Equations of State Theories and Applications - trl.lab… Rules... · In the present report we...

Equations of State:Theories and Applications

K.C. Chao and R.L. Robinson, Jr., Editors, Developed from a symposium sponsored by theDivision 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

Publication Date:

1986

Mixing Rules for Cubic Equations of Stateby

G.Ali Mansoori

Chapter 15of

Mixing Rules for Cubic Equations of State

G.Ali Mansoori

Department of Chemical Engineering, University of Illinois at Chicago

(M/C 063), Chicago, IL [email protected]

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.

There exists a wealth of information in the literature abouiy--)cubic equations of state applicable to varieties of fluids o '-chemical and engineering interest. A !though cubic equations of state are generally empirical modifications of the van der Waals equation of state, they have found widespread applications in process design calculations because of their simplicity. Extension of their applicability to mixtures is generally acieved by introduction of mixing rules for their parameters. Mixing rules are expressions relating parameters of a mixture equation of state to pure fluid parameters through, usually, some composition dependent expressions. Except for the van der Waals equation of state the mixing rules for cubic equations of state are empirical expressions. In the present report we introduce a statistical mechanical conformal solution technique through which we can derive varieties of sets of mixing rules applicable to cubic equations of state. This pressure, energy, and compressibility equations of statistical mechanics. In Part II of the present report we introduce the conformal solution theory of polar fluid mixtures (1) and its relationship to the idea of mixing rules. In Part Ill we introduce the concept of the conformal

Chapter 15 of , 1986

MANSOORI Mixing Rules for Cubic Equations of State 315

solution mixing rules and we produce different sets of mixing rules based on different approximations for the -mixture radial distribution functions. In Part IV we reviewthe existing forms of the cubic equations of state formixtures and the deficiencies . of their mixing rules andcombining rules. Finally, in Part IV we introduce guidelinesfor the use of conformal solution mixing rules andcombining rules in equations of state and we demonstrateapplication of such mixing rules and combining rules forthree representative cubic equations of state.

II. Conformal Solution Theory of MixturesConformal solutions refer to substances whose intermolecular potential energy function, flij• are related toeach other and to those of a reference fluid, usually designated by sub-script (oo), according to (1...2.)

( 1)

For substances whose intermolecular potential energy function can be represented by a n equation of the form

(2)

and for which exponents m and n are the same as for the reference substance, conformal parameters fij and hij willbe defined by the following relations with respect to the intermolecular potential energy parameters Eij and Lif

(3)

Thus the configurational thermodynamic properties of a pure substance of type (a) are related to those of the reference substance according to the following relations:

Pa(V, T) = (faa1haa)P0

(V/haa• T/faa>

Sa(V, T) = S0

(V/haa• T/faa> + Nk!nh aa

and

(4)

(5)

(6)

(7)

(8)

where F, P, S, G, and H are the Helmholtz free energy, pressure, entropy, Gibbs free energy, and enthalpy,


316 EQUATIONS OF STATE: THEORIES AND APPLICATIONS

respectively. According to the above equations, all the thermodynamic properties of substance (a) can be expressed in term s of the properties of a reference puresubstance (o) through the conformal parameters faa and haa· The conformal solution treatment of fluids composed of polar molecules. is more complicated than for non-polar fluids. This is mainly due to electrostatic interactions which cause a departure of the intermolecular potential from spherical sym metry. The electrostatic potential between two otherwise neutral molecules arises from

permanent asymmetry in the charge distribution within the molecules. For any pair of localized charge distribution, the mutual electrostatic interaction energy c an be written in terms of an infinite series of inverse powers of separation of any two points. For no overlap between the charge distributions the series converges(l_). Thus the true pair-potential of polar molecules is orientation-dependent and is the sum of dispersion force as well as electrostatic interactions. In order to extent utility of the above form ulation of the conformal solution theory t o polar fluids we have proposed the following angle-averaged potential function for polar molecular interactions which represents the first order contribution to the anisotropic forces (1)

flij(r,T) = KEij[(oilr) n- (oij/r) m] - µ?µ//(3kTr 6)

+ 7µi4µ/1[ 4 50(kT) 3r 12 1 - (µ?Q/+µ/a12)/(2kTr 8)

- a-2Q-2/(1 4kTr 10) - (oc-µ- 2 +oc-µ- 2)tr6 (9) l J • lj J l

where K = [ n/(n- m)](n/ m)ml(n-m), and where µi, Qi, andoci are the dipole moment, quadrupole mom ent, and polarizability of molecule i, respectively. For a polar fluid, whose intermolecular potential energy function can be represented by eq. 9 the conformal parameters f aa and haa will have the following forms:

haa = (LaaCT,r)/L0 0(T,r)] (10)

where Eij(T,r) = KE1jAij(T,r)[Hi/T,r)]n/ m

Lij(T,r) = oij[Hij(T,r)r l/ m

Hij(T,r) = [Cij(T,r)/Ai/T,r)]ml(n-m)

Aij(T,r) = 1 + 7µ14µj 4/[1 800(kT) 3r lZ-mo1j"KEij]


15. MANSOORI Mixing Rules for Cubic Equations of State

and Cij(T,r) = 1 + µi2 µl/[12 1clr 6-moijmKEij1

+ (7/20)Qi2 o/t[Hr 10-moij6Keij

+ (µ•2Q.2+µ.2 a.2)/[8kTr 8-m0 .. mKE·.]i J J i iJ iJ

+ (cx-µ-2+cx-µ-2)/[4r6-mo·· mKE··]i J J i iJ iJ

317

The basic concept of the CST of mixtures is the same as for pure fluids, except that faa and haa in eqs.4-8 should be replaced with fxx and hxx• the mixture conformal parameters, as given below

( 11 )

Eqs.11 are called the conformal solution mixing rules. Functional forms of these mixing rules will be different for different theories of mixtures as it will be demonstrated later in this report. In the formulation of a mixture theory we also need to know the combining rules for unlike-interaction potential parameters which are usually expressed by the following expressions

h··=(1- l··H( h--113 +h--113)12]3 (12) iJ lj 11 JJ

where k ij and tij are adjustable parameters.

Ill. Statistical Mechanical Theory of Mixing Rules The most important requirement in the development of the CST of mixtures are mixing rules. In the discussion presented here we have introduced a new technique to re-derive the existing mixing rules and derive a number of new mixing rules some of which are density- and temperature-dependent. According to statistical mechanics the macroscopic thermodynamic properties of a pure fluid are related to its microscopic molecular characteristics by the following three equations (3, 4)

00

u = uig + 2rrpJ ;(r)g(r)r2dr. 0 00

P = pRT + (2/3)rrpJr;·(r)g(r)r2dr 0

KT= 1/pRT - (4TT/RT)J[g(r)-1]r2dr 0

( 13)

( 14)

( 15)

where u is the internal energy, P is the pressure and KT is the isothermal compressibility, ;(r) is the pair intermolecular potential energy function, and g( r) is the



radial (or pair) distribution function. Eqs.13-15 are commonly called the energy equation, the virial (or pressure) equation, and the compressibility equation, respectively. For a multicomponent m ixture these equations assume the following forms (3-5)

00

u = uig + 2rrp}:i}:jxixjJflij(r)gij(r)r2 dr 0

00

P = pRT + (2/3)rrp}:i}:jxixjJrfl'ij(r)gij(r)r2dr 0

icT = ( 1 /pRT)IBI/ }:i�jxixjlBhj

( 16)

( 17)

( 18).

In the above equations summations are over all the (c) components of the mixture, xi and Xj are the mole fractions, and IBI is a cxc determinant with its representative terms in the following form

00

Gij = 4rrJ[gij(r)-1Jr2dr 0

where 6ij is the Kroneeker delta, and IBlij is the cofactor of term Bij in determinant IBI. Eqs. 13-18 can be used in the . , manner presented below in order to derive mixing rules ( )based on different mixture theory approximations:

Ill. I. One-Fluid Theory of Nixing Rules: For the development of one-fluid mixing rules we introduce a pseudo-pure fluid which can represent the configurational properties of a mixture provided that the pseudo-pure fluid and the mixture molecular interactions obey eq.1. By replacing eq.1 in eqs.13, 14, 17, and 18 and then equating configurational internal energy, pressure, and isothermal compressibility of the pseudo-pure fluid and the mixture we will obtain the following equations

f xxhxxf fl oo< y)�o< y )y2dy=}:i}:jxix/ ijhijJ fl oo< y )9t/ y)y2dy ( 19)

f xxhxxf Yfl' oo< Y >�o< y)y2dy=}:i}:jxixjf ijhij J Yfl' oo< Y)9jj ( y)y2dy (20)

{ 1-4rrph xxJCg00( y)-l]y2dy }-1 =}:i}:jxixj IBlij/l BI ( 21)

It should be pointed out that for the case of the hard-sphere fluid eq.19 vanishes, eq.21 remains the same, while eq.20 reduces to the following form

(22)


(

15. MANSOORI Mixing Rules for Cubic Equations of State 319

Solution of eqs. 19-21 should produce the two necessary expressions (mixing rules) relating fxx and hxx of the pseudo-pure fluid to fij and hij of components of the mixture. For this purpose we should use an approximation technique relating the radial distribution functions (RDF) in the mixture to the pure reference fluid RDF. However, at a first glance it seems that we have in our hand three equations and two unknowns. As it will be demonstrated below for most of the approximations of the mixture RDFs which are used here these three equations produce two mixing rules. In the previous investigations for the development of mixing rules (5-11) all the investigators have used only eq. 19 and/or eq.20. Our studies indicate that while eqs. 19 and 20 are essential in the development of mixing rules, eq.21 can add a new dimension which could be significant in the calculation of properties of mixtures. In what follows different approximations will be used for relating gij to g00 in order to derive different sets of mixing rules.

111.1.i. Random Mixing Approximation (RMA) for Mixture RDFs: In this approximation it is assumed that the non-scaled RDF of all the components -of the mixture and the interaction RDFs are identical (�). i.e.

(23)

When this approximation is replaced in eqs.19-21, eq.21 will vanish and eq. 19 and 20 will produce the following mixing rules

l'xx<r) = l'.il'.jxixjltij(r)

lt'xx<r) = l'.il'.jxixjlt'ij(r)

(24)

(25)

For example, in the case of the Lennard-Jones ( 12-6) intermolecular potential function we will derive the following mixing ,rules (l.Z.) from eqs. 13 and 14. fxxhxx2 = l'.il'.jxix/ijhi? ( 26)

(27)

For a hard-sphere potential we will derive only one mixing rule through the RMA and that is derived by replacing eq.23 in 22. The resulting mixing rule will be

h 1/3=""-"'·x·x·h·· l /3 XX L.1L.J 1 J lJ (28)



111.1.ii. Conformal Solution Approximation (CSA) for Mixture RDFs: This approximation technique seems more logical for use in the development of mixing rules than RMA. According to this approximation the scaled RDFs in a mixture are all identical (�), i.e.

g11<Y> = g22 <v> = •• • =

gij<Y> = • • •(29)

When we use this approximation in eqs.19 and 20 they both produce the same mixing rule which is

fxxhxx = }:i}:jxix/ijhij (30)

Now, by replacing eq. 29 in 21 an additional mixing rule will be produced which is the following

(31)

where IB*hj = xi[bij + x/hi/hxxH.PRTICTxx-1 )]. Eq.30 isactually the second van der Waals mixing rule which is well known, but eq.31 is a new mixing rule for hxx which is replacing the first van der Waals mixing rule. This new mixing rule, in principle, is a composition-, temperature-, and density-dependent mixing rule. This is because "Txx c ~

)which appears in the right and left hand sides of this equation is generally temperature- and density-dependent. For example, for a binary mixture eq.31 can be written in the following form (�_)

hxx = {}:i}:jxixjhij + X1X2(h11h22-h12 2)(pRTic:Txx-1) }/

{1+x1x2(h11+h22- 2h 12 HpRTic:Txx-1 )} (31-1)

By using the hard-sphere potential (by replacing eq. 29 in 22) we will derive the following mixing rule

(32)

This mixing rule is the first van der Waals m1xrng rule which, in conjunction with eq.30 is usually used for calculation of mixture thermodynamic properties (7,8, 10, 11 ). It should be pointed out that eq.32 constitutes another mixing rule for hard-sphere mixtures. As a result, while the CSA approximation produces two mixing rules for potential functions with two parameters, it also produces two mixing rules for a hard-sphere potential which is a one-parameter potential function.


(

15. MANSOORI Mixing Rules/or Cubic Equations of State 321

111.1. iii. Hard-Sphere Expansion (HSE) Approximation for Mixture RDFs: It is demonstrated that the RDF of a pure fluid (x) can be expanded around the hard-sphere (hs) RDF in the form (�)

9xx<Y)= ghS (y)+ (fxxllo*>g1(y)+ (fxxlTo*>2gz(y)+ ••• (33)

Let us also assume that we could make a similar expansion for RDFs in a mixture around the hard-sphere mixture RDFs as the following

The justification behind this expansion is given elsewhere (6, 9 ). Now by rep lacing eqs. 33 and 34 in either of eqs. 19 or 20 we will be able to derive the following two mixing rules by equating the coefficients of the second and third order inverse temperature terms of the resulting expression.

f xxhxx = }:i}:jxix/ijhij

fx/hxx = }:i}:jxix/ij2hij

(35)

(36)

These mixing rules are used for calculation of excess properties of a mixture over the hard-sphere mixture (U) at the same thermodynamic conditions rnJ. Application of the HSE approximation in eq.21 will not produce any additional mixing rule.

Ill. 1. iv. Density Expansion (DEX) Approximation for Mixture RDFs: It has been demonstrated that the RDF of a pure fluid can be expanded around the dilute gas RDF, exp[-fl(r)/kT], in the form (11.)

gxx<Y) = [1 + Fxx<Y)] exp[-flxx<r)/kT] (37)

Let us also assume that we could make a similar expansion for RDFs in a mixture around the dilute gas mixture RDFs as the following

(38)

Now by replacing eqs.37 and 38 in eq.19 and after a number of algebraic manipulations we will derive the following mixing rule

f xxhxx=}:i}:jxixjf ijhij { 1-( fij/f xx-1 )[u-uig )/kl

(39)



The latter m1xrng rule can be used, joined with another mixing rule, for calculation of mixture properties. Similar approximations can be used in order to derive other mixing rules from the virial and compressibility equations.

111. 2. Multi-Fluid Theory of Mixing Rules: The basic assumption in developing the multi-fluid mixing rules is the same as the one-fluid approach except that in this case we will search for a hypothetical multicomponent ideal mixture which could represent the configurational properties of a multicomponent real mixture, both with the same number of components and at the same thermodynamic conditions. In this case eqs.19-21 will be replaced by the following set of equations

f xihxif fl oo< Y >9oo< y)y2 dy=}:jx/ijhijJ fl oo ( Y )9jj ( y)y2dy ( 40)

f xihxif Yfl' 00( y)9o0( y)y2dy=}:jxjf ijhij J Yfl' 00( y )9jjC y )-/dy ( 41)

{ 1-4rrph xiJlg00( y)-1 J-/dy }-1 =2'.jxj IBli/1B1 ( 4 2)

Expressions for Bij and Gij will be the same as in eq. 18. In the case of the hard-sphere fluid eq.40 will reduce to the following form

( 40-1)

eq.41 will vanish and eq.43 will remain the same.

111.2. i. Average Potential Model (APM) for Mixture RDFs: In this approximation it is assumed that (�_),

( 43)

When this approximation is replaced in eqs.40-42, eq.42 will vanish and eq.40 and 41 will produce the following mixing rules

flxi(r) = LjXjflij(r)

fl'xi(r) = LjXjfl'i/r)

For example, in the case of the intermolecular potential function following mixing rules(.1.ZJ .

f xihxi2 = LjXjfijhij2

( 44)

( 45)

Lennard-Jones (12-6) we will derive the

( 46)


( 15. MANSOORI Mixing Rules for Cubic Equations of State 323

( 47)

For a hard-sphere potential we will derive only one mixing rule

h . 1 /3= °" ·X·h .. 1/3 XI L. J J IJ (48)

111.2.ii. Multi-fluid CSA Approximation for Mixture RDFs: According to this approximation the scaled RDFs in a mixture are related as the following

(49)

When we use this approximation in eqs.40 and 41 they both produce the same mixing rule which is

(50)

Now, by replacing eq.49 in 42 an additional mixing rule will be produced which is the following

IB*ltpRlKTxi = 2'.jxjlB*lij ( 51)

( where

IB*lij = xi{6ij + (xjhi/2 H(pRTKTxi-1 )/hxi+(pRlKTxrl )/hxj1l

Eq.50 is actually the second van der Waals multi-fluid mixing rule, but eq.51 is a new mixing rule for hxi· By using the hard-sphere potential (by replacing eq.49 in 40- 1 ) we will derive the following mixing rule

(52)

This mi)!:ing rule is the first multi-fluid van der Waals mixing r.ule which, in conjunction with eq.50 is usually used for calculation of mixture thermodynamic properties. It should be pointed out that eq.52 constitutes another mixing rule for hard-sphere mixtures.

111.2. iii. Multi-Fluid HSE Mixing Rules: In a similar manner as the one-fluid case we can derive the following mixing rules

fxihxi = LjX/ijhij

f xi2hxi = LjX/i/hij

(53)

(54)

These mixing rules are used for calculation of excess


324 EQUATIONS OF S"IATE: THEORIES AND APPLICATIONS

properties of a mixture over the hard-sphere mixture.

111.2. iv. Multi-Fluid DEX Mixing Rules: In a similar manner as the one-fluid case we can derive the following mixing rule

fxihxi = }:jxjfijhij{l-(fi/fxi-1 )[ui-uig)/kT

+T(Cvi-Cvig)/(ui-uig)]} (55)

This mixing rule can be used, joined with another mixing rule, for calculation of mixture properties.

IV. Application of Mixing Rules for Cubic Equations of StateIn order to apply the varieties of the conformal solutionmixing rules which are introduced here for cubic and otherequations of state the following considerations should betaken into account:

(i) Conformal solution mixing rules are for the molecularconformal volume parameter, h, and the molecularconformal energy parameter, f.

(ii) Conformal solution mixing rules are applicable forconstants of an equation of state only. Before using a set ofmixing rules for an equation of state one has to express the c· \parameters of the equation of state with respect to the molecular conformal parameters h and f. This will then ·make it possible to write the combining rules and mixingrules for the equation of state. In what follows mixing rulesand combining rules for three representative cubicequations of state are derived and tabulated.

IV.1. Mixing Rules for the van der Waals Equation of State:The van der Waals equation of state (IT) can be written in the following form

Z = Pv/lH = v/(v-b)- a/vRT (56)

Parameter b of this equation of state is proportional to molecular volume (b0<h) and parameter a is proportional to (molecular volume)(molecular energy (a0<fh). Then, in order to apply the mixing rules introduced in this report for the van der Waals equation of state we must replace h with b and f with a/b in all the mixing rules. In Table I mixing rules for the van der Waals equation of state based on different theories of mixtures are reported. The combining rules for aij and bij (ioej) of this equation of state,consistent with eqs. 12 will be


(


Table I: Mixing Rules for the van der Waals Equation of State

One-Fluid Mixing Rules

a = [I12jXiXjaijbijJ 312/[212jXiXj8ijbij3 J 1 /2 --

RHA Theory{ ____________ b = [Li2jXjXj8jjbij3/2i2jXiXjllijbijJ 1 /2

-------------a = 212·x1x·ar

vdW Theory{ J J J

b = 2i2jXiXjb jj

a = 2i2jXjXjaij HSE Theory{

b = [Ii2jX jXjaijl 2 1Ii2jXiXjail/b ij

a = [avdw+(b /vRT) 2i2jX jXjaij2 1b ijl/[ 1+avdW/vRT]DEX Theory{

b = 2i2jX jXjbij

'

--------------------------------------------------- -----

a = 2i2jXjXjllij CSA Theory{

l+llxx= IB•Jt 2i2jXiXjlB•liji B•u=xi(6ij+X jllxxb i/b)

Multi-Fluid Mixing Rules----------------------------------------------------------

ai = [IjXjaijbijJ 3l2/[}:jXjaijbij3J 1 /2

APM Theory{ ____________ bi =

_[LjXjaijbij3/IjXjaijbij J 1 /2 ___________________a· = }:·x·a··

vdW Theory{ 1 J J lJ

b j = 2jXjbij

ai = 2jXjaij HSE Theory{

bi = [LjXjlljjl2l:Z:jXjail/b ij

ai = [a1vdw+(bjlvRT) 2jXja1ltb1jl/[ 1 +a ivdw/vRTJDEX Theory{

bj = 2jXjbij

ai = 2jXjaij CSA Theory{

1 +ll xi= IB• 11 2jXj IB· liji B. u=x1I6 u+x jbij ( llx/b j+llx/b j )/2)

flxx= pRTKTxx- 1 = [2a(v-b) 2-RTb( 2v-b)l/lRTv 2-za(v-b) 2]

t.x1= pRTqx1-1 = [2a 1Cv-b1)2-RTb 1(2v-b1)J/[RTv 2-za 1(v-b1)2 J


326 EQUATIONS OF STATE; THEORIES AND APPLICATIONS

{

( 1-k.ij )bij ( aiia jj/biibjj) 112;

( 1-tij )[ ( bii 1 /3+bjj 1 /3)/2] 3 (57)

IV.2. Mixing Rules for the Redlich-Kwong Equation of State:The Redlich-Kwong equation of state ( 16) which is an empirical modification of the van der Waals equation can be written in the following form

Z Pv/RT = v/(v-b) - a/[RT312(v+b)] (58)

Parameter b of this equation of state is proportional to molecular volume (b0<h) and parameter a is

2 proportional to

(molecular volume)(molecular energy)31 or (a0<f.h 312). Then, in order to apply the mixing rules introduced in this report for the Redlich-Kwong e(luation· of state we must replace h with b and f with (a/b)2 1 3 in all the mixing rules. In Table II mixing rules for the Redlich-Kwong equation of state based on different theories of

jmixtures are reported(-\

The combining rules for aij and bij (i"" ) of this equation o"--

;of state, consistent with eqs.12 will be the same as eqs.57.

IV.3. Mixing Rules for the Peng-Robinson Equation of State:The Peng-Robinson equation of state ( 17) which is anotherempirical modification of the van der Waals equation can bewritten in the following form

Z = Pv/RT = v/(v-b) - a(T)v/{RT[v(v+b)+b(v-b)]} (59)

Parameter b of this equation of state is a constant which is proportional to the molecular volume (b0<h). However, parameter a of the Peng-Robinson equation of state is not a constant and it is a function of temperature as the following.

(60)

where

8 = 0.37464+1.54226w-0.26992w 2



Table 11: Mixing Rules for the Redlich-Kwong Equation of State


a = [I i2jXiXjllij2/3 bij4/3 JS/2 /2i2jXiXjllij2 /3 b ij I 0/3 RHA Theory{

------------b

= [I i2jXjXjlli/l3 b;j 10/3 ILj2jXiXjll ij2 /3 bij 4/3 I 1 /2 _a = [I-2·x·x·a .. 2/3b .. 1/3 ]312 ;r2 ·I·x·x·b .. ] 1/2

vdW Theory{ 1 J 1 J 11 lJ 1 J 1 J JJ

b = 2i2jXiXjbij

a = [(2;2jXjXjaij2/3 b ij 1 /3 H2i2jXiXjllij 4/3 b;j-1 /3)] 1 /2HSE Theory{

b = [I i2jXiXjllij2/3 bij 1 /3 ]2 /I;2jXj Xjllij4/3 b ij- I /3 -----------------------------------------------

-----------

a = Li2jXiXjllij(a/b) l/3{1-[Caij/bij)2/3 (b/a)213-1 ]t) DEX Theory{

b = 2i2jXiXjbjj

a= [I;2jXiXjaij2/3bij1/3J312;b1/2

CSA Theory{ l+llxx= IB*lt 2i2jXiXjlB*hJ; B*u=

x;C6ij+Xj6xxbij/b)

Multi-Fluid Mixing Rules

a;= [(2jXja}l3b;j1/3)(2jXjllij4/3bij-1/3 )]112

HSE Theory{ b; = [IjXjaij2 /3 b;j 1/3 J2 /LjXjaij4/3bij-1/3

a;= 2jXjllij(a/b) 1/3{1-[(aij/bijJ2/3 (b; /a1)213-1]�;} DEX Theory{

bi = 2jXjbij

a;= [IjXjaij2 /3bijl/3J312 /bi1/2CSA Theory{

l+l:lxi= IB*I/ 2jXjlB*lij; B*u= x;C6ij+Xjbij[t.x;/b;+t.x/bjl)

t = (3/2)(a/bR)T 372 2n[v/(v+b)]-1/2; t; = (3/2Ha; /b; R)r-3/2 tn[v/(v+b 1)H/2 llxx= pRTKTxx-1 = -l+RT 3/2 (v2 -bt'.)2 /[RT{v(v+b)) 2-a(2v+b)(v-bl 2] t. x;= pRTqx;-1 = -l+RT 3l2(v2 -b12)2 /[RT{v(v+b;))2-ai(2v+bi)(v-b;)2J



and Tc and Pc are the critical point temperature and pressure, respectively; and w is the acentric factor. In order to utilize the statistical mechanical mixing rules for the Peng-Robinson equation of state we must first separate thermodynamic variables from constants of this equation of state. For this purpose we may write this equation of state in the following form

Z = Pv/RT = v/(v-b)- [(A/RT+C- 2(AC/RT)112]/

[(v+b)+(b/v)(v-b)] (61)

where A= ac< 1+8>2 and C = ac82/RT c· Thts new form of the Peng-Robinson equation of state indicates that there exist three _ indepen_dent _ constant parameters in this equation whkh ar·e- A, b, -and C. Parameter-s :_b - and C are:.p-roport:ioflal -to the molecular volume (bo<h and Co<h) while parameter A is proportional to (molecular volume)(molecular energy) or (Ao<fh). Based on different theories of mixtures mixing rules for this new form of the Peng-Robinson equation of state are reported in Table Ill. The combining rules for thee··\ unlike interaction parameters of this equation of state are as the following

(62)

(63)

(64)

Similar procedures to those demonstrated above can be used for derivation of conformal solution mixing rules for other cubic equations of state.


(


Table Ill: Mixing Rules for the Peng-Robinson Eg. of State


A = [}: i2jXjXjAijbij ]312 /[}: i2jXiXjA;jb ii 3 ) 1 /2 ---

RMA Theory{ b = [}:;2jXiXjAijb;i3t2;2jXtXjAijb;jl l /c

___________ C = [}: i2jXjXjAijCij3 /2;2jXiXjAijCij I l /2 _____________A = Li2jXiXjAij

vdW Theory{ b = Z:iZ:jXiXjbij C = 2i2jXjXjC ij

A= 2i2jXiXjAij{1-[(Aij/bij)(b/A)-1 JO DEX Theory{ b = Li2jXJXjbij

C = 2i2jXiXjC ij

Multi-Fluid Mixing Rules ----------------------------------------------------------

A; = [}:jXjA;jb;jJ312 t[}:jXjAjibjj 3J1/2 APM Theory{ b; = [}:jXjAjjb;i3l2jXjAijbijJll<" ____________ C; = _[LjXjAijC;i

3l2jXjAijCij J 1/2 ___________________A; = 2jXjAij

vdW Theory{ b; = Z:jXjbij C; = 2jXjCij

llxx= pRlKTxx-1 = -l+RT/{Rlv 2t(v-b)2-2Av3/(v2+b2)2} llxi = pRlqx;-1 = -l+RT/{RTv 2 /(v-b;)2-2A;v3 /(v2 +b;2)2} € = {[A-v'( ACRT) ]/ ( 2bRT ✓2)}en[ ( y+b-bv'2)/( v+b+b✓2)]

+✓(ACRT)/{2[../(ACRT)-AJ} € ;= {[A ;-✓( A1C 1RT) J/(2b 1RT ✓ 2)}.e,n[ ( v+b 1-b; ✓2)/( v+b ;+b;../2) l

+✓(A1C;RT)/{21-v'(A ;C1RT>-A1]}



Acknowledgment: The author thanks Professor Carol Hall for her helpful comments and corrections. This research is supported by the U.S. Department of Energy Grant No. DE-FG02-84ER13229.

Literature Cited 1. Massih, A.R.; Mansoori, G.A. Fluid Phase Equillbria

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 Ill, 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

Mixtures· 3rd Ed., Butterworths, Wolborn, Mass. 12. Scott, R.L. J. Chem. Phys. 1956, 25,193.

Liquid 1982. (_)

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 gas en

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. Fun dam. 1976,

15, 59.


Date post:	01-Sep-2018
Category:	Documents
Upload:	doanbao
View:	213 times
Download:	0 times

Equations of State Theories and Applications - trl.lab… Rules... · In the present report we...

Documents