b = 0.0778 R T / P

(3)

c

and

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

c

c

where α i s g i v e n by α = [ l + κ (1 - T

0 R

5

- )]

2

(4)

Peng and Robinson (6) c o n s i d e r e d κ t o be a f u n c t i o n of the a c e n t r i c f a c t o r ω o n l y . T h i s e q u a t i o n performs w e l l f o r h y d r o carbons and s l i g h t l y p o l a r compounds at reduced temperatures of the o r d e r of 0.7 and above. At low reduced t e m p e r a t u r e s , however, i t f a i l s to reproduce the vapor p r e s s u r e s of pure compounds even f o r nonpolar s u b s t a n c e s . P r e l i m i n a r y s t u d i e s at reduced temperatures below 0.7 showed that d i f f e r e n t v a l u e s of κ could be o b t a i n e d f o r the same compound depending on the reduced temperature range of the vapor p r e s s u r e d a t a used i n the f i t t i n g p r o c e d u r e . In a d d i t i o n , f o r d i f f e r e n t compounds w i t h a l a r g e v a r i a t i o n i n c r i t i c a l t e m p e r a t u r e , v a l u e s of κ had a smooth v a r i a t i o n w i t h the a c e n t r i c f a c t o r o n l y when a l l compounds were c o n s i d e r e d i n the same reduced temperature range. These r e s u l t s suggested t h a t i t was n e c e s s a r y t o c o n s i d e r a f i x e d reduced temperature f o r a l l compounds i n o r d e r t o g e n e r a t e a s e l f c o n s i s t e n t s e t of κ v a l u e s . In t h i s work we s e l e c t e d T = 0.7. Since t h i s temperature i s c l o s e t o the normal b o i l i n g p o i n t f o r most s u b s t a n c e s , r e l i a b l e vapor p r e s s u r e d a t a are a v a i l a b l e i n i t s v i c i n i t y f o r most compounds. C r i t i c a l p r e s s u r e s and c r i t i c a l temperatures r e p o r t e d i n the l i t e r a t u r e were c a r e f u l l y examined and the most p r e c i s e v a l u e s were s e l e c t e d f o r the e v a l u a t i o n of b o t h ω and κ. The a c e n t r i c f a c t o r ω was o b t a i n e d from i t s d e f i n i t i o n . The v a l u e of κ, here c a l l e d K , was o b t a i n e d so as t o reproduce e x a c t l y the e x p e r i m e n t a l vapor p r e s s u r e when the f u g a c i t y of the vapor phase i s e q u a l t o the f u g a c i t y of the l i q u i d phase at T = 0.7. V a l u e s of K SO o b t a i n e d f a l l over a s i n g l e curve when p l o t t e d a g a i n s t ω, i r r e s p e c t i v e of the p o l a r i t y , degree of a s s o c i a t i o n or g e o m e t r i c a l c o m p l e x i t y of the m o l e c u l e s . The f o l l o w i n g c o r r e l a t i o n reproduces w e l l the v a r i a t i o n of K w i t h ω f o r over e i g h t y compounds. R

q

R

Q

q

K

2

Q

= 0.378893 + 1.4897153ω - 0.17131848ω + 0.0196554ω

3

(5)

The use of e q u a t i o n (5) i n s t e a d of the form f o r the depen dence of κ on ω proposed by Peng and Robinson (6) produced a f i r s t improvement i n the r e s u l t s . Not o n l y the r e p r o d u c t i o n of vapor

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

562

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

p r e s s u r e s was improved i n t h e reduced temperature range from 0.7 to 1.0, but a l s o d e v i a t i o n s between c a l c u l a t e d and e x p e r i m e n t a l vapor p r e s s u r e s f o r d i f f e r e n t compounds presented a more r e g u l a r v a r i a t i o n a t reduced temperatures below 0.7. However, f o r com pounds w i t h ω * 0.1 and ω « 0.4 b o t h forms g i v e t h e same v a l u e of κ and t h u s , the same r e s u l t s . Maximum d i f f e r e n c e s between b o t h c o r r e l a t i o n s a r e produced f o r low and h i g h a c e n t r i c f a c t o r s . In o r d e r t o reproduce s a t u r a t i o n p r e s s u r e s a t reduced tem p e r a t u r e s below 0.7 i t i s n e c e s s a r y t o i n t r o d u c e a temperature dependence on κ w i t h a t l e a s t one a d j u s t a b l e parameter c h a r a c t e r i s t i c of each compound. A f t e r some t r i a l s , the f o l l o w i n g form was found t o g i v e a good f i t of vapor p r e s s u r e s down t o low reduced temperatures

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

κ = κ

0

(1 + τ£· )(0.7 - T ) 5

+

(6)

R

Although values of may be o b t a i n e d from a s i n g l e vapor p r e s s u r e datum a t reduced temperature below 0.7, i n t h i s work v a l u e s of were o b t a i n e d by the f i t t i n g of low reduced temperature vapor p r e s s u r e d a t a . Some r e p r e s e n t a t i v e v a l u e s of κ-^ a r e g i v e n i n Table I t o g e t h e r w i t h c r i t i c a l p r o p e r t i e s and a c e n t r i c f a c t o r s used i n t h i s work. F o r water and lower a l c o h o l s t h e r e i s a s m a l l advantage i n u s i n g e q u a t i o n (6) i n a l l the range from low reduced temperature up t o t h e c r i t i c a l temperature. F o r a l l o t h e r com pounds we recommend t o use κ = κ f o r Τ , > 0.7. ο κ T a b l e I . Pure Compound

Compound Ethane Cyclohexane Benzene Biphenyl Acetone Ammonia Water Methanol Ethanol 1-Propanol 2-Propanol Pyridine Thianaphthene

T ,K C

302.43 553.64 562.16 769.15 508.10 405.55 647.29 515.58 513.92 536.71 508.40 620.00 752.00

Parameters

P , kPa

ω

4879.76 4074.96 4897.95 3120.78 4695.95 11289.52 22089.75 8095.79 6148.33 5169.55 4764.25 5595.26* 3880.71

0.0978 0.2088 0.2093 0.3810 0.3067 0.2517 0.3438 0.5653 0.6444 0.6201 0.6637 0.2372 0.2936

c

K

l

0.02814 0.07023 0.07019 0.11487 -0.00888 0.00100 -0.06635 -0.16816 -0.03374 0.21419 0.23264 0.06946 0.06043

* e s t i m a t e d by group c o n t r i b u t i o n method.

F i g u r e 1 compares the performance of the m o d i f i e d PengRobinson e q u a t i o n , from here on c a l l e d the PRSV e q u a t i o n , w i t h t h e m o d i f i e d c u b i c e q u a t i o n s of s t a t e o f Soave (3) and o f Gibbons and Laughton ( 4 ) . I t should be observed t h e forms of Soave ( 3 ) and o f

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

STRYJEK AND VERA

Improved Cubic Equation of State

563

Gibbons and Laughton (4) c o n t a i n two a d j u s t a b l e parameters p e r compound w h i l e the PRSV e q u a t i o n c o n t a i n s o n l y one. Table I I p r e s e n t s a comparison of r e s u l t s obtained w i t h the PRSV e q u a t i o n and those r e p o r t e d by Mathias ( 1 ) u s i n g a m o d i f i e d Redlich-Kwong e q u a t i o n of s t a t e . C l e a r l y , the PRSV e q u a t i o n g i v e s a b e t t e r r e p r e s e n t a t i o n of pure compound vapor p r e s s u r e s than any o t h e r m o d i f i e d c u b i c e q u a t i o n of s t a t e . F o r water, f o r example, vapor p r e s s u r e s a r e c o r r e l a t e d w i t h a percent d e v i a t i o n of l e s s than 0.03 from i t s t r i p l e p o i n t t o a reduced temperature of 0.75. S l i g h t l y h i g h e r d e v i a t i o n s a r e obtained i n the h i g h reduced tem p e r a t u r e range b u t , i n any case, d e v i a t i o n s a r e s m a l l e r than those o b t a i n e d w i t h o t h e r c u b i c e q u a t i o n s of s t a t e . Similar r e s u l t s a r e o b t a i n e d f o r ammonia. Heavy o r g a n i c compounds, a r e e q u a l l y w e l l represented by the PRSV e q u a t i o n down t o v e r y low vapor p r e s s u r e s . F i g u r e 2 p r e s e n t s the performance of the PRSV e q u a t i o n f o r some compounds. Observe t h a t f o r b i p h e n y l vaporp r e s s u r e s a r e w e l l r e p r e s e n t e d by the PRSV e q u a t i o n down t o a p r e s s u r e of 0.004 k P a . R e s u l t s f o r p y r i d i n e present an average p r e s s u r e d e v i a t i o n of about 0.1%. Table I I .

Comparison of R e s u l t s Obtained w i t h the PRSV E q u a t i o n and Those Reported by Mathias (1)

ΔΡ %

Compound Water Acetone Methanol Ethanol 1-Pentanol 1-Octanol

T,,K

Mathias

273- -647 259- -508 288- -513 293- -514 348- -512 386- -554

0.3 0.4 0.4 0.7 0.7 2.2

T h i s work 0.1 0.2 0.6 0.5 0.7 1.6

Binary Mixtures As f o r the Peng-Robinson e q u a t i o n of s t a t e , the e x p r e s s i o n f o r t h e f u g a c i t y c o e f f i c i e n t of a component i i n a m i x t u r e o b t a i n e d from the PRSV e q u a t i o n has the g e n e r a l form

A

*ηψ. - ^ i ( z - l ) - *n(z-B) b

£k + 1 - !îi)£n ζ+(1+/¥)Β ( 7 ) a b z+(l-/2)B

2/2 Β 2

where ζ = Pv/RT, A » P a / ( R T ) , obtained as

â, - (2ïïL)

(8)

Β = Pb/RT and a

±

b

±

and b

- (IBL)

i

are

(9)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

564

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

ETHANE

WATER Δρ (%)

JE. AMMONIA

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

m p

0-4

0-5

-ι

0-6

0-7

0-8

I 0-9

ι

T

I R

1-0

F i g u r e 1. P e r Cent D e v i a t i o n s i n C a l c u l a t e d Pure Compound Vapor P r e s s u r e s f o r Ethane, Water and Ammonia. PRSV; Soave (1984); - · - — Gibbons and Laughton (1984). BIPHENYL 0004 (kPa) 0-413 +

g

+

(a) THIANAPHTHENE

*

-15-4

1

(kPa) 175

CL

< (b)

PYRIDINE

0-5 0 -0-5 .(c) 0-4

^

0-5

0 0

0

0

0

0

° 0 0 0 o

I

I

0-6

0-7

To

F i g u r e 2. P e r Cent D e v i a t i o n s i n C a l c u l a t e d Pure Compound Vapor P r e s s u r e s f o r B i p h e n y l , Thianaphtene and P y r i d i n e u s i n g the PRSV Equation of State.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Improved Cubic Equation of State

27. STRYJEK AND VERA

565

The p a r t i c u l a r e x p r e s s i o n s of a^ and depend on the m i x i n g r u l e s chosen f o r a and b, r e s p e c t i v e l y . I n t h i s work we have used the c o n v e n t i o n a l mixing r u l e f o r b i n a l l cases b = Y x, b,

(10)

j which g i v e s = b ^ ( p u r e compound) For

a, we t e s t e d

(11)

f i r s t the c o n v e n t i o n a l m i x i n g r u l e

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

(12) with 0.5 a

ij

=

(

a

a

i j>

( 1 3 )

which g i v e s a i

= 2 £ χ a -a j J iJ

(14)

These m i x i n g r u l e s proved to be s a t i s f a c t o r y f o r m i x t u r e s of the type hydrocarbon/hydrocarbon and a r o m a t i c h y d r o c a r b o n / p o l a r compound. However they gave o n l y f a i r r e s u l t s f o r systems of the type s a t u r a t e d h y d r o c a r b o n / p o l a r compound. For b i n a r y systems the b i n a r y parameters were f i t t e d m i n i m i z i n g the o b j e c t i v e f u n c t i o n : Q = £(ΔΡ/Ρ) , where Ρ i s the bubble p r e s s u r e . R e s u l t s presented i n the t a b l e s are those w i t h the optimum v a l u e of the parameter a t each temperature. R e s u l t s f o r the b i n a r y system b e n z e n e / b i p h e n y l are presented i n Table I I I . For comparison we have a l s o i n c l u d e d i n Table I I I r e s u l t s o b t a i n e d by Gmehling et a l . (7) u s i n g s t a n dard γ-ψ approach. I t i s i n t e r e s t i n g t o observe t h a t at the lowest i s o t h e r m b i p h e n y l i s below i t s normal m e l t i n g p o i n t . Table IV p r e s e n t s a s i m i l a r comparison f o r d i f f e r e n t i s o t h e r m s of the system benzene/acetone. A g a i n here the one parameter m i x i n g r u l e produces s u r p r i s i n g l y good r e s u l t s . N o t a b l y , v a l u e s of the k j ~ parameter are almost independent of temperature f o r the system benzene/ acetone and p r e s e n t a r e g u l a r temperature dependence f o r the system b e n z e n e / b i p h e n y l . No attempt was done t o smooth the temperature dependence of k ^ s i n c e γ-ψ approaches use a l s o p a r t i c u l a r v a l u e s of two a d j u s t a b l e parameters a t each temperature. For systems of the type s a t u r a t e d h y d r o c a r b o n / p o l a r compound such as c y c l o h e x a n e / acetone the use of the one-parameter m i x i n g r u l e g i v e n by e q u a t i o n (13) produced l a r g e r d e v i a t i o n s i n the c a l c u l a t e d p r e s s u r e s . These d e v i a t i o n s were not random, they presented a s y s t e m a t i c trend w i t h r e s p e c t to c o m p o s i t i o n f o r d i f f e r e n t systems s t u d i e d . The use of two b i n a r y parameters was c l e a r l y r e q u i r e d . In t h i s work we have used the two-parameter m i x i n g r u l e

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

566

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Table I I I . Vapor-Liquid

Equilibria

f o r the System Benzene/Biphenyl ΔΡ, mm Hg

PRSV-1

3

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

T,K

k «10 1 2

— ΔΡ

Wilson*

— 3 Ay«10

ΔΡ

NRTL*

A y 10

ΔΡ

UNIQUAC*

— ΔΡ

Ay«10

— 3 Ay«10

318.15

A.420

0.60

0.2

7.73

0.1

3.16

0.0

12.57

328.15

5.785

0.50

0.3

15.74

0.3

4.50

0.1

18.78

0.4

338.15

7.100

0.73

0.4

18.68

0.4

5.84

0.1

24.75

0.5

0.1

*As reported by Gmehling et a l . (1980)

Table IV. Vapor-Liquid E q u i l i b r i a f o r the System Benzene/Acetone "AT, mm Hg

PRSV-1 T,K 298.15 303.15 313.15 318.15 323.15

k

1 2

«10

2.716 2.836 2.877 2.512 2.985

2

ΔΡ

AylO

0.75 1.23 2.46 0.68 2.28

5.0 9.9 6.8 4.2 7.2

3

ΔΡ 0.50 1.78 2.14 0.71 3.19

AylO 2.8 6.4 4.2 3.4 7.6

UNIQUAC*

NRTL*

Wilson* 3

ΔΡ

AylO

0.51 1.78 2.13 0.76 3.43

2.8 6.4 4.2 3.5 7.8

3

ΔΡ 0.48 1.78 2.08 0.60 3.06

*As reported by Gmehling et a l . (1980)

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

AylO 2.7 6.4 4.1 3.4 7.5

3

Improved Cubic Equation of State

27. STRYJEK AND VERA

567

proposed by Huron and V i d a l (5) f o r the a term, which f o r the PRSV e q u a t i o n takes the form, a = b Ci x

±

a /b i

E

i

- c g )

(15)

w i t h c = 2/2/An [ ( 2 + / 2 ) / ( 2 - / 2 ) ] .

With t h i s m i x i n g r u l e f o r

a, the term of e q u a t i o n (7) c o n t a i n i n g a^ takes the form

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

Λ — 2/2B

(ii +

1 - %

a

b

= - i - [U L

- b.RT 2/2

- c An γ .] »

(16)

±

1

1

For the " i n f i n i t e p r e s s u r e " excess f u n c t i o n we have used i n equa t i o n (15) the e x p r e s s i o n f o r t h e excess Gibbs energy proposed by Renon and P r a u s n i t z (8) as suggested by Huron and V i d a l ( 5 ) . Thus,

x

»

RT ν α . χ ,

R T

i r G, . χ ,

I k i k k G

x

G

x

J

J j kj k k

^

I kj k k G

x

(17)

with 0.3 Ag.. . G.. = exp ZlJ_ (18) ij RT W a t e r / a l c o h o l systems p r e s e n t complex m o l e c u l a r i n t e r a c t i o n s and thus they a r e a s e v e r e t e s t f o r c o r r e l a t i n g methods. Table V shows r e s u l t s o b t a i n e d w i t h the o n e - b i n a r y parameter PRSV-1 form of e q u a t i o n (13) f o r the system water/methanol. These r e s u l t s a r e p e r f e c t l y comparable w i t h those o b t a i n e d by Gmehling e t a l . ( 7 ) u s i n g t h e γ-ψ approach. F o r w a t e r / e t h a n o l , w a t e r / l - p r o p a n o l and w a t e r / 2 - p r o p a n o l systems, the PRSV-1 e q u a t i o n gave poor r e s u l t s . T a b l e VI shows the r e s u l t s o b t a i n e d w i t h the PRSV-HV model of e q u a t i o n s (15) t o ( 1 8 ) , f o r the systems w a t e r / e t h a n o l and water/ p r o p a n o l s . A d r a m a t i c improvement i n the r e s u l t s i s o b t a i n e d w i t h the two-binary-parameter PRSV-HV approach i n comparison w i t h the p r e v i o u s t r e a t m e n t . For these h i g h l y non i d e a l systems the PRSV e q u a t i o n of s t a t e i s v e r y s e n s i t i v e t o the m i x i n g r u l e a p p l i e d . V a l u e s of the b i n a r y parameters d ^ reported i n Table V I . r

a n

a

r

e

a

s o

Conclusions The PRSV e q u a t i o n i s a v a l u a b l e t o o l f o r computation of v a p o r l i q u i d e q u i l i b r i a f o r systems t h a t p r e v i o u s l y could o n l y be t r e a t e d by the γ-ψ approach. I n t h i s p r e l i m i n a r y p r e s e n t a t i o n o f the PRSV e q u a t i o n we have attempted t o g i v e a g e n e r a l overview of i t s p o t e n t i a l i t i e s . More d e t a i l e d s t u d i e s w i l l be presented i n forthcoming p u b l i c a t i o n s .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

568

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Table V.

Vapor-Liquid E q u i l i b r i a for the System Water/Methanol ΔΡ, mm Hg

PRS\J-l

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

Τ

Ν

298.15 298.15 308.15 312.91 313.05 322.91 323.15 333.15 333.15 338.15 373.15

k »10

2

ΔΡ

1 2

10 10 14 21 10 12 13 7 12 12 16

8.866 9.450 8.555 9.172 8.240 9.033 8.500 7.680 8.385 8.225 8.360

0.86 1.56 1.62 3.49 1.27 6.12 3.12 2.16 6.45 5.19 15.59

ΔΡ%

AylO

1.39 3.56 1.95 2.85 0.75 2.44 1.59 0.49 2.37 1.29 0.91

7.6 13.9 14.4 42.5 7.0 18.8 11.4

3

-

18.2 9.5 10.4

VLE data were taken from Gmehling et a l . (1980)

Table VI. Vapor-Liquid E q u i l i b r i a for the Systems Water/Ethanol, Water/2-Propanol with the PRSV-HV Model+

T,K

Δ

8 ι 2

Δ

«21

ΔΡ%

A y 10

3

Τ,Κ

Water/Ethiinol 283.15 288.15 293.15 298.15 303.15 303.15 313.15 323.15 323.15 323.15 323.15 323.15 328.15 333.15 343.15 343.15 363.15 423.15 523.15 548.15 573.15 598.15

Σ

15.389 17.069 12.873 12.242 13.956 11.757 12.463 7.649 9.465 9.339 10.002 9.890 9.876 9.242 10.290 9.394 10.818 6.330 7.367 26.043 44.183 54.582

37.330 38.054 45.070 49.783 49.483 52.320 54.900 64.892 65.629 64.331 64.692 64.534 67.724 66.508 73.020 72.342 76.827 94.911 98.509 73.547 68.324 68.014

1.30 0.78 0.96 1.02 1.07 0.72 2.43 1.33 0.50 0.64 0.68 0.49 0.57 0.60 0.33 0.40 0.15 1.31 0.81 1.99 1.51 0.59 0^2

Δ

Δ8

8 ι 2

Water/1-Propanol, and

ΔΡ%

A y 10

0.63 1.42 0.89 0.58

12.19 12.02 10.97 8.38 10.89

2 1

3

Water/1-Propanol

6.78 8.18

333.15 333.15 333.15 363.15

30.025 26.341 30.479 27.979

113.080 111.025 111.773 122.111

ÏÏ^SS*

ι

Water/2-Propanol

12.39

8.11

14.37 8.14

9.13

5.88

308.15 318.15 318.20 328.18 333.15 338.15 348.14 423.15 473.15 523.15 548.15

Σ

22.282 25.538 23.154 23.502 24.862 24.109 24.138 22.444 24.068 21.999 22.386

85.950 93.873 91.787 97.766 101.854 103.587 109.184 144.766 152.649 159.728 165.561

0.43 0.87 0.46 0.44 0.87 0.50 0.52 0.94 0.64 0.25 0.41 0.58

4.18 22.23 4.45 5.19 27.06 4.64 4.09 15.50 6.10 7.92 16.96 10.76

6.22 6.28 8.55

+For Water/Ethanol, sets 1-17 from Gmehling et a l . (1980) and 18-22 from Barr-David and Dodge (1959), for Water/l-Propanol from Gmehling et a l . (1980), and for Water/2-Propanol, sets 1-7 from Gmehling et a l . (1980) and 8-11 from Barr-David and Dodge (1959).

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

STRYJEK AND VERA

Improved Cubic Equation of State

569

I n g e n e r a l , t h e d r a m a t i c improvement i n t h e r e p r o d u c t i o n of the pure compound vapor p r e s s u r e s a l l o w s t o o b t a i n l e s s tempera t u r e dependent b i n a r y parameters. A one-parameter m i x i n g r u l e s u f f i c e s t o r e p r e s e n t systems w i t h symmetric excess Gibbs energy curves and a two-parameter m i x i n g r u l e i s r e q u i r e d when a system i s h i g h l y asymmetric.

Acknowledgements The a u t h o r s are g r a t e f u l t o NSERC and M c G i l l U n i v e r s i t y f o r f i n a n c i a l s u p p o r t , and one of us (R.S.) t o the P o l i s h Academy of S c i e n c e s , f o r a l e a v e of absence.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

Legend of Symbols a,b

e q u a t i o n of s t a t e parameters

I^jbj^

f o r d e f i n i t i o n see e q u a t i o n s (8) and ( 9 ) ,

A,Β

d i m e n s i o n l e s s terms: A = P a / ( R T ) ; Β = Pb/RT

c

n u m e r i c a l c o n s t a n t , 2/2/λη [ ( 2 + / 2 ) / ( 2 - / 2 ) ]

respectively

2

g

excess Gibbs energy a t i n f i n i t e p r e s s u r e 00

Ag.. Gj,j k η Ν Ρ R Τ T ν χ y ζ α

b i n a r y parameter temperature dependent b i n a r y parameter b i n a r y parameter number of moles number of d a t a p o i n t s pressure gas c o n s t a n t a b s o l u t e temperature reduced temperature molar volume mole f r a c t i o n v a p o r , phase mole f r a c t i o n compressibility factor f u n c t i o n of reduced temperature and a c e n t r i c value

ΔΡ

average d e v i a t i o n i n p r e s s u r e

Ay κ K <^ ψ ω

average d e v i a t i o n i n vapor phase c o m p o s i t i o n see e q u a t i o n (6) see e q u a t i o n ( 5 ) pure compound parameter. See Table I . fugacity coefficient acentric factor

R

Q

factor

Subscripts c 00

c r i t i c a l property compounds value at i n f i n i t e pressure

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch027

570

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Literature Cited 1. Mathias, P.M., Ind. Eng. Chem., Proc. Des. Develop., 22, 385-391 (1983). 2. Mathias, P.M. and Copeman, T.W., Fluid Phase Equil., 13, 91-108 (1983). 3. Soave, G., Chem. Eng. Sci., 39, 357-369 (1984). 4. Gibbons, R.M. and Laughton, A.P., J. Chem. Soc., Faraday Trans. 2, 80, 1019-1038 (1984). 5. Huron, M.J. and Vidal, J., Fluid Phase Equil., 3, 255-271 (1979). 6. Peng, D.Y. and Robinson, D.B., Ind. Eng. Chem. Fundam., 15, 59-64 (1976). 7. Gmehling, J., Onken, V. and Arlt, W., "Vapor Liquid Equilibrium Data Collection", Chemistry Data Series, Dechema, Frankfurt/Main 1980. 8. Renon, H. and Prausnitz, J.M., AIChE J., 14, 135-144 (1968). 9. Barr-David, F. and Dodge, B.F., J. Chem. Eng. Data, 4, 107 (1959). RECEIVED November 5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

28 New Mixing Rule for Cubic Equations of State for Highly Polar, Asymmetric Systems A. Z. Panagiotopoulos and R. C. Reid Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge,

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

M A 02139

A new two-parameter mixing rule for van der Waals-type cubic equations of state was developed by making the normally used single binary interaction parameter k a linear function of composition. A significant improvement was observed in the representation of binary and ternary phase equilibrium data for highly polar and asymmetric systems. Results are presented for systems with water and supercritical fluids at high pressures, as well as for low-pressure non-ideal systems. Ternary phase equilibrium data at high pressures, including LLG three-phase equilibria, were successfully correlated using parameters regressed from binary data only. ij

C u b i c e q u a t i o n s o f s t a t e (EOS) have become i m p o r t a n t t o o l s i n t h e a r e a o f p h a s e e q u i l i b r i u m m o d e l l i n g , e s p e c i a l l y f o r systems a t p r e s s u r e s c l o s e t o o r above t h e c r i t i c a l p r e s s u r e o f one o r more o f the system components. Among t h e more common o f t h e c u r r e n t l y u s e d c u b i c EOS, a r e t h e Soave m o d i f i c a t i o n o f t h e Redlich-Kwong (1) and the Peng-Robinson EOS ( 2 ) . The f u n c t i o n a l form o f b o t h e q u a t i o n s , as w e l l as s e v e r a l o t h e r p r o p o s e d c u b i c s , c a n be r e p r e s e n t e d i n a g e n e r a l manner as shown i n E q u a t i o n 1 (3.) : RT

a m 2 V + uVb

V -b m

(

1

)

2 + wb m m

where u a n d w a r e n u m e r i c a l c o n s t a n t s . T a b l e I l i s t s t h e v a l u e s o f u and w f o r some common EOS. F o r a m i x t u r e , parameters a and b a r e r e l a t e d t o t h e pure component parameters and t h e m i x t u r e c o m p o s i t i o n t h r o u g h a m i x i n g r u l e . E q u a t i o n s 2 and 3 show one common c h o i c e f o r t h e m i x i n g r u l e , the v a n d e r Waals 1 - f l u i d m i x i n g r u l e : m

a

= m b

m

Υ Υ r . ι J -

m

χ. χ. a. . î j i i J

(2)

J

X x. b . h ι ι ι

0097-6156/86/0300-0571 $06.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

(3)

572

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

T a b l e I . Parameters

f o r a few c u b i c E q u a t i o n s o f S t a t e

Equation o f State

u

w

van d e r W a l l s (1873)

0

0

Redlich-Kwong (1949)

1

0

Soave (1972)

1

0

Peng-Robinson (1976)

2

-1

The c r o s s - p a r a m e t e r s a ^ a r e r e l a t e d i n t u r n t o t h e pure-compo nent parameters by a "combining r u l e " . E q u a t i o n 4 shows a common form o f t h e combining r u l e f o r a : £ j

7 a. a.

a.. =

(1-k. .)

(4)

I n E q u a t i o n 4, k i s c a l l e d a b i n a r y i n t e r a c t i o n parameter, and was o r i g i n a l l y i n t r o d u c e d so t h a t t h e e q u a t i o n c a n b e t t e r reproduce e x p e r i m e n t a l c o m p o s i t i o n d a t a i n systems t h a t c o n t a i n components o t h e r than t h e l i g h t hydrocarbons. There has been c r i t i c i s m d i r e c t e d toward t h e o v e r s i m p l i c i t y o f the c u b i c e q u a t i o n form, and r i g h t l y so. N e v e r t h e l e s s t h i s r e p r e s e n t a t i o n does d e s c r i b e , a t l e a s t q u a l i t a t i v e l y , a l l the important c h a r a c t e r i s t i c s o f v a p o r - l i q u i d equilibrium behavior. Alternative e q u a t i o n s o f s t a t e have (and a r e ) b e i n g suggested, b u t , t o d a t e , none have been w i d e l y used and t e s t e d . A l s o , t o o o f t e n , a l t e r n a t e EOS a r e s i g n i f i c a n t l y more complex and b r i n g w i t h them a d d i t i o n a l pure-compo nent and m i x t u r e parameters which must be e v a l u a t e d by r e g r e s s i n g experimental data. We f e e l t h a t t h e k e y t o s u c c e s s i n employing t h e c u b i c e q u a t i o n form t o model phase e q u i l i b r i a i s i n t h e c h o i c e o f t h e m i x i n g and combining r u l e s . Our g o a l , then, was t o s e l e c t t h e s i m p l e s t forms, w i t h the fewest i n t e r a c t i o n parameters t h a t c o u l d be used t o c o r r e l a t e complex phase b e h a v i o r i n b o t h n o n - p o l a r and h i g h l y p o l a r systems. t j

Proposed

Method

We p r o p o s e an e m p i r i c a l m o d i f i c a t i o n o f t h e combining r u l e shown i n E q u a t i o n 4. I n o u r approach, we r e l a x t h e assumption k - k. thus i n t r o d u c i n g a second i n t e r a c t i o n parameter p e r b i n a r y : t j

a.. IJ

Ja.

a. [1-k..+(k..-k..)x.] ι

J

1 J 1 J

j i i

J

±

(5) v

'

E q u a t i o n 5 has t h e f o l l o w i n g c h a r a c t e r i s t i c s : I f k.. - k. recovered.

±

, t h e o r i g i n a l m i x i n g r u l e g i v e n by E q u a t i o n 4 i s

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

28. PANAGIOTOPOULOS A N D REID

Highly Polar, Asymmetric Systems

573

The " e f f e c t i v e " i n t e r a c t i o n parameter between components i and j a p p r o a c h e s k ^ a s x , t h e m o l e f r a c t i o n o f component i , approaches zero. I t a l s o approaches i fx approaches u n i t y . The apparent asymmetry under an i n t e r c h a n g e o f i and j i s c o r r e c t e d by t h e f a c t t h a t b o t h a ^ and a ^ e n t e r i n t h e c a l c u l a t i o n o f t h e m i x t u r e parameter a s y m m e t r i c a l l y . £

L

i

A

m

A p p l i c a t i o n o f t h e r u l e g i v e n by E q u a t i o n 2 f o r t h e c a l c u l a t i o n o f t h e m i x t u r e parameter a r e s u l t s i n a c u b i c e x p r e s s i o n f o r the mole f r a c t i o n dependence o f t h e m i x t u r e parameter a . T h i s i s d i f f e r e n t from t h e case o f the conventional mixing rule (Equation 4 ) , which r e s u l t s i n a q u a d r a t i c e x p r e s s i o n f o r a . m

m

m

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

U s i n g t h i s m i x i n g r u l e w i t h E q u a t i o n 1 , we f u g a c i t y c o e f f i c i e n t o f a component i n a m i x t u r e a s :

ο A *k

i n

o -

I

N

f

k ~X7P~

b

=

k —

, FV - . RT " > (

0

1

"

i

n

can o b t a i n the

P(V-b ) RT

^

m

+

2 Yx ( a +a

^ ?

i

ik

) - ΥΥχ χ j (ki „ k i '

1

J

i j

-k j J l ) 7 a1 aJ + χk Yx ^ i (k k i -ki k) "y ka a i J i

i

J

k^

ι

k i

i k

/

v

k

i

bk k

b~ 2

in Ju

z

- 4w b R T

2V + b (u-7u -4w) = 2V + b ( u + y u - 4 w )

(6)

2

m

An a l g o r i t h m f o r t h e d e t e r m i n a t i o n o f t h e phase e q u i l i b r i u m p r o p e r t i e s u s i n g t h e e q u a t i o n s d e s c r i b e d above was developed. The main c h a r a c t e r i s t i c s o f t h e method used a r e : P u r e component p a r a m e t e r e v a l u a t i o n : A r e c e n t l y p r o p o s e d technique ( 7 ) was used f o r t h e s u b c r i t i c a l components o f t h e m i x t u r e s under c o n s i d e r a t i o n . T h i s t e c h n i q u e , s i m i l a r t o t h e J o f f e a n d Z u d k e v i t c h m e t h o d (8.) , p r o v i d e s p u r e component p a r a m e t e r s t h a t e x a c t l y r e p r o d u c e vapor p r e s s u r e and l i q u i d volume o f a compound a t a temperature o f i n t e r e s t . The r e a s o n t h i s approach was p r e f e r r e d over t h e c o n v e n t i o n a l a c e n t r i c f a c t o r c o r r e l a t i o n i s t h a t t h e l a t t e r does n o t work w e l l f o r h i g h l y p o l a r o r a s s o c i a t e d c o m p o n e n t s . F o r s u p e r c r i t i c a l components though, t h e normal a c e n t r i c f a c t o r c o r r e l a t i o n was used. Multicomponent e q u i l i b r i a were c a l c u l a t e d as f o l l o w s : We s t a r t by assuming t h a t t h e component and i n t e r a c t i o n parameters a r e known a t a g i v e n temperature, and p o s t u l a t e t h e e x i s t e n c e o f a g i v e n number o f phases (2 o r 3 f o r t h e examples g i v e n b e l o w ) . Then, Newton's method i s a p p l i e d f o r t h e s o l u t i o n o f t h e system o f n o n - l i n e a r e q u a t i o n s g i v e n by E q u a t i o n 7. -

f f ( » f[

where t h e s u p e r s c r i p t s p e c i f i e s component number.

) ,

i « 1,2 ... Ν

(7)

t h e phase and i r e f e r s t o t h e

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

574

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

F o r t h e c a l c u l a t i o n s p r e s e n t e d i n t h i s paper, we e l e c t e d t o use the Peng - R o b i n s o n EOS. The v a l u e s o f t h e two b i n a r y i n t e r a c t i o n parameters were r e g r e s s e d from e x p e r i m e n t a l b i n a r y phase e q u i l i b r i u m d a t a on a g i v e n i s o t h e r m . I t was found t h a t a l t h o u g h t h e temperature v a r i a t i o n o f t h e parameters i s weak, i t must be t a k e n i n t o account i f a q u a n t i t a t i v e agreement w i t h e x p e r i m e n t a l d a t a over a wide temper a t u r e range i s d e s i r e d .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

Results P o l a r - S u p e r c r i t i c a l F l u i d Systems. As s u g g e s t e d e a r l i e r , t h e p r i n c i p a l m o t i v a t i o n b e h i n d t h e development o f t h e new m i x i n g r u l e , h a s b e e n t h e r e p r e s e n t a t i o n o f phase e q u i l i b r i u m i n systems t h a t c o n t a i n a s u p e r c r i t i c a l component and one o r more h i g h l y p o l a r compounds, such as water. The inadequacy o f t h e c o n v e n t i o n a l method f o r such systems i s demonstrated i n F i g u r e 1, t h a t shows t h e p r e d i c t e d c o m p o s i t i o n o f t h e two phases f o r t h e system c a r b o n d i o x i d e - water a t 323 K, f o r b o t h the one-parameter and t h e two-parameter m i x i n g r u l e s . Parameter k was f i t t e d t o t h e s u p e r c r i t i c a l f l u i d phase c o n c e n t r a t i o n o f water (Y2) f o r b o t h models. Parameter k f o r t h e two parameter model was f i t t e d t o t h e l i q u i d phase c o m p o s i t i o n o f C 0 ( X I ) . I t i s c l e a r t h a t when t h e a d j u s t a b l e parameter i n t h e s i n g l e - p a r a m e t e r c o r r e l a t i o n i s f i t t e d t o t h e c o m p o s i t i o n o f one phase, t h e r e s u l t s f o r t h e o t h e r p h a s e a r e v e r y p o o r . I n c o n t r a s t , t h e two-parameter c o r r e l a t i o n p r e d i c t s t h e c o m p o s i t i o n o f b o t h phases e s s e n t i a l l y w i t h i n e x p e r i m e n t a l a c c u r a c y t h r o u g h o u t a p r e s s u r e range from a t m o s p h e r i c t o 1000 b a r . An i m p o r t a n t f e a t u r e o f t h e new method i s t h a t t h e two p a r a meters a r e e s s e n t i a l l y u n c o r r e l a t e d i n many c a s e s , as shown i n t h e p r e v i o u s example, i n w h i c h t h e parameters were d e t e r m i n e d from d a t a i n d i f f e r e n t phases. T h i s i s g e n e r a l l y t r u e o n l y f o r systems i n w h i c h t h e c o m p o s i t i o n s o f t h e c o e x i s t i n g phases a r e v e r y d i f f e r e n t . I n t h i s r e s p e c t , t h e proposed method i s s i m i l a r t o a p r e v i o u s l y s u g g e s t e d t e c h n i q u e f o r h i g h l y asymmetric systems (9) i n w h i c h a d i f f e r e n t i n t e r a c t i o n parameter i s used f o r each phase. The advantage o f t h e p r o p o s e d method i s b e s t seen i n systems t h a t approach a c r i t i c a l p o i n t ( f o r example a l i q u i d - l i q u i d phase ' s p l i t ) , and t h e r e f o r e cannot be a d e q u a t e l y m o d e l l e d i f a d i f f e r e n t e q u a t i o n i s used f o r d i f f e r e n t phases. I n F i g u r e 2, a comparison i s made between model p r e d i c t i o n s and the e x p e r i m e n t a l l y o b s e r v e d b e h a v i o r f o r t h e C 0 - water system over a temperature range from 298 Κ t o 348 Κ (pure C0 i s s u b c r i t i c a l below 304.2 K) . V e r y good agreement i s o b t a i n e d f o r a l l p r e s s u r e s f o r t h e c o m p o s i t i o n o f b o t h phases. An i n t e r e s t i n g q u e s t i o n a r i s e s i f we c o n s i d e r t h e l o w e r p r e s s u r e range examined. A t those c o n d i t i o n s , t h e d e n s i t y o f t h e gas phase i s low and d e v i a t i o n s from i d e a l i t y a r e s m a l l . One o f t h e reasons f o r the c o m p l e x i t y o f p r e v i o u s l y proposed m o d i f i c a t i o n s o f t h e m i x i n g r u l e s f o r c u b i c EOS (4,5,6) i s t h e r e q u i r e m e n t t h a t t h e m i x i n g r u l e s h o u l d reduce t o a q u a d r a t i c form f o r a m i x t u r e second v i r i a l c o e f f i c i e n t B a t l o w d e n s i t i e s , whereas t h e o b s e r v e d b e h a v i o r a t t h e h i g h d e n s i t y l i m i t c a n o n l y be m o d e l l e d w i t h a h i g h e r o r d e r m i x i n g r u l e . F o r a c u b i c EOS o f t h e type shown i n E q u a t i o n 1, t h e e x p r e s s i o n 1 2

2 1

2

2

2

m

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

28.

PANAGIOTOPOULOS AND REID

-1.0

Η

•

Wîebe and Gaddy

+

Coan and King

— -1.5

Η

-2.0

Η

Highly Polar, Asymmetric Systems

PR

575

(1941)

(1971)

EOS

CM >CP

ο

k21 tr-

"2.5

k12 -

a -0.198

k21 -

+0.160

h0.160

Η 200

400

600

Pressure (bar) F i g u r e 1. E x p e r i m e n t a l and p r e d i c t e d phase e q u i l i b r i u m b e h a v i o r f o r t h e system c a r b o n d i o x i d e - water a t 323 Κ. X I i s t h e mole f r a c t i o n o f C0 i n t h e l i q u i d phase and Y2 t h e mole f r a c t i o n o f water i n t h e f l u i d phase. 2

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

576

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

f o r t h e m i x t u r e second v i r i a l c o e f f i c i e n t i s a Β m

b m

m

(9)

RT

The proposed m i x i n g r u l e does n o t obey t h e r e q u i r e m e n t f o r a q u a d r a t i c dependence o f B , s i n c e t h e dependence o f a on t h e mole f r a c t i o n s o f t h e components o f a m i x t u r e i s c u b i c . Because o f t h i s , i t was s u r p r i s i n g a t f i r s t t h a t such a good agreement was found a t the l o w p r e s s u r e range between model p r e d i c t i o n s and e x p e r i m e n t a l results. To compare t h e performance o f t h e p r o p o s e d m i x i n g r u l e f o r the m i x t u r e v o l u m e t r i c p r o p e r t i e s , we c a l c u l a t e d t h e m i x t u r e second v i r i a l c o e f f i c i e n t a t 323 Κ u s i n g E q u a t i o n 7 w i t h t h e v a l u e s o f k and k c a l c u l a t e d e a r l i e r , as w e l l as u s i n g a common i n t e r a c t i o n parameter e q u a l t o t h e i r a r i t h m e t i c average. I n t h e l a t t e r c a s e , t h e m i x i n g r u l e r e s u l t s i n a q u a d r a t i c f u n c t i o n a l i t y f o r B . As c a n be seen from F i g u r e 3, t h e d i f f e r e n c e i n t h e c a l c u l a t e d dependence o f B on t h e m i x t u r e c o m p o s i t i o n between t h e two cases i s i n d e e d s m a l l . A d d i t i o n a l examples o f t h e model performance f o r p o l a r - super c r i t i c a l f l u i d systems a r e shown i n F i g u r e s 4 and 5 f o r t h e c a r b o n d i o x i d e - e t h a n o l and t h e c a r b o n d i o x i d e - acetone systems. A g a i n , agreement between model p r e d i c t i o n s and experiment i s e x c e l l e n t , even quite c l o s e t o mixture c r i t i c a l p o i n t s . I t i s i n t e r e s t i n g t o note, t h a t f o r t h e system c a r b o n d i o x i d e - a c e t o n e , t h e o p t i m a l v a l u e s o f the two parameters a r e q u i t e c l o s e t o each o t h e r . U s i n g t h e conven t i o n a l m i x i n g r u l e f o r t h i s p a r t i c u l a r system would have r e s u l t e d i n a l m o s t as good agreement between experiment and p r e d i c t i o n as f o r the two - parameter c o r r e l a t i o n . I t i s e n c o u r a g i n g t o n o t e t h a t t h e proposed c o r r e l a t i o n n a t u r a l l y leads i t s e l f t o the e l i m i n a t i o n o f the second parameter i f t h e systems b e i n g m o d e l l e d a r e r e l a t i v e l y s i m p l e . m

m

1 2

2 1

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

m

m

Low P r e s s u r e VLE. Up t o t h i s p o i n t , we have f o c u s e d o u r d i s c u s s i o n on s y s t e m s a t r e l a t i v e l y h i g h p r e s s u r e s . F o r such systems, t h e e q u a t i o n o f s t a t e approach has a l r e a d y been t e s t e d e x t e n s i v e l y . I f we a r e i n t e r e s t e d i n m o d e l l i n g t e r n a r y phase b e h a v i o r , we a l s o need i n t e r a c t i o n parameters between t h e l e s s v o l a t i l e components o f a m i x t u r e t h a t do n o t form two-phase systems a t h i g h p r e s s u r e s . The e a s i e s t way t o o b t a i n such parameters i s t o u t i l i z e VLE d a t a a t low pressures that are widely available. As a t e s t o f t h e a p p l i c a b i l i t y o f t h e method f o r l o w p r e s s u r e s y s t e m s , we c o r r e l a t e d i s o t h e r m a l VLE d a t a f o r t h e b i n a r y system e t h a n o l - water a t a range o f temperatures ( 1 8 ) . The r e s u l t s a r e p l o t t e d i n F i g u r e 6. Agreement i s a g a i n v e r y good, even c l o s e t o t h e a z e o t r o p i c r e g i o n , and t h e a c c u r a c y o f t h e p r e d i c t i o n s i s c l e a r l y comparable t o t h e a c c u r a c y o f excess Gibbs Free Energy models w i t h the same number o f a d j u s t a b l e parameters. T e r n a r y Systems One o f t h e most s t r i n g e n t t e s t s f o r a proposed c o r r e l a t i o n i s i t s a b i l i t y t o p r e d i c t t e r n a r y b e h a v i o r when o n l y t h e b i n a r y b e h a v i o r i s known. I n F i g u r e 7 t h e model p r e d i c t i o n s f o r t h e t e r n a r y system c a r b o n d i o x i d e - e t h a n o l - water a r e p r e s e n t e d . The m o d e l p r e d i c t i o n s a r e based s o l e l y on v a l u e s o f t h e i n t e r a c t i o n parameters r e g r e s s e d from b i n a r y d a t a a t t h e same temperature ( t h e v a l u e s o f t h e pure component and m i x t u r e parameters u s e d a r e a l s o shown on t h e same f i g u r e ) .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

28.

Highly Polar, Asymmetric Systems

PANAGIOTOPOULOS AND REID

577

Mole fraction C02 F i g u r e 2. E x p e r i m e n t a l and p r e d i c t e d phase e q u i l i b r i u m b e h a v i o r f o r t h e system c a r b o n d i o x i d e - water a t s e v e r a l t e m p e r a t u r e s . E x p e r i m e n t a l d a t a a r e from r e f e r e n c e s 12-15.

θ η ? ο

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-40

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,

, 0.2

,

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1 0.6

1

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1

1

Mole Fraction C02 F i g u r e 3. P r e d i c t e d second v i r i a l c o e f f i c i e n t v e r s u s c o m p o s i t i o n f o r t h e c a r b o n d i o x i d e - water system a t 323 K.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

578

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

120

0

0.2

0.4

0.6

0.8

Mole fraction C 0 2 F i g u r e 4. Experimental (10) and p r e d i c t e d phase b e h a v i o r f o r t h e system c a r b o n d i o x i d e - e t h a n o l .

0

0.2

0.4

0.6

equilibrium

0.8

Mole Fraction Carbon Dioxide F i g u r e 5. Experimental (16) and p r e d i c t e d phase e q u i l i b r i u m b e h a v i o r f o r t h e system c a r b o n d i o x i d e - a c e t o n e a t 313 K.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

1

PANAGIOTOPOULOS AND REID

Highly Polar, Asymmetric Systems

579

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

28.

Figure 6. Experimental (18) and p r e d i c t e d b e h a v i o r f o r the system e t h a n o l - water.

phase

equilibrium

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

580

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

As c a n b e s e e n f r o m F i g u r e 7, t h e a c c u r a c y o f t h e m o d e l p r e d i c t i o n s a t t h e low e t h a n o l c o n c e n t r a t i o n range i s v e r y good. The c a l c u l a t e d t i e - l i n e s d e v i a t e from t h e e x p e r i m e n t a l d a t a as t h e p l a i t p o i n t i s approached, b u t s t i l l t h e c o r r e c t q u a l i t a t i v e b e h a v i o r i s p r e d i c t e d . One more i l l u s t r a t i v e example o f t h e c a p a b i l i t i e s o f the model i s g i v e n i n F i g u r e 8 f o r t h e t e r n a r y system carbon d i o x i d e - acetone - w a t e r a t 313 K. The most prominent c h a r a c t e r i s t i c o f t h e model p r e d i c t i o n s f o r t h e system i s an e x t e n s i v e three-phase r e g i o n . A t t h i s temperature, t h e s o l u b i l i t y o f carbon d i o x i d e i n t h e acetone phase i s v e r y h i g h even a t moderate p r e s s u r e s . T h i s c a n be seen a l r e a d y i n F i g u r e 8a t h a t corresponds t o a p r e s s u r e o f 20 b a r . As p r e s s u r e i s i n c r e a s e d t o approx. 22 b a r ( F i g u r e 8b), t h e l i q u i d phase undergoes a phase s p l i t , i n t o a lower l i q u i d phase r i c h i n w a t e r and a m i d d l e l i q u i d phase r i c h i n acetone, w h i c h c o e x i s t w i t h a f l u i d phase r i c h i n C 0 . The t h r e e phase r e g i o n i n c r e a s e s i n s i z e as p r e s s u r e i s i n c r e a s e d ( F i g u r e s 8c, 8d) and t h e n g r a d u a l l y s h r i n k s a g a i n ( F i g u r e 8e) . A t approx. 82 b a r (not shown on F i g u r e 8 ) , t h e system passes through a c r i t i c a l s t a t e a g a i n , and t h e m i d d l e l i q u i d phase becomes i d e n t i c a l t o t h e s u p e r c r i t i c a l phase. A t even h i g h e r p r e s s u r e s , o n l y two phases c o e x i s t , and t h e e f f e c t o f p r e s s u r e on t h e c o m p o s i t i o n o f t h e phases i s much s m a l l e r ( F i g u r e 8 f ) . T h i s c o m p l e x b e h a v i o r was o b s e r v e d e a r l i e r f o r t h e system e t h y l e n e - acetone - water by E l g i n and W e i n s t o c k ( 1 1 ) . T h e i r " Type 2 " q u a l i t a t i v e phase diagrams b e a r a s t r i k i n g resemblance t o our m o d e l p r e d i c t i o n s . The m o d e l p r e d i c t i o n s a r e f u l l y s u p p o r t e d by e x p e r i m e n t a l e v i d e n c e from o u r l a b o r a t o r y as i n d i c a t e d on F i g u r e s 8c, 8d and 8e, i n which t h e measured c o m p o s i t i o n s o f t h e t h r e e phases a r e shown i n a d d i t i o n t o t h e model r e s u l t s . The c a l c u l a t e d phase c o m p o s i t i o n s a r e n o t i n e x a c t agreement w i t h t h e e x p e r i m e n t a l d a t a , b u t t h e c o r r e c t p r e d i c t i o n o f t h e appearance and d i s a p p e a r a n c e o f t h e t h i r d phase s t r o n g l y i m p l i e s t h a t t h e model c a p t u r e s t h e s u b s t a n t i a l f e a t u r e s o f t h e p h y s i c a l r e a l i t y . A more complete p r e s e n t a t i o n o f t h e p e r t i n e n t e x p e r i m e n t a l r e s u l t s and comparison w i t h model p r e d i c t i o n s i s g i v e n elsewhere ( 1 0 ) .

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

2

Conclusions A new t w o - p a r a m e t e r m i x i n g r u l e i s p r o p o s e d f o r use i n c u b i c E q u a t i o n s o f S t a t e and i s shown t o be e s p e c i a l l y u s e f u l i n c o r r e l a t i n g t h e phase e q u i l i b r i u m b e h a v i o r i n h i g h l y p o l a r systems t h a t c a n n o t b e c o r r e c t l y r e p r e s e n t e d by a c o n v e n t i o n a l one-parameter m i x i n g r u l e . The i n t r o d u c t i o n o f E q u a t i o n 5, i s a t t h i s p o i n t a p u r e l y e m p i r i c a l c o r r e c t i o n t o t h e o r i g i n a l form o f t h e m i x i n g r u l e . The m o d i f i c a t i o n i s r e l a t e d , b u t i s n o t d i r e c t l y d e r i v e d from, the i d e a o f l o c a l c o m p o s i t i o n s , t h a t has been shown i n t h e p a s t t o r e s u l t i n improved r e p r e s e n t a t i o n o f t h e phase e q u i l i b r i u m b e h a v i o r i n h i g h l y p o l a r and asymmetric m i x t u r e s . Among t h e a d v a n t a g e s o f t h e p r o p o s e d m i x i n g r u l e , a r e t h e relative simplicity of the r e s u l t i n g expressions f o r the derived thermodynamic p r o p e r t i e s , t o g e t h e r w i t h t h e f a c t t h a t t h e model c a n be e a s i l y reduced t o t h e c o n v e n t i o n a l one-parameter m i x i n g r u l e f o r w h i c h a s u b s t a n t i a l amount o f r e g r e s s e d i n t e r a c t i o n parameters e x i s t s . The model i s shown t o a c c u r a t e l y reproduce d a t a f o r b o t h h i g h

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

28.

PANAGIOTOPOULOS A N D REID

Highly Polar, Asymmetric Systems

581

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

Ethanol (2)

Water (3)

C02C1) Τ = 313 Κ . Ρ = 103 ba

F i g u r e 7. T e r n a r y phase e q u i l i b r i u m b e h a v i o r f o r t h e system c a r b o n d i o x i d e - e t h a n o l - water ( e x p e r i m e n t a l d a t a from 12,12.) .

Wator

P - 20.• bar

Ρ - 40. 1 bar

C

0

W

a

t

e

r

2

^

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Ρ - 22. 3 bar

C

m

5 7 m l

b

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r

W

o

t

e

2

C p

0

0 2

*

Q t e r

r

Ρ - 30.0 bar

Ρ - 100 bar

F i g u r e 8. T e r n a r y phase e q u i l i b r i u m b e h a v i o r f o r t h e system c a r b o n d i o x i d e - acetone - water a t 313 Κ ( e x p e r i m e n t a l d a t a (10) f o r t h e t h r e e - p h a s e e q u i l i b r i u m c o m p o s i t i o n s i n c , d and e a r e shown as f i l l e d t r i a n g l e s ) .

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

582

p r e s s u r e p o l a r - s u p e r c r i t i c a l f l u i d , and l o w p r e s s u r e p o l a r - p o l a r b i n a r y phase e q u i l i b r i u m . I n addition, predictions f o r ternary systems based on c o e f f i c i e n t s r e g r e s s e d from b i n a r y d a t a o n l y , a r e q u a l i t a t i v e l y c o r r e c t f o r the systems s t u d i e d . One p o s s i b l e weakness o f t h e p r o p o s e d model, i s t h a t i t does not reproduce t h e c o r r e c t f u n c t i o n a l dependence o f a m i x t u r e second v i r i a l c o e f f i c i e n t on the c o m p o s i t i o n .

Acknowledgments

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

F i n a n c i a l s u p p o r t f o r t h i s r e s e a r c h was p r o v i d e d b y t h e N a t i o n a l S c i e n c e F o u n d a t i o n (CPE-8318494). One o f t h e a u t h o r s (AZP) was a l s o s u p p o r t e d by a f e l l o w s h i p from H a l c o n I n c . We g r a t e f u l l y acknowledge both sponsors.

Literature Cited 1. Soave, G., Chem. Eng. Sci. 1972, 27, 1197-1203. 2. Peng, D.-Y. and Robinson, D.B., Ind. Eng. Chem. Fundam. 1976, 15(1), 59-64. 3. Schmidt G. and Wenzel Η., Chem. Eng. Sci., 1980, 35, 1503-1512. 4. Whiting, W.B. and Prauznitz, J.M., Fluid Phase Equil. 1982, 9, 119-147. 5. Mollerup, J., Fluid Phase Equil. 1981, 7, 121-138. 6. Mathias, P.M. and Copeman, T.W., Fluid Phase Equil. 1983, 13, 91-108. 7. Panagiotopoulos, A.Z. and Kumar, S.K., Fluid Phase Equil. 1985, 22, 77-88. 8. Joffe, J. and Zudkevitch, D., Ind. Eng. Chem. Fundam. 1970, 9(4), 545-548. 9. Robinson, D.B., Peng, D.-Y. and Chung C.Y.-K., " The Development of the Peng - Robinson Equation and its Application to Phase Equilibrium in a System containing Methanol", paper presented at the Annual AIChE meeting in San Fransisco, CA, November 1984. See also Peng, D.-Y. and Robinson, D. Β., ACS Symposium Series, 1980, 133, 393-414. 10. Panagiotopoulos, A.Z. and Reid, R.C., 1985, " High Pressure Phase Equilibria in Ternary Fluid Mixtures with a Supercritical Fluid ", ACS Division of Fuel Chemistry Preprints, vol.30 No 3, 46-56. 11. Elgin, J.C. and Weinstock, J.J., J. Chem. Eng. Data 1959, 4(1), 3-12. 12. Wiebe, R. and Gaddy, V.L., J. Am. Chem. Soc. 1941, 63,475-77 and Wiebe, R., Chem. Rev. 1941, 29, 475-81. 13. Coan, C.R. and King, A.P., J. Am. Chem. Soc. 1971, 93, 1857-62. 14. Matous, J. et al., Coll. Czech. Chem. Commun. 1969, 34, 3982-85. 15. Zawisza, A. et al., J. Chem. Eng. Data 1981, 26, 388-391. 16. Katayama, T., et al., J. Chem. Eng. Jpn. 1975, 8(2) , 89-92. 17. Kuk, M.S. and Montagna, J.C., Chapter 4 in Paulaitis et al. (ed.), " Chemical Engineering at Supercritical Fluid Conditions ", Ann Arbor Science Publishers, 1983. 18. Pemberton, R.C. and Mash, C.J., J. Chem. Thermodynamics 1978, 10, 867-888. RECEIVED November 5, 1985

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix001

Author Index A b b o t t , M i c h a e l M. Adachi, Y. B a r r u f e t , M a r i a A. Blum, L e s s e r B r a i n a r d , A. J . BurCham, Andy F . Calado, J . C . G. Celmins, Ilga Chabak, J o s e p h J . Chang, R . F . Chao, K . C . C h o r n g , H. Twu Chung, F . T . H . Copeman , Thomas W. D e i t e r s , U l r i c h Κ. Donahue, Marc D. E l y , James F . E n i c k , R. M . E r i c k s o n , D a l e D. E s p e r , Gunter J . Eubank, P h i l i p T . G e o r g e t o n , Gus K . G r e e n k o r n , R. A. G r e n k o , J . A. H a l l , C . K. H a l l , K e n n e t h R. Han, S. J . H a s e l o w , J . S. Heidemarm, R o b e r t A. H e n d e r s o n , Douglas H o l d e r , G. D. H o l s t e , James C . Honnell, K.G. Kanchanakpan, S . B . K i m , H. Lee, K . - H .

Lee. L . L . Lee, R . J . L e i a n d , Thomas W. Levelt Sengers, J.M.H. L i , M.H. Lu, B.C.-Y. Majeed, A l i I. M a n l e y , D a v i d B. Mansoori, G. A l i M a t h i a s , Paul M . M o r r i s , R a n d a l l W. Morrison, G. Nass, Kathryn K. P a l e n c h a r , Robert M . P a n a g i o t o p o u l o s , A. Z. P o l i n g , Bruce E . Reid, R.C. R i j k e r s , M a r i nus P.W. Rowlinson, John S. Rubio, R.G. Sandler, S.I. Smith, Richard Lee, J r . So, C . - K . Starling, K.E. S t r e e t t , W.B. S t r y j e k , R. T a n i , Alessandro T e j a , Am yη S . Trampe, Mark D. T u r e k , Edward A. Vera, J . H . V i m a l c h a n d , P. Wagner, Jan W i l s o n , Grant M. Yu, J . - M .

Subject Index A c i d g a s - o i l system phase d i s t r i b u t i o n s f o r San Andres o i l s t u d y , 432f s a t u r a t i o n pressure l o c i , 428f,429f A c t i v i t y c o e f f i c i e n t , d e f i n i t i o n , 21 A c t i v i t y c o e f f i c i e n t model a p p l i c a t i o n s , 352 F l o r y - H u g g i ns mode1, 195 Adhesive i n t e r m o l e c u l a r p o t e n t i a l , l o c a l compositions for m i x t u r e s , 356

A

Accelerated Successive S u b s t i t u t i o n (ACSS) c o n v e r g e n c e b e h a v i o r of a m i x t u r e , 484 COSS c o m p a r i s o n , 484 f l a s h c a l c u l a t i o n s , 484,486f K v a l u e s u p d a t e d , 480-481 s o l u t i o n r e g i o n f o r a m i x t u r e , 486f

583

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

Author Index A b b o t t , M i c h a e l M. Adachi, Y. B a r r u f e t , M a r i a A. Blum, L e s s e r B r a i n a r d , A. J . BurCham, Andy F . Calado, J . C . G. Celmins, Ilga Chabak, J o s e p h J . Chang, R . F . Chao, K . C . C h o r n g , H. Twu Chung, F . T . H . Copeman , Thomas W. D e i t e r s , U l r i c h Κ. Donahue, Marc D. E l y , James F . E n i c k , R. M . E r i c k s o n , D a l e D. E s p e r , Gunter J . Eubank, P h i l i p T . G e o r g e t o n , Gus K . G r e e n k o r n , R. A. G r e n k o , J . A. H a l l , C . K. H a l l , K e n n e t h R. Han, S. J . H a s e l o w , J . S. Heidemarm, R o b e r t A. H e n d e r s o n , Douglas H o l d e r , G. D. H o l s t e , James C . Honnell, K.G. Kanchanakpan, S . B . K i m , H. Lee, K . - H .

Lee. L . L . Lee, R . J . L e i a n d , Thomas W. Levelt Sengers, J.M.H. L i , M.H. Lu, B.C.-Y. Majeed, A l i I. M a n l e y , D a v i d B. Mansoori, G. A l i M a t h i a s , Paul M . M o r r i s , R a n d a l l W. Morrison, G. Nass, Kathryn K. P a l e n c h a r , Robert M . P a n a g i o t o p o u l o s , A. Z. P o l i n g , Bruce E . Reid, R.C. R i j k e r s , M a r i nus P.W. Rowlinson, John S. Rubio, R.G. Sandler, S.I. Smith, Richard Lee, J r . So, C . - K . Starling, K.E. S t r e e t t , W.B. S t r y j e k , R. T a n i , Alessandro T e j a , Am yη S . Trampe, Mark D. T u r e k , Edward A. Vera, J . H . V i m a l c h a n d , P. Wagner, Jan W i l s o n , Grant M. Yu, J . - M .

Subject Index A c i d g a s - o i l system phase d i s t r i b u t i o n s f o r San Andres o i l s t u d y , 432f s a t u r a t i o n pressure l o c i , 428f,429f A c t i v i t y c o e f f i c i e n t , d e f i n i t i o n , 21 A c t i v i t y c o e f f i c i e n t model a p p l i c a t i o n s , 352 F l o r y - H u g g i ns mode1, 195 Adhesive i n t e r m o l e c u l a r p o t e n t i a l , l o c a l compositions for m i x t u r e s , 356

A

Accelerated Successive S u b s t i t u t i o n (ACSS) c o n v e r g e n c e b e h a v i o r of a m i x t u r e , 484 COSS c o m p a r i s o n , 484 f l a s h c a l c u l a t i o n s , 484,486f K v a l u e s u p d a t e d , 480-481 s o l u t i o n r e g i o n f o r a m i x t u r e , 486f

583

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

584

EQUATIONS OF STATE: THEORIES AND

A l k a n e m o d e l , Monte C a r l o s i m u l a t i o n o f l o c a l f l u i d s t r u c t u r e , 202-203 Alkanes properties cubic equations performance s t u d i e d , 539-542 percent d e v i a t i o n s i n c u b i c e q u a t i o n s s t u d y , 541t predictive a b i l i t y of cubic e q u a t i o n s compared, 541-542 t r e a t m e n t i n Monte C a r l o s i m u l a t i o n o f f l u i d s , 204 A r g o n , m o l e c u l a r volume, C a r n a h a n - S t a r l i n g and R a c k e t t e q u a t i o n s compared, 53Of Average p o t e n t i a l model (APM), m i x t u r e radial distribution f u n c t i o n s , 322 - 323 A z e o t r o p e s , p r e d i c t i o n by t h e p e r t ur be d - a n i s ot r o p i c - c h a i n t h e o r y , 305

Β Bead-bead i n t e r m o l e c u l a r r a d i a l distribution function a l k a n e c o m p a r i s o n s , 208f bead d i a m e t e r e f f e c t , 207,208f butane c o m p a r i s o n i n f l u i d s t r u c t u r e s t u d y , 206f c h a i n l e n g t h e f f e c t , 207 c h a i n l e n g t h i n c r e a s e , 205 e q u a t i o n , 204-205 l o c a l c o r r e l a t i o n s of h a r d - s p h e r e monatomi c s , 20 6f Monte C a r l o s i m u l a t i o n c a l c u l a t i o n f o r f l u i d s , 204 Monte C a r l o s i m u l a t i o n p r e d i c t i o n f o r b u t a n e , 206f range i n f l u i d s t r u c t u r e s t u d y , 209-212 RISM and Monte C a r l o c o m p a r i s o n f o r butane,

21 Of

RISM and Monte C a r l o c o m p a r i s o n f o r o c t a n e , 21 Of temperature e f f e c t , 207,208f Binary c r i t i c a l l o c i d e f i n i n g e q u a t i o n s , 133-134 d i f f i c u l t i e s of p r o p e r t y p r e d i c t i o n , 132 m a t h e m a t i c a l approach used i n p r e d i c t i o n , 136 p r e d i c t i o n error for cubic equations of s t a t e , I48t p r e d i c t i v e a b i l i t y of cubic equations of s t a t e , 132-154 r e s u l t s i n p r e d i c t i o n s t u d y , 137-138 systems s e l e c t e d f o r s t u d y , 136-137

APPLICATIONS

Binary c r i t i c a l l o c i — C o n t i n u e d u n i i k e - p a i r parameters e n h a n c i n g p r e d i c t a b i l i t y , 150 van der Waals q u a l i t a t i v e p r e d i c t i o n , 133 van Konyenburg and S c o t t c l a s s i f i c a t i o n , 139-141 B i n a r y f l u i d m i x t u r e , thermodynamic c o n d i t i o n s of phase e q u i l i b r i u m , 375 B i n a r y i n t e r a c t i o n parameters d e f i n i t i o n and p u r p o s e , 572 new l o c a l c o m p o s i t i o n model u s e , 262 o n e - f l u i d m i x i n g r u l e s i m p r o v e d , 353 p e r t ur be d - a n i s o t r o p i c - c h a i n t h e o r y a p p l i c a t i o n , 305, 306t second i n t e r a c t i o n parameter i n t r o d u c e d , 572 Binary mixtures c r i c o n d e n t h e r m l o c a t i o n , 48 e q u a t i o n - o f - s t a t e mixing r u l e s s t u d i e d , 366t l o c a l compositions defined f o r f l u i d s , 252 phase e q u i l i b r i u m s t u d y r e s u l t s , 378-380 phase e q u i l i b r i u m p r e d i c t e d , 378-380 van der Waal s o n e - f l u i d t h e o r y , 256 van d e r Waals t h r e e - f l u i d t h e o r y , 255 van der Waals t w o - f l u i d t h e o r y , 255-256 vapor-liquid equilibrium calculations from m i x i n g r u l e s , 263-26 8t B i n a r y p a r a m e t e r s , a p p l i c a t i o n t o phase e q u i l i b r i u m c a l c u l a t i o n s , 378 B o i l i n g p o i n t s , e n t r o p y d e v i a t i o n , 521 B o u b l i k - A l d e r - C h e n - K r e g l e w s k i (BACK) equation accuracy for nonspherical m o l e c u l e s , 244 model b a s i s , 228 t e t r a f l u o r o m e t h a n e a p p l i c a t i o n , 76-78 Bring-back correction factors, four-phase f l a s h calculations u s e , 504 Bubble-point pressures , p r e d i c t i o n f r o m PFGC e q u a t i o n f o r s y n t h e t i c o i l , 469f B u t a n e , c a l c u l a t e d volume e r r o r u s i n g R e d l i c h - K w o n g p a r a m e t e r s , 400t

C

Carbon d i o x i d e " a " parameter equation, " b " parameter equation,

f o r Redlich-Kwong 391f f o r Redlich-Kwong 391f

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

585

INDEX Carbon

dioxide—Continued

Conformai s o l u t i o n a p p r o x i m a t i o n (CSA) mixture r a d i a l d i s t r i b u t i o n f u n c t i o n s , 320 R e d l i c h - K w o n g parameters , 398t See a l s o M u l t i f l u i d c o n f o r m a i m o l a r volume e r r o r f o r R e d l i c h - K w o n g s o l u t i o n approximation equation Conformai s o l u t i o n m i x i n g r u l e s new p a r a m e t e r s , 396f Soave p a r a m e t e r s , 393f compared w i t h new l o c a l c o m p o s i t i o n s t a n d a r d parameters , 392f m o d e l s , 262 Y a r b o r o u ^ i p a r a m e t e r s , 394t density calculations for binary vapor p r e s s u r e e r r o r f o r m i x t u r e s , 269t Conventional Successive S u b s t i t u t i o n R e d l i c h - K w o n g e q u a t i o n , 397f (COSS) Carbon d i o x i d e - o i l system ACSS c o m p a r i s o n , 484 s a t u r a t i o n l o c i comparison of convergence b e h a v i o r of a p r e d i c t e d and compared v a l u e s , 41 8t m i x t u r e , 487 s a t u r a t i o n l o c u s i n t e r s e c t i o n , 426f Κ v a l u e s u p d a t e d , 479 Carnahan-Starling equation s o l u t i o n r e g i o n f o r a m i x t u r e , 485f c o m p r e s s i b i l i t y f a c t o r , 526 Convergence, speed w i t h f l a s h c o n t a c t v a l u e s f o r f l u i d s , 256 c a l c u l a t i o n s , 488-492 Chemical p o t e n t i a l s , b i n a r y f l u i d C o o r d i n a t i o n number l o c a l c o m p o s i t i o n model f o r b i n a r y m i x t u r e , 375 m i x t u r e s , 252 C l a s s i c a l behavior predictions for square-well f l u i d m i x t u r e s , 115-116 f l u i d s , I86f i m p l i c a t i o n s , 115 C o o r d i n a t i o n number models s c a l e d behavior c r o s s o v e r , 113-115 square-well f l u i d C l a s s i c a l e q u a t i o n s , d i s a d v a n t a g e s , 119 comparison, 183-I85t Classical fluids van der Waals p a r t i t i o n f u n c t i o n , 184 d e f i n i t i o n , 111 Correlation hole, explanation i n f l u i d s t r o n g divergences i n behavior s t r u c t u r e e x p e r i m e n t , 20 5 o b t a i n e d , 111 C o s t a i d c o r r e l a t i o n , s p e c i f i c volume C l a s s i c a l s o l u t i o n thermodynamics, p r e d i c t i o n , 93 independent v a r i a b l e s , 2 Cricondentherm, c o l l i n e a r isochores for Combining r u l e s e m p i r i c a l m o d i f i c a t i o n , 572 v a r i o u s s y s t e m s , 51 e x p r e s s i o n , 572 C r i t i c a l l i n e , i n i t i a l slope e x p r e s s i o n , 126 new l o c a l c o m p o s i t i o n m o d e l s , 261 C r i t i c a l point C o m p r e s s i b i l i t y equation, mixing r u l e thermodynamics, 133-134 d e v i a t i o n f o r f l u i d s , 317-318 See a l s o M i x t u r e c r i t i c a l p o i n t Compressibility factor C r i t i c a l s t a t e s , d e s c r i b e d by e q u a t i o n s d e v i a t i o n contours f o r c u b i c of s t a t e , 109-178 equations, 555f,556f C r i t i c a l volume, c u b i c e q u a t i o n s hard spheres predictive a b i l i t y i n binary C a r n a h a n - S t a r l i n g e q u a t i o n , 528f m i x t u r e s , 151 d i s r u p t e d by m i x i n g r u l e s , 335-336 C r o s s o v e r f u n c t i o n , d e f i n i t i o n , 113 m i x t u r e h a r d - s p h e r e e r r o r , 339f C r y o g e n i c m i x t u r e s , f l u i d - f l u i d and mixtures of Lennard-Jones s o l i d - f l u i d e q u i l i b r i a , 371-385 m o l e c u l e s , 259t Cubic Chain of Rotators equation predictions for square-well d e s c r i p t i o n of binary f l u i d s , I88f s u p e r c r i t i c a l p r o p e r t i e s , 171 p r o p a n e - p r o p a d i e n e system r e s u l t s , s o l u b i l i t y of a c r i d i n e i n 98 t r i f l u o r o c h l or cm e t h a n e , 1 76f van der Waals m i x i n g r u l e s compared s o l u b i l i t y o f phenol i n c a r b o n w i t h new, 337 d i o x i d e , 1 76f Computer s i m u l a t i o n , l o c a l c o m p o s i t i o n supercritical solubility f o r m u l a t i o n s c o m p a r i s o n , 357-358 d e s c r i p t i o n , 164 Conf i g u r a t i o n a l e n e r g i e s , d e f i n e d f o r C u b i c e q u a t i o n s of s t a t e van der Waals p a r t i t i o n f u n c t i o n , 191 binary system's c r i t i c a l l o c i Conformai p a r a m e t e r s , d e f i n i t i o n , 315 p r e d i c t e d , 153-154 Conformai s o l u t i o n design, 542-546 d e f i n i t i o n , 315 d i s a d v a n t a g e s of two m i x i n g r u l e s , 31 7 t em per at ur e - de pen dent p o l a r f l u i d t r e a t m e n t d i f f i c u l t y , 316 parameters, 546-548

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

calculated

volume e r r o r

using

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

586

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

Cubic equations

of

state—Continued

e q u a t i o n s examined i n d e s i g n s t u d y , 538-539 e v a l u a t i o n , 538-539 e x p r e s s i o n , 537 f e a t u r e s , 540t g e n e r a l , c h o i c e s of c o n s t a n t s , 436t g e n e r a l f o r m , 434,571 mixing rules a p p l i c a t i o n s , 324 p o l a r , asymmetric systems d e r i v e d , 571-582 g e n e r a l i z e d , 314-328 m i x t u r e a p p l i c a t i o n s , 314 parameter v a l u e s i n b i n a r y c r i t i c a l l o c i s t u d y , I48t parameters , 572 Peng-Robinson e q u a t i o n m o d i f i e d t o low r e d u c e d t e m p e r a t u r e s , 560 -56 9 p o l a r f l u i d s a p p l i c a t i o n , 434-451 p r e d i c t i n g binary c r i t i c a l l o c i , 132-154 quadratic temperature f u n c t i o n u s e d , 435 s e l e c t i o n and d e s i g n , 537-538 t em per at ur e - de pen dent par am et e r s , 43 5, 53 9 van d e r Waals m i x i n g r u l e s a p p l i c a t i o n , 361-362 r e s u l t s , 331

D

D e i t e r ' s equation a p p l i e d t o t et r a f l u o r cm e t h a n e , 71-73 d i s a d v a n t a g e s , 82 m i x i n g r u l e s , 374 D e v i a t i o n entropy a r g o n , l i q u i d s a t u r a t i o n c u r v e , 527f hard-sphere equation-of-state p r o p e r t y c o r r e l a t i o n , 520-521 heats of v a p o r i z a t i o n o b t a i n e d , 520 i d e a l gas h y d r o c a r b o n s w i t h r a r e g a s e s , 523f l i g h t g a s e s , h a l o g e n s , and a m i n e s , 523f v a r i o u s compounds, 521f, 524f liquid-volume variation with t e m p e r a t u r e , 52 5 Deviation functions d e f i n i t i o n , 3-4 rationale, 3 Dew-bubble p o i n t curve (DBC), m i x t u r e c r i t i c a l p o i n t l o c a t i o n , 42-44 D i e l e c t r i c continuum m o d e l , i o n i c f l u i d s , 281 Diffusional s t a b i l i t y homogeneous phase c r i t e r i a , 481

Diffusional stability—Continued l i m i t evaluated i n s i n g l e - s t a g e c a l c u l a t i o n s t u d y , 483 D i l u t e m i x t u r e s , c l a s s i c a l and non c l a s s i c a l t r e a t m e n t , 125 D i l u t e n e a r - c r i t i c a l mixtures d i v e r g i n g p a r t i a l m o l a r volumes t h e s o l u t e , 121 non c l a s s i c a l a n a l y s i s , 122-125 o s m o t i c s u s c e p t i b i l i t y , 124

flash

of

Ε

E f f e c t i v e carbon number (ECN) d e f i n i t i o n , 443 P a t e l - T e j a e q u a t i o n , parameters o b t a i n e d , 443 E n e r g y — S e e G i b b s energy — S e e Helmholtz f r e e energy Energy e q u a t i o n , m i x i n g r u l e d e v i a t i o n f o r f l u i d s , 317-318 E n e r g y o f i n t e r a c t i o n , van der Waals p a r t i t i o n f u n c t i o n , 181 Entropy c a l c u l a t i o n f o r m e t h a n e - c a r bon s y s t e m , 55t See a l s o D e v i a t i o n e n t r o p y E q u a t i o n s of s t a t e Adachi e q u a t i o n , 135 a p p l i c a t i o n s , 352 c l a s s i c a l s o l u t i o n thermodynamics c o n n e c t i o n , 1-40 c o m b i n i n g r u l e s f o r new l o c a l c o m p o s i t i o n m o d e l s , 261 c o m p a r i s o n of e q u a t i o n s a p p l i e d t o t et r a f l u o r cm e t h a n e , 81-82 composition mixing rules studied for b i n a r y s y s t e m s , 366t d a t a , phenomena, and c r i t i q u e , 41-107 D e i t e r ' s e q u a t i o n i n phase e q u i l i b r i a s t u d y , 372 d e n s i t y dependence o f a t t r a c t i v e terra, 363f d e s c r i p t i o n f o r San Andres o i l s d e v e l o p e d , 407-408 d i s t r i b u t i o n function theory basis for nonspherical m o l e c u l e s , 227-244 examined f o r b i n a r y c r i t i c a l l o c i p r e d i c t i o n , 134-135 f l u i d m i x t u r e s , 259 f l u i d - f l u i d and s o l i d - f l u i d e q u i l i b r i a , 371-385 formalism concepts r e l a t i n g t o s o l u t i o n the rm o d y na m i c s, 35 g e n e r a l i z e d van der Waals p a r t i t i o n f u n c t i o n , 180-200 h a r d - s p h e r e a p p l i c a t i o n , 521

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

587

INDEX

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

Equations of

state—Continued

hard-sphere mixtures f o r f l u i d r e p r e s e n t a t i o n , 255-258 i n i t i a l l o c a l composition i m p o r t a n c e , 354 l i q u i d - m i x t u r e p r o p e r t i e s , 10 l o c a l composition a c t i v i t y c o e f f i c i e n t model a s s i m i l a t i o n , 251 m i x i n g r u l e s development, 3 53-355 parameter c o n s t r a i n t s , 58 P e n g - R o b i n s o n e q u a t i o n , 135 p e r t u r b a t i o n t h e o r y , 64-70 perturbed-anisotropic-chain theory e q u a t i o n , 299 phase e q u i l i b r i u m c a l c u l a t i o n b a s i s , 371 quantum c o r r e c t i o n o f van der Waals t y p e , 373 quantum e f f e c t s contributions a v e r a g e d , 374 radial-distribution function a p p l i c a t i o n , 361 R e d l i c h - K w o n g e q u a t i o n , 134 selected i n supercritical extraction s t u d y , 157-159b Soave e q u a t i o n , 134 s o l u b i l i t y data deviations for binary m i x t u r e s , 166-16 9t supercritical extraction calculation summary, 170t,171 supercritical solubility calculation u s e , 157-159,171 T e j a e q u a t i o n , 135 volume r o o t s , 476 See a l s o C u b i c e q u a t i o n s of s t a t e E t h a n e , c a l c u l a t e d volume e r r o r u s i n g R e d l i c h - K w o n g parameters , 399t Excess f r e e energy, l o c a l c o m p o s i t i o n model a p p l i c a t i o n s , 250 Excess f u n c t i o n s c a l c u l a t i o n o f van der W a a l s , 25 definition, 6 r e s i d u a l Helmholtz energy r e l a t i o n s h i p , 1 4t E x c e s s volumes , s i m u l a t e d and c a l c u l a t e d compared, 344f Expansion c o e f f i c i e n t s , l i q u i d - l i q u i d m i x t u r e s , 119-120

F

F i e l d - d e n s i t y space , c r i t i c a l exponents p a t h , 112f F i e l d - f i e l d s p a c e , c r i t i c a l exponents p a t h , 112f First-contact studies, d e f i n i t i o n , 406-407

Flash calculations c h a r a c t e r i s t i c p r o b l e m s , 495 flow diagram of c o m p u t a t i o n a l a l g o r i t h m , 508f m u l t i phase c o m p u t a t i o n a l a l g o r i t h m , 496 -504 d i s t r i b u t i o n d e t e r m i n e d , 500-501 f o u r - p h a s e s y s t e m r e s u l t s , 494-51 5 o v e r v i e w , 495 s o l u t i o n s e a r c h s t r a t e g y , 505f water a d d i t i o n e f f e c t s t o f o u r - p h a s e s y s t e m , 515 w a t e r - f r e e d e s c r i p t i o n , 504 - 507 s i n g l e stage convergence b e h a v i o r , 476-492 COSS r e s u l t s , 4 8 4 , 4 8 7 f slow convergence p r o b l e m s , 488 two phase e q u i l i b r i u m c o n s t a n t s d e f i n e d , 497 Κ v a l u e i n i t i a t i o n , 498 water e f f e c t on phase distribution, 512f-514f Flory-Huggins model, a c t i v i t y c o e f f i c i e n t m o d e l , 195 Fluid descriptions development f o r o i l - c a r b o n d i o x i d e s y s t e m s , 415-42 5 o v e r v i e w , 407-408 See a l s o F l u i d - f l u i d e q u i l i b r i a Fluid-fluid equilibria e q u a t i o n - o f - s t a t e a p p l i c a t i o n to c r y o g e n i c m i x t u r e s , 371-385 thermodynamic c o n d i t i o n s , 375-376 See a l s o F l u i d d e s c r i p t i o n s F l u i d mixtures c l a s s i c a l c r i t i c a l behavior, 115-116 n o n c l a s s i c a l b e h a v i o r e v i d e n c e , 119 nonclassical c r i t i c a l behavior, 116-119 See a l s o F l u i d s Fluids a t t r a c t i v e and r e p u l s i v e f o r c e s i n m o d e l s , 228 c r i t i c a l b e h a v i o r of p u r e , 110-113 l o c a l s t r u c t u r e v i a Monte C a r l o s i m u l a t i o n , 201-213 t e s t e d by t h e per t ur be d - a n i sot r ο p i c - c h a i η t h e o r y , 301-302 See a l s o F l u i d m i x t u r e s See a l s o R e a l f l u i d s Free energy expression derived for i o n i c f l u i d s , 292-295 W h i t i n g - P r a u s n i t z e x p r e s s i o n , 253-254 See a l s o G i b b s energy See a l s o H e l m h o l t z f r e e e n e r g y F u g a c i t y , mixture behavior, 21-25 Fugacity coefficient activity coefficient c o n v e r s i o n , 21-22 d e f i n i t i o n , 21,479

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND

588

Fugacity coefficient—Continued d e r i v e d from new c u b i c e q u a t i o n m i x i n g r u l e , 573 e v a l u a t i o n at i n f i n i t e d i l u t i o n , 23 e x p r e s s i o n f o r PRSV e q u a t i o n , 563 e x p r e s s i o n f o r use i n p r o p a n e - p r o p a d i e n e s y s t e m , 97 e x p r e s s i o n from van d e r Waals e q u a t i o n , 34 P e n g - R o b i n s o n e q u a t i o n , 354 PFGC e q u a t i o n d e r i v a t i o n , 456 p r e s s u r e - e x p l i c i t e q u a t i o n - o f - s t ate e x p r e s s i o n , 19

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

G

G a s e s , quantum e f f e c t s a t l o w t e m p e r a t u r e , 372-373 Gibbs-Duhem e q u a t i o n , d i f f e r i n g f o r p a r t i a l p r o p e r t i e s , 18 G i b b s energy b i n a r y f l u i d m i x t u r e , 375 c a l c u l a t i o n f o r pure s u b s t a n c e s , 379f i n f i n i t e pressure excess f u n c t i o n d e r i v e d , 567 m u l t i p h a s e e f f e c t s , 506 s i m u l a t e d and c a l c u l a t e d v a l u e s compared, 346f Goodwin e q u a t i o n parameter p r e c i s i o n i n pro p a n e - p r o pa d i en e s y s t e m , 93 propane and p r o p a d i en e c o n s t a n t s , 911 pro p a n e - p r o pa d i en e system a p p l i c a t i o n , 89

H

Haar

and K o h l e r e q u a t i o n , a p p l i e d t o t e t r a f l u o r cm e t h a n e , 70-71 Han-Cox-Bono-Kwok-Starling equation s o l u b i l i t y of b e n z o i c a c i d i n c a r b o n d i o x i d e , 175f s o l u b i l i t y of f l u o r i n e i n e t h y l e n e , 175f supercritical solubility d e s c r i p t i o n , 164 Hard-sphere e q u a t i o n , property c a l c u l a t i o n at h i g h d e n s i t i e s , 526 H a r d - s p h e r e e x p a n s i o n (HSE) approximation m e a n - d e n s i t y a p p r o x i m a t i o n , 332 mixture r a d i a l d i s t r i b u t i o n f u n c t i o n s , 321 Hard spheres conformai s o l u t i o n m i x i n g r u l e s , 335-337

APPLICATIONS

Hard s p h e r e s — C o n t i n u e d e q u a t i o n s of s t a t e f o r f l u i d s , 255-258 Harmen-Knapp e q u a t i o n , s u p e r c r i t i c a l s o l u b i l i t y d e s c r i p t i o n , 164 Heats of v a p o r i z a t i o n e n t r o p y d e v i a t i o n s used f o r e s t i m a t i o n , 520 e n t r o p y e f f e c t s used f o r e s t i m a t i n g , 52 5 Heavy h y d r o c a r b o n d i s t r i b u t i o n s , f l u i d s from San Andres o i l s , 412f Helmholtz f r e e energy e q u a t i o n , 453 m i x t u r e e r r o r , 34Of p e r t u r b a t i o n t h e o r y d e s c r i p t i o n , 282 PFGC e q u a t i o n - o f - s t a t e f o r m u l a t i o n , 453 van d e r Waals m i x i n g r u l e s compared w i t h new, 337 W h i t i n g - P r a u s n i t z e x p r e s s i o n used f o r f l u i d m i x t u r e s , 260 See a l s o R e s i d u a l H e l m h o l t z energy Henry's constants, mixing rules a s s e s s e d f o r e q u a t i o n s of s t a t e , 35 H e n r y ' s law a c t i v i t y c o e f f i c i e n t , d e f i n i t i o n , 22 Heyen e q u a t i o n s o l u b i l i t y of benzoic a c i d i n carbon d i o x i d e , 1 73f s o l u b i l i t y of fluorine i n e t h y l e n e , 173f Hydrocarbon systems water a d d i t i o n t o s i n g l e - p h a s e s y s t e m , 507, 508t water a d d i t i o n t o t h r e e - p h a s e s y s t e m , 509 water a d d i t i o n t o t w o - p h a s e s y s t e m , 509 Hydro c a r bon-water system A CSS t e c h n i q u e d e r i v e d , 502-503 d i v e r g e n c e problems p r e v e n t e d , 502 i n i t i a l e s t i m a t e o f Κ v a l u e s , 502 m u l t i component system des c r i b e d , 511 multiphase flash c a l c u l a t i o n s , 504 -50 7 Hydrocarbons m o l e c u l a r f l e x i b i l i t y i n c r e a s e as c h a i n l e n g t h i n c r e a s e s , 522 See a l s o L i g j i t h y d r o c a r b o n systems Hydrogen s u l f i d e , c a l c u l a t e d volume error u s i n g Redlich-Kwong p a r a m e t e r s , 400t

I

I n f l e c t i o n l o c u s , i s o c h o r i c , 58 I n t e r a c t i o n e n e r g y , PFGC e q u a t i o n ,

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

455

589

INDEX

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

I n t e r a c t i o n parameters c a r b o n d i o x i d e , d e s c r i b e d by R e d l i c h - K w o n g e q u a t i o n , 410 determined f o r propane-propadiene system, 93-95 h y d r o c a r b o n - w a t e r s y s t e m , 51 0t R e d l i c h - K w o n g e q u a t i o n f o r San Andres F o r m a t i o n o i l s , 41 It I n t e r n a l energy l o c a l c o m p o s i t i o n model of b i n a r y m i x t u r e , 2 53 s i m p l e f l u i d , 534f Ionic fluids d i e l e c t r i c continuum m o d e l , 281 e q u a t i o n of s t a t e , 2 8 1 - 2 9 5 f r e e energy e x p r e s s i o n d e r i v e d , 292-295 model a p p r o x i m a t i o n s used i n p e r t u r b a t i o n e x p a n s i o n , 287 models u s e d , 281 p e r t u r b a t i o n t h e o r y a d v a n t a g e s , 282 I s e n t r o p e s , c o l l i n e a r i t y at c r i c o n d e n b a r , 51-53 I s o b u t a n e , c a l c u l a t e d volume e r r o r using Redlich-Kwong p a r a m e t e r s , 399t Isochores c o l l i n e a r i t y f o r f l u i d s , 42-59 c o l l i n e a r i t y p r o o f of G r i f f i t h s , 48-51 I s o p l e t h — S e e Dew-bubble p o i n t c u r v e (DBC) I s o t h e r m a l m e l t i n g , d i a g r a m , 383f I t e r a t i o n counts ACSS a p p l i e d t o m i x t u r e , 490f ,491f COSS a p p l i e d t o m i x t u r e , 490f Newton-Raphson p r o c e d u r e a p p l i e d t o m i x t u r e , 491f

Κ

Ko h i e r e q u a t i o n , d i s a d v a n t a g e s , 82 Kubic equation, s u p e r c r i t i c a l s o l u b i l i t y d e s c r i p t i o n , 164

L

L a t t i c e f l u i d t h e o r i e s , approximations f o r s h o r t - c h a i n m o l e c u l e s , 202 L a t t i c e gas model c o o r d i n a t i o n number model d e r i v e d , 184 n o n c l a s s i o a l model a p p l i c a t i o n , 118 See a l s o L a t t i c e models Lattice models, a p p l i e d t o tetrafluoromethane, 78

L a t t i c e t h e o r i e s , l o c a l composition m i x i n g r u l e s d e r i v e d , 355-356 Lennard-Jones mixtures compressibility factor for binary m i x t u r e s , 259t c o n f o r m a i s o l u t i o n c a l c u l a t i o n s , 342 e q u a t i o n s of s t a t e f o r b i n a r y m i x t u r e s , 258 l o c a l c o m p o s i t i o n compared w i t h s q u a r e - w e l l f l u i d s , 22Ot new m i x i n g r u l e s t e s t e d , 342 compressibility factor for binary m i x t u r e s , 259t Lennard-Jones p o t e n t i a l a p p r o x i m a t i o n by s q u a r e - w e l l p o t e n t i a l , 230 six-step square-well potential a p p r o x i m a t i o n , 23Of Leung-Griffiths equation, gas-liquid m i x t u r e s m o d e l i n g , 118 Lewis-Randall standard s t a t e s , a c t i v i t y c o e f f i c i e n t d e f i n i t i o n , 21 L i g h t h y d r o c a r b o n s y s t e m s , PFGC e q u a t i o n a p p l i c a t i o n , 462-469 L i q u i d d e n s i t i e s , P e n g - R o b i n s o n and P a t e l - T e j a compared, 442t Li quid-liquid equilibrium m e t h a n o l - h e p t a n e s y s t e m , 449f l i q u i d - l i q u i d mixtures, divergence p r e d i c t e d f o r s p e c i f i c h e a t , 119 L i quid-vapor e q u i l i b r i u m , compositions f o r San Andres o i l - a c i d gas m i x t u r e , 42Ot Local composition adhesive i n t e r molecular p o t e n t i a l for m i x t u r e s , 356 b u l k c o m p o s i t i o n d i f f e r e n c e s , 214 bulk composition e f f e c t s i n s q u a r e - w e l l m o l e c u l e s s t u d y , 217 d e f i n i t i o n for binary f l u i d s y s t e m s , 252 density effects i n square-well m o l e c u l e s s t u d y , 217 d e t e r m i n a t i o n , 214 f o r m u l a t i o n c o m p a r i s o n t o computer s i m u l a t i o n , 357-358 equations f o r s q u a r e - w e l l m o l e c u l e s , 215 e x t e n t i n s q u a r e - w e l l m i x t u r e , 198f G i e r y c z and N a k a n i s h i e q u a t i o n , 218 H u , L u d e c k e , and P r a u s n i t z e q u a t i o n , 21 8 mixing r u l e s , density calculations f o r b i n a r y m i x t u r e s , 269t model c o m p a r i s o n f o r s q u a r e - w e l l m o l e c u l e s , 219 o r i g i n , 354 s q u a r e - w e l l m o l e c u l e s by Monte C a r l o s i m u l a t i o n , 214-225 s t a t i s t i c a l mechanical d e f i n i t i o n f o r f l u i d s , 251-252

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

590

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

Local

EQUATIONS OF STATE: THEORIES AND

composition—Continued

temperature e f f e c t s i n s q u a r e - w e l l m o l e c u l e s s t u d y , 217 L o c a l c o m p o s i t i o n models c o m b i n i n g r u l e s f o r f l u i d s , 261 c o m p a r i s o n among d e n s i t y de p e n d e n t , 196-199 comparison with conformai s o l u t i o n m i x i n g r u l e s , 262 c o m p o s i t i o n dependence of m o d e l , 199 c o o r d i n a t i o n number f o r b i n a r y m i x t u r e s , 252 d e n s i t y - d e pendent model d i s c u s s i o n , 196 e r r o r s , 358 W h i t i n g and P r a u s n i t z , 195-199 e q u a t i o n of s t a t e f o r m i x t u r e s , 260 e x c e s s f r e e energy a p p l i c a t i o n s , 250 i n t e r n a l energy of binary m i x t u r e , 253 mixing r u l e f o r low-density m i x t u r e , 360 summary and c o m p a r i s o n , 197t See a l s o L o c a l c o m p o s i t i o n L o c a l c o o r d i n a t i o n numbers , d i f f e r e n t models l e a d i n g t o d i f f e r e n t e q u a t i o n s of s t a t e , 191-192

M

Mean-density approximation (MDA) compared t o m o d i f i e d m e a n - d e n s i t y approximation for argon-krypton m i x t u r e , 334f Lennard-Jones molecules s t u d i e d f o r f l u i d s , 258-259 van der W a a l s , 332 See a l s o M o d i f i e d m e a n - d e n s i t y a p p r o x i m a t i o n (MMDA) Mean s p h e r i c a l a p p r o x i m a t i o n (MSA) , expansion of i o n i c f l u i d m o d e l i n g , 290-291 M e c h a n i c a l s t a b i l i t y , homogeneous phase c r i t e r i a , 481 M e t h a n e , c a l c u l a t e d volume e r r o r f o r R e d l i c h - K w o n g p a r a m e t e r s , 398t Mixing rules a t t r a c t i v e e n e r g i e s e f f e c t on b i n a r y f l u i d s , 258-259 binary c r i t i c a l l o c i prediction u s e d , 136 conformai s o l u t i o n s d e r i v a t i o n , 332-335 c o n f o r m a i s o l u t i o n s e q u a t i o n s , 335 d e f i n i t i o n , 314 d e n s i t y d e p e n d e n t , 195-196 d e n s i t y - d e pen dent l o c a l c o m p o s i t i o n , 354

Mixing

APPLICATIONS

rules—Continued

e q u a t i o n - o f - s t ate composition dependence, 254-255 e q u a t i o n - o f - s t ate a p p l i c a t i o n s , 324 i n d u s t r i a l p e r s p e c t i v e , 352- 3 67 L e n n a r d - J o n e s m i x t u r e s t e s t e d , 342 l o c a l c o m p o s i t i o n d e r i v a t i o n , 355-357 l o c a l composition e f f e c t , 358-360 l o c a l c o m p o s i t i o n model a p p l i e d t o l o w - d e n s i t y m i x t u r e , 360 l o c a l c o m p o s i t i o n r u l e s and c o n f o r m a i s o l u t i o n r u l e s compared, 262-272 van d e r Waals o n e - f l u i d , m a c h i n e - s i m u l a t i o n r e s u l t s , 360 molecular size effects, 255-258 Monte C a r l o s i m u l a t i o n used t o t e s t L e n n a r d - J o n e s f l u i d s , 342 m u l t i f l u i d hard-sphere e x p a n s i o n , 323 - 324 m u l t i f l u i d t h e o r y , 322 new l o c a l c o m p o s i t i o n r u l e a p p l i e d t o t e r n a r y s y s t e m s , 272 o n e - f l u i d conformai s o l u t i o n s i m p r o v e d , 331-349 o n e - f l u i d d i a m e t e r s c a l c u l a t e d , 337 o n e - f l u i d t h e o r y , 318-319 P a t e l - T e j a e q u a t i o n , 446 Peng-Robinson e q u a t i o n , 326-329t PRSV e q u a t i o n , 565-566 r e c e n t d e v e l o p m e n t s , 331 R e d l i c h - K w o n g e q u a t i o n , 326, 329t s t a t i s t i c a l mechanical t h e o r y , 317-318 s u p e r c r i t i c a l extraction study a p p l i c a t i o n s , 157 systems s t u d i e d by l o c a l c o m p o s i t i o n , 355 van der Waals and random m i x i n g compared, 359-360 van der Waals compared w i t h MMDA, 337 van d e r Waals e q u a t i o n , 324-326 van der Waals one f l u i d a s s u m p t i o n s , 359-360 i m p r o v e d , 353 m a c h i n e - s i m u l a t i o n r e s u l t s , 360 p r e d i c t i v e power, 360 r a d i a l d i s t r i b u t i o n f u n c t i o n , 359 summary, 353 See a l s o C o n f o r m a i s o l u t i o n m i x i n g rules See a l s o van der Waals o n e - f l u i d mixing rules Mixture c r i t i c a l point, r e l a t i o n s h i p to t h e c r i con den bar and c r i c o n d e n t h e r m , 42-44 M i x t u r e p r o p e r t i e s , two major formalisms, 2 Mixture r a d i a l d i s t r i b u t i o n functions c o n f o r m a i s o l u t i o n a p p r o x i m a t i o n , 320 d e n s i t y expansion a p p r o x i m a t i o n , 321-322

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

591

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

INDEX Mixture r a d i a l d i s t r i b u t i o n f u n c t i o n s —Continued hard-sphere expansion a p p r o x i m a t i o n , 321 m u l t i f l u i d conformai s o l u t i o n a p p r o x i m a t i o n s , 323 random m i x i n g a p p r o x i m a t i o n s , 319-320 See a l s o R a d i a l d i s t r i b u t i o n function Mixtures c r i t i c a l exponents p a t h , 117f e q u a t i o n of s t a t e f r o m l o c a l c o m p o s i t i o n m o d e l s , 260 equations of s t a t e f o r f l u i d s , 259-260 i m p u r i t y e f f e c t s on e x t e n s i v e p r o p e r t i e s , 128 non c l a s s i c a l b e h a v i o r n e a r a c r i t i c a l p o i n t , 123f perturbed-anisotropic-chain theory d i s c u s s e d , 300-301 t e s t e d f o r f l u i d s , 302-306 van d e r Waals p a r t i t i o n f u n c t i o n e q u a t i o n , 189 See a l s o B i n a r y m i x t u r e s See a l s o M u l t i c o m p o n e n t m i x t u r e s Models characteristics, 3 phase e q u i l i b r i u m d e s c r i p t i o n , 352 Modified mean-density approximation (MMDA) argon-krypton mixture a p p l i c a t i o n , 333 c o m p a r i s o n w i t h van der Waals o n e - f l u i d p r e d i c t i o n s , 347t e q u a t i o n , 333 h a r d - s p h e r e diameter comparison, 34Of,341f h a r d - s p h e r e diameter r a t i o , 338f,339f Lennard-Jones mixtures r e s u l t s , 344t-345t phase e q u i l i b r i a a p p l i c a t i o n , 377-378 phase e q u i l i b r i a c a l c u l a t i o n f o r b i n a r y s y s t e m s , 380 s i m u l a t i o n r e s u l t s compared, 348t van der Waals compared f o r b i n a r y s y s t e m s , 380 M o l a r heat of v a p o r i z a t i o n , d e t e r m i n e d from an e q u a t i o n of s t a t e , 26 M o l a r volumes d i v e r g e n c e s i n m i x t u r e s , 213f PFGC e q u a t i o n , 457 p r e d i c t i o n e r r o r s f o r e q u a t i o n s of state, 46lt p r e d i c t i o n e r r o r s f o r PFGC e q u a t i o n , 462 p r e d i c t i o n from PFGC e q u a t i o n f o r i s o p e n t a n e , 463f M o l e f r a c t i o n s — S e e Phase m o l e fractions M o l e c u l a r dynamics s i m u l a t i o n , m o l e c u l a r model e x a m i n a t i o n , 202

Molecules—See Nonspherical molecules Monte C a r l o program d e s c r i p t i o n , 203 m o l e c u l a r movements, 203 r e l i a b i l i t y t e s t s , 204 Monte C a r l o s i m u l a t i o n f l u i d s t r u c t u r e c a l c u l a t i o n , 204-205 mixing r u l e s t e s t e d f o r Lennard-Jones m i x t u r e s , 342 models h i s t o r y , 202 procedure i n s q u a r e - w e l l molecules s t u d y , 216 r e s u l t s compared w i t h RISM, 209 results i n fluid structure e x p e r i m e n t , 212 r e s u l t s i n square-well molecules s t u d y , 216-217 M u l t i component m i x t u r e s , i s o c h o r i c and i s e n t r o p i c c o l l i n e a r i t y , 58 M u l t i f l u i d conformai s o l u t i o n approximation, mixture r a d i a l d i s t r i b u t i o n f u n c t i o n s , 323 M u l t i f l u i d density expansion, mixing r u l e s , 324 M u l t i f l u i d hard-sphere expansion, m i x i n g r u l e s , 323-324 M u l t i f l u i d t h e o r y , m i x i n g r u l e s , 322 Multiphase e q u i l i b r i u m , governing e q u a t i o n s , 500 Multiphase s o l u t i o n s , search s t r a t e g y , 504

Ν

N - f l u i d t h e o r i e s , compared, 256-258 Near-critical fluids d e s c r i p t i o n o f i m p u r e , 125-129 i m p u r i t y e f f e c t s c a l c u l a t e d , 126 i m p u r i t y e f f e c t s on d e n s i t y , 126 See a l s o N e a r - c r i t i c a l m i x t u r e s N e a r - c r i t i c a l mixtures c l a s s i c a l a n a l y s i s , 120-122 non c l a s s i c a l d e s c r i p t i o n , 110-131 p a r t i a l molar p r o p e r t i e s behavior, 120-121 See a l s o N e a r - c r i t i c a l f l u i d s Near-critical region ACSS d e f i n e d and u s e d , 499 Κ v a l u e e s t i m a t i o n by e v a p o r a t i o n t e c h n i q u e , 499 N e a r - c r i t i c a l s t a t e s , d e s c r i b e d by e q u a t i o n s of s t a t e , 109-178 N e a r e s t n e i g h b o r s , d e f i n i t i o n , 356-357 Newton-Raphson p r o c e d u r e d e f i n e d , 481 Κ v a l u e s u p d a t e d , 481 N i t r o g e n , c a l c u l a t e d volume e r r o r f o r R e d l i c h - K w o n g parameters , 398t

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

592

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

N o n c l a s s i c a l f l u i d s , d i s c u s s i o n , 111 Ν on p o l a r f l u i d s , l o c a l c o m p o s i t i o n model a p p l i c a t i o n , 2 5 0 - 2 7 5 Nonspherical molecules a c c u r a c y of e q u a t i o n of s t a t e f o r h y d r o c a r b o n s , 240-244 equation-of-state a p p l i c a t i o n to p o l a r f l u i d s , 244 e q u a t i o n s of s t a t e based o n t h e distribution function t h e o r i e s , 227-244 e q u a t i o n s of s t a t e compared, 240-244 parameter r e d u c t i o n i n e q u a t i o n s of s t a t e , 239-240

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

0 O n e - f l u i d conformai s o l u t i o n , m i x i n g r u l e s i m p r o v e d , 331-349 O n e - f l u i d t h e o r y , d e f i n i t i o n , 23 One-step square-well potential a n g l e - d e pendent G a u s s i a n o v e r l a p e q u a t i o n , 235-238 c o n t a c t v a l u e s of c o r r e l a t e d f u n c t i o n s , 236-237t i n d e x v a r i a t i o n w i t h i n t e n s i t y , 234f p r e s s u r e p r e d i c t i o n , 237t Osmotic s u s c e p t i b i l i t y d i v e r g e n c e i n m i x t u r e s , 115,116 n e a r - c r i t i c a l m i x t u r e s , 124

Ρ

P a r a f f i n s , h a r d - c o r e volume v a r i a t i o n w i t h c r i t i c a l volume, 243f Parameters c o r r e c t i o n parameter c o r r e l a t e d f o r P a t e l - T e j a e q u a t i o n , 445f correlation for Patel-Teja e q u a t i o n , 444f degrees of freedom v s . b v a l u e o f paraffins, 532f development f o r San Andres F o r m a t i o n o i l s , 410-413 no η p o l a r f l u i d s t e s t e d f o r p e r t ur be d - a n i s o t r o p i c - c h a i η t h e o r y , 302 Pen g-R o b i ns o n - S t r yj ek-V er a e q u a t i o n , 562t P e n g - R o b i n s o n and P a t e l - T e j a compared, 437f Redlich-Kwong equation carbon d i o x i d e , 402-403 c r i t i c a l t e m p e r a t u r e s and r a n g e s f i t , 204t d e f i n e d , 390 d e t e r m i n a t i o n , 390

in

Redlich-Kwong e q u a t i o n — C o n t i n u e d v a r i o u s s u b s t a n c e s , 403-404 v o l u m e t r i c e r r o r comparison f o r c a r b o n d i o x i d e , 398t v o l u m e t r i c e r r o r comparison f o r e t h a n e , 399t v o l u m e t r i c e r r o r comparison f o r hydrogen s u l f i d e , 400t v o l u m e t r i c e r r o r comparison f o r i s o b u t a n e , 399t v o l u m e t r i c e r r o r comparison f o r methane, 398t v o l u m e t r i c e r r o r comparison f o r n i t r o g e n , 398t t em per at ur e - de pen dent c u b i c e q u a t i o n s , 539 f o r t h e Redlich-Kwong e q u a t i o n , 389-400 two t e m p e r a t u r e - d e p e n d e n t , d e v i a t i o n curves o b t a i n e d from c u b i c e q u a t i o n s , 547f See a l s o C o n f o r m a i parameters See a l s o I n t e r a c t i o n parameters Parameters from group c o n t r i b u t i o n s (PFGC) e q u a t i o n b a s i s , 453 b i n a r y i n t e r a c t i o n c o e f f i c i e n t s , 457 des c r i p t i o n , 4 52 d i f f i c u l t i e s , 462 g r o u p parameters o b t a i n e d , 456-4 57 l i m i t a t i o n , 457 p a r a m e t e r s , 455 phase e q u i l i b r i a , 452-470 w r i t t e n i n t e r m s of c o m p r e s s i b i l i t y f a c t o r , 4 54 Partial equation-of-state parameters determination, 23-24 van der Waals e q u a t i o n e x a m p l e , 2 4 - 2 5 P a r t i a l properties d e f i n i t i o n o f g e n e r a l i z e d , 16 e q u a t i o n - o f - s t ate p r o p e r t i e s , conversion of laboratory pro p e r t i e s , 18-19 d e f i n i t i o n , 17 l a b o r a t o r y p r o p e r t i e s , d e f i n i t i o n , 17 l i s t , 15b r a t i o n a l e and d e f i n i t i o n s , 13-20 r e l a t i o n s h i p of d i f f e r e n t t y p e s , 17-20 s u m m a b i l i t y f e a t u r e , 15 Patel-Teja equation c o n s t a n t s o b t a i n e d from the c r i t i c a l p o i n t , 438 c r i t i c a l l o c i p r e d i c t i o n s , 448 m i x i n g r u l e s , 446 p o l a r m o l e c u l e vapor p r e s s u r e s , 439 r e d u c t i o n t o Peng-Robinson e q u a t i o n , 438 vapor-liquid equilibria results c o m p a r e d , 446-448 Peng-Robinson equation accuracy for nonspherical m o l e c u l e s , 244

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

593

INDEX

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

Peng-Robinson e q u a t i o n — C o n t i n u e d c r i t i c a l l o c i p r e d i c t i o n s , 448 e x p r e s s i o n , 561 f u g a c i t y c o e f f i c i e n t s , 354 m i x i n g r u l e s , 326 - 3 2 9 t , 4 96 - 4 97t parameter i n f l u e n c e o n s a t u r a t e d l i q u i d volume p r e d i c t i o n , 550 - 5 5 2 f parameter v a r i a t i o n f o r n - h e x a n e , 54 9f p o l a r m o l e c u l e vapor p r e s s u r e s , 439 solubility benzoic a c i d i n carbon d i o x i d e , 1 74f f l u o r i n e i n e t h y l e n e , 174f s u p e r c r i t i c a l d e s c r i p t i o n , 164 Peng-Ro b i n s o n - S t r y j e k - V e r a e q u a t i o n f u g a c i t y c o e f f i c i e n t e x p r e s s i o n , 563t r e s u l t s compared, 563t P e r t u r b a t i o n expansion i o n i c f l u i d modeling approximations u s e d , 287 mean s p h e r i c a l a p p r o x i m a t i o n s (MSA) result obtained f o r i o n i c f l u i d s , 287 Perturbation theory a d v a n t a g e s , 64 applied to tetrafluoromethane, 79-81 integral evaluation for ionic f l u i d s , 287-290 i o n i c f l u i d m o d e l i n g a d v a n t a g e s , 282 mixture equation-of-state development, 3 53 - 3 54 Perturbed-anisotropic-chain theory a c c u r a c y , 302 anisotropic multipolar interactions c a l c u l a t e d , 300 a z e o t r o p e p r e d i c t i o n i n m i x t u r e s , 305 binary i n t e r a c t i o n p a r a m e t e r s , 305, 306t c a n o n i c a l ensemble p a r t i t i o n f u n c t i o n , 298 e q u a t i o n of s t a t e , 299 f l u i d m i x t u r e s , 302-306 fluid-mixture properties c a l c u l a t e d , 303-305 h a r d - c h a i n r e p u l s i o n c a l c u l a t i o n , 299 h i g h e r o r d e r terms i n p e r t u r b a t i o n e x p a n s i o n , 300 isotropic interactions c a l c u l a t i o n , 299 Κ factors c o m p a r i s o n f o r hydrogen s u l f i d e - p e n t a n e s y s t e m , 308f comparison f o r methyl i o d i d e - a c e t o n e s y s t e m , 309f comparison f o r s u l f u r d i o x i d e - a c e t o n e s y s t e m , 307f Lennard-Jones i s o t r o p i c i n t e r a c t i o n s c a l c u l a t e d , 299 m i x t u r e p r o p e r t y p r e d i c t i o n from pure-component p a r a m e t e r s , 305

Perturbed-anisotropic-chain

theory

—Continued parameters f o r no η p o l a r f l u i d s , 302 phase e q u i l i b r i a c o m p a r i s o n f o r a cet o n e - c y c l o h e x a n e s y s t e m , 31 Of pure f l u i d s t e s t e d , 3 0 1 - 3 0 2 pure-component f l u i d parameters o b t a i n e d , 301t pure-component p a r t i t i o n f u n c t i o n e x t e n s i o n , 300-301 s a t u r a t e d l i q u i d m o l a r volume c o m p a r i s o n , 304f thermodynamics of m u l t i p o l a r m o l e c u l e s , 297-311 vapor p r e s s u r e c o m p a r i s o n of d i p o l a r f l u i d s , 303f Perturbed-hard-chain theory approximations for s h o r t - c h a i n m o l e c u l e s , 202 e q u a t i o n - o f - s t a t e model b a s i s , 228 p r o p e r t y p r e d i c t i v e a b i l i t i e s , 298 Perturbed-soft-chain theory, d i f f e r e n c e s from h a r d - c h a i n t h e o r y , 298 Petroleum r e s e r v o i r s , water i n c l u s i o n i n phase b e h a v i o r s t u d i e s , 494 Phase b e h a v i o r , p r e d i c t e d by PFGC e q u a t i o n , 462 Phase diagram a r g o n - w a t e r s y s t e m at c r i t i c a l r e g i o n , 450f e t h e n e - n a p h t h a l e n e s y s t e m , 384f t e t r a f luorom ethane-alkane s y s t e m , 386f Phase d i s t r i b u t i o n c o e f f i c i e n t s c o m p u t a t i o n of f e e d m i x t u r e s e p a r a t i o n i n t o two p h a s e s , 478 d e f i n i t i o n , 477 i n i t i a l values v a r i e d f o r mixtures i n f l a s h c a l c u l a t i o n s t u d y , 483 Newton-Raphson a l g o r i t h m employed, 489 single-stage flash calculations u p d a t e d , 479 t r i v i a l s o l u t i o n convergence a f t e r i n i t i a t i o n , 492 Phase e q u i l i b r i a behavior new m i x i n g r u l e a p p l i e d t o b i n a r y system, 575f,577f p r e d i c t e d behavior f o r b i n a r y s y s t e m u s i n g new m i x i n g r u l e , 578f,579f t e r n a r y phase b e h a v i o r u s i n g new m i x i n g r u l e , 58lf c a l c u l a t i o n d i f f i c u l t i e s at h i g h p r e s s u r e s , 371 data, s c a r c i t y for high molecular w e i g h t h y d r o c a r b o n s , 297 e q u a t i o n - o f - s t a t e c a l c u l a t i o n , 371 PFGC e q u a t i o n r e p r e s e n t a t i o n , 452-470

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

594

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

Phase e q u i l i b r i u m — C o n t i n u e d p r o p e r t i e s , algorithm developed f o r c u b i c e q u a t i o n s , 573-574 t h e o r i e s , f a i l u r e f o r high molecular w e i # i t h y d r o c a r b o n s , 297 Phase mole f r a c t i o n s , d e t e r m i n a t i o n f o r single-stage flash c a l c u l a t i o n s , 478 Polar f l u i d s cubic equation-of-state a p p l i c a t i o n , 434-451 equation-of-state application to n o n s p h e r i c a l m o l e c u l e s , 244 l o c a l c o m p o s i t i o n model a p p l i c a t i o n , 250-275 Polar s u p e r c r i t i c a l f l u i d properties c o n v e n t i o n a l method i n a d e q u a c i e s , 574 l o w - p r e s s u r e range e x a m i n e d , 574 new m i x i n g r u l e a p p l i e d , 574 P r e s s u r e c o m p o s i t i o n diagram h y d r o g e n - c a r bo η monoxide s y s t e m , 382f h y d r o gen-methane s y s t e m , 379f t et r a f l u o r cm et h a n e - b u t a n e s y s t e m , 386f Pressure composition l o c i , c a l c u l a t e d f o r San Andres Formation o i l s , 417,425 Près sure-temperature diagram, h y d r o gen-met hane s y s t e m , 3 8 l f P r o p a d i en e a n a l y s i s i n v a r i o u s compounds, 89t p r o p e r t i e s i n pro p a n e - p r o pa d i en e e x p e r i m e n t , 94t vapor p r e s s u r e v a l u e s i n propane-propadiene experiment compared, 94f Propane c a l c u l a t e d volume e r r o r u s i n g R e d l i c h - K w o n g p a r a m e t e r s , 399t relative volatilities i n p r o p a d i ene, 1 04t vapor pressure values i n pro pan e - pro pa d i ene e xperim ent c o m p a r e d , 91 f Pro p a n e - p r o p a d i ene s y s t e m b i n a r y i n t e r a c t i o n p a r a m e t e r s , 99t c e l l and t r a n s d u c e r u s e d , 88f e q u a t i o n s u s e d , 87 e x p e r i m e n t a l c o n d i t i o n s , 87-89 f u g a c i t y c o e f f i c i e n t e x p r e s s i o n , 97 i n t e r a c t i o n p a r a m e t e r s , 96f liquid compressibility factors i n e x p e r i m e n t , 98 n o n l i n e a r b e h a v i o r of r e l a t i v e v o l a t i l i t i e s , 98 phase e q u i l i b r i a , 86-105 p r e s s u r e d a t a , 90t p r e s s u r e measurements used t o d e t e r m i n e p a r a m e t e r s , 89-93 relative volatilities compared, 102f-104f

Pure f l u i d s , van d e r Waals p a r t i t i o n f u n c t i o n u s e d , 181-189

Q

Quantum c o r r e c t i o n , van der W a a l s - t y p e e q u a t i o n of s t a t e , 372-373 Quantum c o r r e c t i o n f u n c t i o n , e x p a n s i o n c o e f f i c i e n t , 373t

R

Radial d i s t r i b u t i o n functions, s o f t - s p h e r e m i x t u r e , 334f Random m i x i n g a p p r o x i m a t i o n (RMA) , mixture r a d i a l d i s t r i b u t i o n f u n c t i o n s , 319-320 Raoul t ' s l a w , Κ v a l u e s i n i t i a t e d , 477 Real f l u i d s h a r d s p h e r e s as model b a s i s , 227 P - V - T b e h a v i o r , 238-244 Redlich-Kwong e q u a t i o n a t t r a c t i o n parameter, 374 b i n a r y system r e s u l t s , 378 c a l c u l a t e d volume e r r o r from parameters f o r c a r b o n d i o x i d e , 398t description of binary s u p e r c r i t i c a l p r o p e r t i e s , 171 d i s a d v a n t a g e , 380 g e n e r a l i z e d , component p r o p e r t i e s f o r San Andres o i l s , 409t m i x i n g r u l e s , 326,327t,374 m i x t u r e phase b e h a v i o r o f San Andres F o r m a t i o n o i l s , 406-433 parameters d e f i n e d , 134, 390, 408 pro p a n e - p r o p a d i ene s y s t e m , 95-98 Soave m o d i f i c a t i o n , 390 t em per at u r e - d e pen d e n t , 390 t e m p e r a t u r e s above c r i t i c a l , 390-3 9 5 used i n San Andres o i l s s t u d y , 408 P e n g - R o b i n s o n e q u a t i o n compared, 533 phase e q u i l i b r i a e q u a t i o n , 372 solubility b e n z o i c a c i d i n carbon d i o x i d e , 1 72f f l u o r i n e i n e t h y l e n e , 172f three-phase e q u i l i b r i a d e s c r i p t i o n of San Andres o i l s , 419 van d e r Waals e q u a t i o n c o m p a r i s o n , 81 Relative v o l a t i l i t i e s , nonlinear b e h a v i o r of pro p a n e - p r o p a d i ene s y s t e m , 98

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

595

INDEX

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

Residual functions definition, 4 H e l m h o l t z e n e r g y d e r i v a t i o n , 9t pressure-explicit equation-of-state d e r i v a t i o n , 7t r e l a t i o n s h i p t o excess f u n c t i o n s , 12 volume-explicit equation-of-state d e r i v a t i o n , 8t Residual Helmholtz energy, r o l e i n e q u a t i o n - o f - s t a t e thermodynamics, 5

S o l i d volume, d a t a and s o u r c e s f o r v a r i o u s compounds, 1 63t S o l u b i l i t y , deviations for binary m i x t u r e s from e q u a t i o n s of state, 166-I69t Solute saturated vapor, e q u a t i o n - o f - s t a t e c a l c u l a t i o n , 161 S o l u t i o n thermodynamics, p a r t i a l - p r o pert y c o n c e p t , 13 Square-well fluids c o o r d i n a t i o n number models compared, 183-184 c o o r d i n a t i o n number models f o r S v a r i o u s e q u a t i o n s of s t a t e , I85t deviation in compressibility f a c t o r , I87t San Andres F o r m a t i o n o i l s l o c a l c o m p o s i t i o n s compared w i t h a c i d g a s - o i l mixture L e n n a r d - J o n e s f l u i d s , 220t d e s c r i p t i o n , 425-43 0 p r e s s u r e c a l c u l a t i o n e q u a t i o n , 235 c a l c u l a t e d and e x p e r i m e n t a l r e s u l t s S q u a r e - w e l l m i x t u r e s , s i m u l a t e d number compared, 41 7t of n e a r e s t n e i g h b o r s , 2211 c o m p a r i s o n i n s o u r gas s t u d i e s , 427t Square-well molecules c o m p o s i t i o n data p r e p a r a t i o n f o r b u l k c o m p o s i t i o n e f f e c t on l o c a l R e d l i c h - K w o n g a p p l i c a t i o n , 410 c o m p o s i t i o n , 219 equation-of-state parameters l o c a l composition a d j u s t e d , 41 4t changing with bulk f l u i d c o m p o s i t i o n s compared, 413t c o m p o s i t i o n , 223f Redlich-Kwong equation a p p l i e d t o f u n c t i o n of d e n s i t y , 22Of phase b e h a v i o r , 406-433 f u n c t i o n o f t e m p e r a t u r e , 222f Saturated l i q u i d d e n s i t i e s , Monte C a r l o s i m u l a t i o n , 2 1 4 - 2 2 5 P e n g - R o b i n s o n and P a t e l - T e j a model c o m p a r i s o n at v a r i o u s c o m p a r e d , 442t d e n s i t i e s , 224 S a t u r a t e d l i q u i d volume, d e v i a t i o n Monte C a r l o s i m u l a t i o n contours for cubic com p a r i s o n , 218-219 e q u a t i o n s , 553f, 554f new l o c a l c o m p o s i t i o n model Scaled p a r t i c l e theory, r i g i d p r o p o s e d , 224 n o n s p h e r i c a l m o l e c u l e s , 228 t e m p e r a t u r e e f f e c t on l o c a l S c a l i n g l a w s , pure f l u i d s a p p l i e d t o c o m p o s i t i o n , 219 f l u i d m i x t u r e s , 116-118 u n l i k e - e n e r g y f a c t o r e f f e c t on l o c a l S c a l i n g v a r i a b l e s , one-component c o m p o s i t i o n , 219 f l u i d , 1l4f Square-well potential Schmidt-Wenzel e q u a t i o n , s u p e r c r i t i c a l c o n t a c t v a l u e , 232 s o l u b i l i t y d e s c r i p t i o n , 164 c o n t a c t v a l u e s of c o r r e l a t e d Second v i r i a l c o e f f i c e n t , p r e d i c t e d v s . f u n c t i o n s , 232-23 8 c o m p o s i t i o n f o r b i n a r y s y s t e m , 577f d e f i n i t i o n , 229 Soave-Redlich-Kwong equation e q u a t i o n , 215 mixture pressure d e v i a t i o n s f o r Gaussian o v e r l a p f o r a n i s o t r o p i c o n e - p a r a m e t e r f i t , 96f m o l e c u l e s , 23Of mixture pressure d e v i a t i o n s f o r G a u s s i a n o v e r l a p m o d e l , 230 two-parameter f i t , 99f See a l s o O n e - s t e p s q u a r e - w e l l parameter i n f l u e n c e o n s a t u r a t e d potential l i q u i d volume See a l s o T w o - s t e p s q u a r e - w e l l p r e d i c t i o n , 550-552f potential parameter v a l u e s , 134 S t a b i l i t y l i m i t s , mixtures i n f l a s h supercritical solubility c a l c u l a t i o n s t u d y , 482f d e s c r i p t i o n , 164 S t a t i s t i c a l mechanics Solid-fluid equilibria d i f f i c u l t i e s i n equation-of-state b i n a r y systems s t u d i e d , 380 a p p l i c a t i o n s , 201-202 d i s c u s s e d f o r b i n a r y m i x t u r e , 376-377 f l u i d m o d e l i n g , 180 equation-of-state application to h i s t o r y of equation-of-state c r y o g e n i c m i x t u r e s , 371-385 a p p l i c a t i o n s , 201

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

596

EQUATIONS OF STATE: THEORIES AND

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

S t a t i s t i c a l mechanics—Continued m i x i n g r u l e s , 31 7-31 8 Strobridge equation c o e f f i c i e n t s and t h e i r e r r o r s , 65t form used i n t e t r a f l u o r omet hane e x p e r i m e n t , 63 Successive s u b s t i t u t i o n algorithms, convergence t o t r i v i a l s o l u t i o n , 489 Supercritical extraction a p p l i c a t i o n s , 156,352 c a l c u l a t i o n summary f o r v a r i o u s e q u a t i o n s of s t a t e , 170t c r i t i c a l p r o p e r t i e s used i n s t u d y , 1 60t e q u a t i o n s of s t a t e e v a l u a t e d , 156-180 e q u a t i o n s of s t a t e s e l e c t e d , 157 r e s u l t s i n b i n a r y m i x t u r e s s t u d y , 164 Supercritical solubility c a l c u l a t i o n , 161-164 data sources for various compounds, I65t e q u a t i o n - o f - s t ate u s e f u l n e s s , 157 S u p e r c r i t i c a l s t a t e s , d e s c r i b e d by e q u a t i o n s of s t a t e , 109-178

Τ

T e r n a r y s y s t e m s , new m i x i n g r u l e performance d e s c r i p t i o n , 576-580 T et r a f l u o r omet hane a p p a r a t u s used i n e x p e r i m e n t , 63 density values from Strobridge e q u a t i o n , 67t e q u a t i o n of s t a t e , 60-82 i s o t h e r m s s t u d i e d , 61 PVT p r o p e r t i e s , 61 thermodynamic b e h a v i o r , 60-61 Thermodynamic m o d e l i n g a p p l i c a t i o n r e q u i r e m e n t s , 182 a p p r o a c h e s , 180 T h r e e - p h a s e s y s t e m , water a d d i t i o n t o h y d r o c a r b o n s y s t e m , 509 T r i v i a l solution convergence i n f l a s h c a l c u l a t i o n s t u d y , 483 m i x t u r e s o c c u r r e n c e , 476 T r o u t o n ' s law c o r r e c t i o n f o r d i f f e r e n c e s i n vapor m o l a r volume, 521 i n t e r p r e t a t i o n s , 52 0-53 3 Two-phase e q u i l i b r i u m g o v e r n i n g e q u a t i o n s , 497 p r e d i c t i o n by i n i t i a l f l a s h c a l c u l at i ons , 50 6 Two-phase f l a s h c a l c u l a t i o n s , p r o c e d u r e , 497 -500 Two-phase s y s t e m , w a t e r a d d i t i o n t o h y d r o c a r b o n s y s t e m , 50 9

APPLICATIONS

Two-step square-well p o t e n t i a l a n g l e - d e pendent G a u s s i a n o v e r l a p , e q u a t i o n , 238 a n g l e - d e pen dent p a r a m e t e r s , 234f n o n s p h e r i c a l m o l e c u l e s , 230

U

Unlike-interaction potential p a r a m e t e r s , e x p r e s s i o n s , 317 Unlike-pair coefficients a p p l i c a t i o n t o projected binary c r i t i c a l l o c i , 138 o b t a i n e d f o r Soave e q u a t i o n f o r b i n a r y m i x t u r e s , 152 obtained for Teja equation for binary m i x t u r e s , 152

V

van d e r Waals e q u a t i o n d i s a d v a n t a g e , 81 l i q u i d d e n s i t y e x p r e s s i o n at z e r o p r e s s u r e , 28 m i x i n g r u l e s , 324, 32 5t p o l a r asymmetric s y s t e m d e s c r i p t i o n , 355 terms d i s c u s s e d , 533 van der Waals o n e - f l u i d m i x i n g r u l e s a p p l i e d t o a l l d e n s i t i e s , 192 compared w i t h m o d i f i e d m e a n - d e n s i t y a p p r o x i m a t i o n , 347t compared w i t h random m i x i n g , 3 5 9 - 3 6 0 Lennard-Jones mixture r e s u l t s , 344t-345t p r e d i c t i v e and c o r r e l a t i v e power, 360 square-well configuration p r e d i c t i o n , 362f van d e r Waals p a r t i t i o n f u n c t i o n accuracy for square-well f l u i d s , 187-189 advantages i n e q u a t i o n - o f - s t a t e d e v e l o p m e n t , 189 c o n f i g u r a t i o n a l e n e r g i e s d e f i n e d , 191 c o o r d i n a t i o n number model u s e d , 184 e q u a t i o n , 1 81 fundamental e q u a t i o n o f s t a t e o b t a i n e d , 182 m i x t u r e s e q u a t i o n , 189 v o l u m e t r i c e q u a t i o n of s t a t e o b t a i n e d , 182 van d e r Waals t h e o r i e s — S e e N - f l u i d theories Vapor-liquid equilibrium c y c l ο he xan e - he x a n o l s y s t e m ,

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

44 7f

597

INDEX

Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ix002

Vapor-liquid equilibrium—Continued l o c a l composition mixing r u l e s a p p l i e d to various s y s t e m s , 270-271f , 2 7 3 - 2 7 4 f l o w - p r e s s u r e r e g i o n s t u d i e d f o r new a p p l i c a t i o n , 576 p r e d i c t e d by PFGC e q u a t i o n , 462 p r e d i c t i o n from l i q u i d - l i q u i d phase s e p a r a t i o n s , 385 p r o p a n e - p r o p a d i e n e system values, 100t,102t t o l u e n e - p e n t a n o l s y s t e m , 447f v a l u e s o b t a i n e d from c u b i c e q u a t i o n s o f s t a t e , 537 vapor-pressure representation i n c u b i c e q u a t i o n s s t u d y , 540 Vapor p r e s s u r e s A n t o i n e e q u a t i o n used f o r es t i m a t i n g , 52 5 compared f o r two e q u a t i o n s of s t a t e , 460t data sources for various compounds, 1 62t e f f e c t of a l p h a c o n s t a n t on c u b i c e q u a t i o n s of s t a t e , 436t P e n g - R o b i n s o n e q u a t i o n f a i l u r e , 561 PFGC e q u a t i o n , 457 p r e d i c t i o n s from PFGC e q u a t i o n f o r isopentane, 46lf Variational inequality minimization t h e o r y — S e e VIM t h e o r y

VIM t h e o r y , a p p l i e d t o t et r a f l u o r omet hane, 79 V i r i a l c o e f f i c i e n t — S e e Second v i r i a l coefficient V i r i a l equation a n i s o t r o p i c m o l e c u l e s , 230-232 mixing r u l e d e v i a t i o n for f l u i d s , 317-318 V o l u m e — S e e Excess volumes Volumetric properties c u b i c e q u a t i o n s r e p r e s e n t a t i o n , 538 hydrogen-methane s y s t e m , 3 8 l - 3 8 2 f new m i x i n g r u l e p e r f o r m a n c e , 576

W Water a d d i t i o n t o two-phase hydrocarbon s y s t e m , 509 a d d i t i o n t o s i n g l e - p h a s e hydrocarbon s y s t e m , 507, 50Qt Whiting-Prausnitz expression, free e n e r g y model f o r pure f l u i d s , 2 54 Wilson equation, l a t t i c e theory d e r i v a t i o n r e s u l t s , 356 Wilson formula comparison t o e v a p o r a t i o n t e c h n i q u e , 504 Κ v a l u e e s t i m a t e s , 498

Production and indexing by Keith B. Belton Jacket design by Pamela Lewis Elements typeset by Hot Type Ltd., Washington, D.C. Printed and bound by Maple Press Co., York, Pa.

In Equations of State; Chao, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

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