Thermodynamics of semicontinuous mixtures using equations of state with group contributions

RglDPUF EQUIUBRIA ELSEVIER

Fluid Phase Equilibria 127 (1997) 29-44

Thermodynamics of semicontinuous mixtures using equations of state with group contributions Uwe Baer, Dieter Browarzik, Horst Kehlen

*

Institute of Physical Chemistry, Martin-Luther-University Halle-Wittenberg, Geusaer Strafle, 06217 Merseburg, Germany

Received 5 March 1996; accepted 9 July 1996

Abstract

In the chemical engineering design of industrial distillation columns data on the vapor-liquid equilibrium of complex mixtures are very important. Because of the expense of the experimental determination of such data, there is interest in their accurate prediction. Multicomponent mixtures such as petroleum or petroleum fractions contain a large number of similar chemical species. The method of continuous thermodynamics is suitable for the description of the vapor-liquid equilibria of these systems. Continuous thermodynamics is based on the application of a continuous distribution function for describing the composition of multicomponent mixtures. Coupled with activity coefficient models, this method is shown to be convenient for the prediction of vapor-liquid equilibria. However, applications have been limited to low pressures. Therefore, in this paper a model for the prediction of vapor-liquid equilibria based on a two-parameter cubic equation of state is developed. These parameters are calculated using the mixing rule of Wong and Sandier [1] which allows the inclusion of group contribution models (e.g. UNIFAC, modified UNIFAC or ASOG). Keywords: Vapor-liquid equilibria; Equations of state; Group contributions; Prediction

I. Introduction

Industrial separation operations for petroleum or petroleum fractions, such as distillation, require an extensive knowledge of the physical properties and of the composition. Vapor-liquid equilibrium data are especially needed for the calculation of petroleum distillation. These data have to be obtained either by experiment or by prediction. Today, there are many methods for the prediction of phase equilibria for mixtures with non-polar or low-polarity substances based mostly on semiempirical models. Often, for polar mixtures empirical methods or the direct use of experimental data are the only possibilities.

* Corresponding author. 0378-3812/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 ( 9 6 ) 0 3 1 5 6 - 1

30

U. Baer et a l . / F l u i d Phase Equilibria 127 (1997) 2 9 - 4 4

The experimental determination of vapor-liquid equilibria of multicomponent mixtures is very complicated. Therefore, prediction may not be avoided. For such predictions it is useful to describe the composition of the mixtures by continuous distribution functions. Such a mixture is called a continuous mixture [2]. A mixture where the concentrations of some of the components are given by discrete values while the concentrations of others are described by continuous distribution functions is called a semicontinuous mixture [2]. Thermodynamics of continuous or semicontinuous mixtures is called by many authors [3-6]) continuous thermodynamics. For petroleum or petroleum fractions the continuous distribution function may be obtained by gaschromatographically simulated distillation. To calculate the activity coefficients of multicomponent mixtures group contribution models ([7,8]) are powerful methods. In recent years interest in the prediction of vapor-liquid equilibria based on a group contribution equation of state has increased. Skjold-J~rgensen [9] developed a group contribution equation of state for the calculation of gas solubilities. Dahl and Michelsen [10] introduced a model based on the R e d l i c h - K w o n g - S o a v e (RKS) equation of state [11]. The equation of state mixture parameters a and b may be calculated by a modified Huron-Vidal mixing rule [12]. This mixing rule allows the combination of an equation of state and the UNIFAC model. Holderbaum and Gmehling [13] developed a model that may be applied over a wide range of temperature and pressure permitting an optimal description of vapor-liquid equilibria at high pressures. Cotterman et al. [14] and Cotterman and Prausnitz [15] combined the method of continuous thermodynamics with an equation of state. The mixing rule parameters were fitted to experimental data on vapor-liquid equilibria. The aim of the present work is the development of a mixing rule for semicontinuous mixtures that allows the use of models with predictive character. For this reason the mixing rule of Wong and Sandier [1] seems to be suitable. This mixing rule permits a direct use of the activity coefficient models of Fredenslund et al. [7] or Kojima and Tochigi [8].

2. T h e r m o d y n a m i c model A semicontinuous mixture is considered [2], i.e. 1. The mixture contains D ensembles of chemically similar species B (B = B~,B 2 . . . . . B D) characterized by intensive distribution functions with respect to the normal boiling-point temperature r. The overall mole fraction of the ensemble B is denoted by x 8. 2. Additionally, the mixture contains I individual components A ( A = A t , A 2 . . . . . A~). The mole fractions of these are denoted by x a. The normalization conditions for a semicontinuous mixtures are

~ X A + )-~x 8 s f~ WB(r)dr = 1

(1)

A

where A/

E = A

E A=A I

BD

;E

= B

E B-B 1

(2)

U. Baer et al. / Fluid Phase Equilibria 127 (1997) 29-44

31

The degree of vaporization ~b is defined as the quotient of the total number of moles n v of substances in the vapor phase to the total number of moles n F of substances in the feed. The mass balance for each discrete component A is given by (V, vapor; L, liquid)

XVa= chxV + ( 1 - ck)x L

(3)

and the mass balance of each species identified by ~- of the continuous ensemble B is

X~WBV(~) = ~xVWBV('r) + (1 -- ¢)x~WnC('r)

(4)

The conditions for vapor-liquid equilibrium are given by L e

v v

(5a)

XA~DA = X A ~ A

xLwL(~')q~L(~ ") = xVwV(T)~oV(z)

(5b)

where q~a and ¢pn(~-) are the fugacity coefficients. The following derivations are based on the general two-parameter cubic equation of state as written by Holderbaum and Gmehling [13]

RT p = - -

a (6)

Wm-- b

(Vm-F ~ l b ) ( V m + t~2b)

Well-known equations of state are obtained specifying 6~ and 62. Important examples are R e d l i c h - K w o n g - S o a v e equation of state: Peng-Robinson equation of state:

61 = 1 61 = 1 + vt2

62 = 0 6z = 1 - v ~

2.1. Mixing rule In applying Eq. (6) to a semicontinuous mixture, the equation of state parameters a and b must be calculated from an appropriate mixing rule. In this paper the mixing rule of Wong and Sandler [ 1] and Wong et al. [16] is used. This mixing rule permits the use of prediction methods for the Gibbs energy (e.g. UNIFAC model) in an equation of state. From Eq. (6) the second virial coefficient B may be calculated from the equation of state parameters a and b by a B =b RT (7) Statistical thermodynamics demands that the second virial coefficient of a mixture is a quadratic function with respect to the mole fractions, which for a semicontinuous mixture results in

b

RT

~" Y"XAXA' bAA' a

A'

aAA' + 2 Y'. ~_,XaX B RT A B

"r bAB('l')

RT

dr

d-r'd,r= F 1 8

B'

J#T

RT

(8)

32

U. Baer et al./ Fluid Phase Equilibria 127 (1997) 29-44

The well-known equations for the binary cross-virial coefficients are

[aa aaa =[ a RT J

2

[ aAB(T)

bAR(T)

aa][ bA--- ~

RT

(9a)

[1 -- kAA']

aR(~')

"JI- bB(T )

-

RT

[ 1 - - kAB(T)]

2

(9b)

and

bB('r)

aRs,(~',T' )

bRR'('r'r')

RT

as(T) + [ bR,(T') aR,(T') RT RT

=

2

[ 1 - k~,(r,~-')]

(9c)

Normally, the coefficients kAa,, kAR(T) and kRa,(~-,C) are not equal to zero, except kaA 0 and knR(T,T) = 0. The coefficients kRR(T,C) describe the interaction between two species of the same ensembles, so that the species are chemically similar; therefore kRR(T,C) ----0. Wong and Sandler [1] calculated the molar excess Helmholtz energy at infinitely large pressure E from an equation of state. In this way, and using Eq. (8), relations are obtained allowing the Am= calculation of the parameters a and b for any mixtures. The application of the algorithm to semicontinuous mixtures [17] results in Fl b= F2 (10) 1 -- - =

RT

a = bF2

(11)

where F 1 is given by Eq. (8) and aA F 2 = EXA'-~A +

a

aR( )

ExjWR(r)--d¢ B J'~ bs(z)

E ,oo Am

A

(12)

E ~ is essentially equal to the molar excess Gibbs energy at As Wong and Sandler [1] showed Am, usual pressure. A is a constant depending on the equation of state used 1 1+8~ A = 6 -, - - ~ l n 6----~ 1+ (13)

2.2. Fugacity coefficients Relations for the fugacity coefficients of a semicontinuous mixture may be obtained from the general two-parameter cubic equation of state (Eq. 6). The fugacity coefficient ~oA of a component A may be calculated by

RTlnq~A= f V

Op

-

|dV - R T l n Z

(1 4)

U. Baer et al. / Fluid Phase Equilibria 127 (1997) 29-44

33

Temperature, volume and amounts of all substances except the one considered and all distribution functions have to be kept constant in taking the derivative, Analogous conditions apply to the following derivatives. Z is the compressibility factor

pV Z = nRT

pV m RT

(15)

V is the (extensive) volume and Vm is the molar volume. Analogously the relation for the fugacity coefficient of a species identified by r and belonging to the ensemble B is [14]

RTlnq~n(r)= f v

Own(r)--vTIdV

(16)

-RTlnZ

wn(r) = nnWn(r) is the extensive distribution function, Op/Own(r) is the continuous version of Op/On a in Eq. (14). Considering any variable ~ the simplest way to calculate O~/Ow(r) is to apply the relation O~(w+ t6w) {

O~

,=0 =

(17)

8w(r)dr

The derivative on the left-hand side of Eq. (17) is obtained using the well-known differentiation rules resulting in the integral shown at the right-hand side. Hence, the term in front of 6 w ( r ) d r equals the derivative O~/Ow(r). From Eqs. (14) and (16), and applying Eq. (17), the fugacity coefficients are given by

RT

a

Vm

a

Vm+61b

nrln~A=-~A Vm--b b(Vm+~ab)(Vm+~2b) b2(~ ~,)lnVm + 62b -

aA Vm + 61b + b(62_3,)lnvm+62b +Rrln RTln q~n(r ) = -bn(r )

RT Vm - b

a

-

Vm RrlnZ

Vm-----~

Vm

(18a)

a

b (Vm -b ~lb)(V m q- 62b)

an(r) Vm+ ~b Vm + b(3 _6,)lnvm+62b +eTln Vm----~

b2(~2 -- ~1)

In

Vm + 61 b Vm+ 62b

RTlnZ

(lab)

with A' "aA ~

"

--RT]

[

~xBf~ WB('r) ba~('r) F2 1 D RT

RT ] ,I

am

I~bA-"-A-In')'A-F2) (19a)

U. Baer et aL / Fluid Phase Equilibria 127 (1997) 29-44

34

aAB(T)

=

+ Ex.f B'

" ~''

RT × bsn'(~"z')

dz' + [ bs(z)

RT

2 ~a,XX[baa,--aaA']+ RTI ~xBLWB(z)ban(r)

A lnyB(~) -- F2

(19b)

RT J ,+-R'T ~A-----Aln3~A-RT (19c)

F2

RT a AB('T )

1-

RT

RT

X b88'(~"T')

b [ an('r , ann'}(r R 'T') T] d~'' + RT[ bB(r)

RT A ln3,B(r)-RT

])

(19d)

In Appendix B, Eq. (18b) and in Appendix C, Eqs. (19b) and (19d) are derived in more detail for a single continuous ensemble. The generalization to semicontinuous mixtures and to individual components is straightforward.

3. Experimental The continuous distribution functions of the multicomponent mixtures investigated were determined by gaschromatographically simulated distillation. The conditions were chosen in such a way that the experimental retention time was a linear function of the normal boiling-point temperature [17]. To obtain separate distribution functions the aliphatics and the aromatics were separated using the method of fluorescent indicator adsorption (ASTM D1319). 1,25

7, , ¢

1 0,75

0.50 0,25

300

350

400

450

500

x/K

Fig. I. Experimental mass distribution function for a heavy petrol fraction. - - , Total distribution; • - . , m a s s - w e i g h t e d contribution o f the aliphatic fraction; - - - , m a s s - w e i g h t e d contribution o f the aromatic fraction.

U, Baer et a l . / Fluid Phase Equilibria 127 (1997) 29-44

2--

35

~100

/ Fig

k 3

from 4

Fig. 2. Dew-Bubble-Pointapparatus: 1, measuring cell; 2, glass tube; 3, bath; 4, thermostat. V~, V2, V3, valves. In phase equilibrium calculations, the integrals involving the distribution function have to be solved numerically. To reduce computer-time requirements, the method of Gaussian quadrature was applied. For this purpose, the experimental distribution functions were smoothed by Spline interpolation. Fig. 1 shows the mass distribution function of the heavy petrol fraction and the mass-weighted contributions of the aliphatics and aromatics (aromatic amount 11.3 Ma-%). The experimental data are summarized by Baer [17]. The experimental EFV curves were determined by a statical method using a Dew-Bubble-Point apparatus. The phase equilibrium pressure of a mixture was determined as a function of the volume of the sample at isothermal conditions. The p V - p isotherm shows two sharp bends dividing the curve into three parts: the vapor region, the two-phase region and the liquid region. From the location of the bends on the curve the bubble-point pressure and the dew-point pressure are obtained. The values of the two-phase region result in the EFV curve (equilibrium pressure as a function of the degree of vaporization). Fig. 2 shows schematically the Dew-Bubble-Point apparatus. The measuring cell (1) and the glass tube (2) are connected and filled with mercury. The measuring cell is thermostated by a bath (3) and the thermostat (4). The sample enters the measuring cell by the valve V3. The phase equilibrium pressure p was calculated by P = PL -- PM q'- Prig,2 -- Prig,1 -t- P~ig(T2) -- prig(T1 )

(20)

where PL is the atmospheric pressure, PM is the manometer pressure, PHg,t and Prig,2 are the hydrostatic pressures of mercury in the measuring cell and in the glass tube, respectively, and PHg(T~) and Prig(7"2) are the corresponding values of the vapor pressure of mercury. The experimental error of the phase-equilibrium pressure (as a consequence of the temperature gradient in the measuring cell and of the error in determining the mercury level in the measuring cell and in the glass tube) amounts to 0.3 kPa. The volume of the sample was determined indirectly from the difference between the mercury levels in the measuring cell and in the valve V3. For this purpose, a calibration function interrelating this difference of heights and the volume was used. For the experimental determination of the T-x curves a dynamical method was applied. Fig. 3 shows the Ebulliometer used. The equilibrium temperature corresponding to a given pressure is measured. With the aid of the heating wire (1) the liquid sample in the vessel (2) is brought to boiling. The rising vapor is enriched with the low boiling species and, together with it, liquid drops enter the

36

U. Baer et a l . / Fluid Phase Equilibria 127 (1997) 29-44

Fig. 3. Schematic representation of the Ebulliometer used: 1, heating wire; 2, vessel for the boiling mixture; 3, Cottrell

pump; 4, counter for the number of drops. Cottrell pump (3). The outlet stream of the Cottrell pump enters a zone, in which the equilibrium between the liquid and the vapor phases is achieved. Here the phase-equilibrium temperature T is measured by a platinum resistance thermometer Pt 100. The liquid phase flows back to the vessel (2) with the boiling mixture. The vapor phase is condensed in the cooler and, passing the counter (4) to determine the number of drops, also flows into the vessel (2).

4. Predictions

For the prediction of vapor-liquid equilibrium the following input quantities are necessary. First, two of the three quantities: temperature T, pressure p and degree of vaporization 4) have to be given. F of the individual An important input quantity is the composition of the feed: the mole fractions x m components A, the (overall) mole fractions x n of the ensembles B and their intensive distribution functions W~(r). For the mixtures discussed below the distribution functions were obtained experimentally by gaschromatographically simulated distillation after separating the petroleum fractions into an aromatic and an aliphatic fraction by liquid chromatography. Hence, in the calculations two ensembles of chemically similar species were considered. To calculate the molar excess Gibbs energy or the activity coefficients the group contribution models UNIFAC [7]) and modified UNIFAC [18]) were used. The number VCH ( r ) of the aliphatic CH2-groups and the number VACH(r) of the aromatic CH-groups in the aliphatic fraction were calculated by the relations vcH2(r) = [ M ( r ) / g m o l - '

- 2.016]/14.026; VACU(r) = 0

(21)

Correspondingly, in the aromatic fraction the relations read Vcn2(r) = [ M ( r ) / g m o l - ' -

78.108]/14.026; VACH(r) = 6.0

(22)

U. Baer et a l . / Fluid Phase Equilibria 127 (1997) 29-44

37

Table 1 Models for the correlation of the molar mass, the critical temperature, the critical pressure and the acentric factor with the boiling-point temperature for the aliphatics and the aromatics

Aliphatics Twu [19] Twu [19] Lin and Chao [20] Chen et al. [21]

Molar mass Critical temperature Critical pressure Acentric factor

Aromatics Hariu and Sage [22] Ambrose [23] Ambrose [24] Chen et al. [21]

To calculate the fugacity coefficients using Eqs. (18a) and (18b) the equation of state parameters a and b of the pure species are needed. These parameters were calculated according to the well-known relations [ 11 ]

RZT2 a(T) = O~

a ( w,T)

(23)

Pc

RL b = Ob---

(24)

Pc ~'~a and Ob are constants depending on the equation of state, ot(o~,T) is a function describing the dependence of the parameter a on the temperature T. To apply Eqs. (23) and (24) to a species of a continuous ensemble its critical quantities and its acentric factor must be known. These quantities were obtained by different correlations using the normal boiling-point temperature ~- [17]. Table 1 summarizes the models used for the correlations of the molar mass, the critical temperature, the critical pressure and the acentric factor. For the aliphatics and for the aromatics different models were applied. Generally, the correlations for aliphatics are more accurate than those for the aromatics. The mean error of these correlations amounts to 0.5-1.0%. The correction parameters of the second cross-virial coefficients (Eq. 9) were determined following the procedure by Orbey et al. [25]. In the following, for some systems the results of the calculations are shown using the RKS equation of state which in most cases gives the best coincidence with the experimental data [17]. For comparison the results of calculations based on the generalized Raoult's law [26,27] using UNIFAC and the vapor pressure equation by Frost and Kalkwarf [28] are also shown.

4.1. Heavy petrol fraction - - N-methylpyrrolidinone (NMP) In Fig. 4 the very good agreement between the calculated and experimental results is seen. The difference between calculated and experimental equilibrium temperatures usually is less than or equal to 1 K. Only in the range XN~w > 0.7 do larger differences occur (up to 5 K). The temperatures calculated using the generalized Raoult's law always are lower than the experimental data. The deviations ( 3 - 5 K) may be accepted, but they are considerably larger than those of the calculations based on the equation of state.

U. Baer et a l . / Fluid Phase Equilibria 127 (1997) 29-44

38

400 375 "-- 350 I.---

325'

. . . .

300

w

v

,~

t

A -

J

/' :::'

.......

. . . .

0,25

•

,

. . . .

i

0,5

. . . .

0,75

XNMP

Fig. 4 . T - x diagram of the mixture heavy petrol fraction and generalized Raoult's law; • • • , e x p e r i m e n t a l d a t a .

NMP

at

p = 7 kPa.

-

-

,

Equation of

state;

- •.,

4.2. Light petrol fractions Fig. 5 shows the equilibrium flash vaporization curve (EFV curve) without a polar component. The calculated equilibrium pressures are in good agreement (deviation 1-2 kPa) with the experimental data for the total range of degrees of vaporization. The largest differences appear for the middle vaporization range. The EFV curve calculated using the generalized Raoult's law and the UNIFAC model of Fredenslund et al. [7] is also shown in Fig. 5. Also, represented are the results of calculations based on the generalized Raoult's law, and on the modified UNIFAC model by Weidlich and Gmehling [18] yielding practically identical results.

4.3. Light petrol fraction - - i-propanol In Fig. 6 the EFV curve of the mixture of light petrol fraction and 4.28 Ma-% i-propanol at the temperature T = 343.15 K is shown. Again, the differences between the predicted and the experimental values are small ( 1 - 2 kPa) for the total range of degrees of vaporization. The largest differences again occur in the middle vaporization range. 40

30

~-

2o

0.

10

0,25

0,5

0,'/5

#

Fig. law;

5. p-tk • • -,

diagram of a light petrol fraction experimental data.

a t T = 3 4 3 15 K .

-

-

,

Equation of

state;

• -.,

generalized Raoult's

U. Baer et a l . / Fluid Phase Equilibria 127 (1997) 29-44

39

60

"".'...

~- 40 20

0

0,25

0,5

0,75

Fig. 6. p - ~ b d i a g r a m m o f the mixture light petrol fraction and 4.28 Ma-% i-propanol at T = 343 15 K. - - , Equation o f state; • • -, generalized R a o u l t ' s law; - - - , generalized Raoult's law (and modified UNIFAC); • • • , experimental data.

The results calculated using the generalized Raoult's law in this example differ markedly (especially for small @values) depending on whether the UNIFAC model of Fredenslund et al. [7] or the modified UNIFAC model of Weidlich and Gmehling [18] is used. Thus, the results of both calculations are shown.

5. Final remarks This work presents a method for the prediction of vapor-liquid equilibria of complex multicomponent mixtures based on a two-parameter cubic equation of state and on the method of continuous thermodynamics. A continuous version of the Wong-Sandler mixing rule has been developed. The method permits the prediction of vapor-liquid equilibria over a wide range of temperatures and pressures. In the examples tested the Redlich-Kwong-Soave equation of state proves to give very good agreement with experimental data. Future improvements seem to be possible in two directions. First, the temperature function o~(to,T) is not satisfactory in some cases. Schmidt and Wenzel [29] modified the temperature function o~(to,T) to account better for polar components. Further investigations on this function were carried out by Zheng [30]. Secondly, the determination of the parameters kaB of the mixing rule should be improved. Here, recent developments by Coutsikos et al. [31 ] could be useful.

Appendix A. List of symbols a akn

A A = A I , A 2. . . . . A l b B

equation of state parameter interaction parameters between the structure groups k and n in the activity coefficient models Helmholtz energy, free energy individual components parameter of the equation of state second virial coefficient

40

B = BI,B 2. . . . . BD F1, /72 G kAB

M n

P

Ok R Rk t

T V w

W x

Z

U. Baer et a l . / Fluid Phase Equilibria 127 (1997) 29-44

ensembles of chemically similar species functions of the mixing rule by Wong and Sandier [1] Gibbs energy integrals correction parameter of the second cross-virial coefficient molar mass amount of substance pressure surface parameter of the structure group k in the activity coefficient models gas constant volume parameter of the structure group k in the activity coefficient models number temperature volume extensive distribution function intensive distribution function mole fraction compressibility factor

A.1. Greek letters ot

Y 61 , 62

6w(r) A A

T

4) qo (.o

Oa' ~ b

temperature function for the calculation of the equation of state parameter a activity coefficient parameters of the general two-parameter cubic equation of state variation of the extensive distribution function difference parameter depending on the equation of state used number of interaction groups in the activity coefficient models quantity not specified normal boiling-point temperature (identification variable) vaporization degree fugacity coefficient acentric factor constants for the calculation of the equation of state parameters

A.2. Subscripts C m O0

ACH CH2 NMP

critical quantity molar quantity infinite aromatic CH-group in the activity coefficient models aliphatic CH2-grou p in the activity coefficient models N-methylpyrrolidinone

U. Baer et a l . / Fluid Phase Equilibria 127 (1997) 29-44

41

A.3. Superscripts E F L V

excess quantity feed phase liquid phase vapor phase

Appendix B Here, Eq. (18b) is derived in more detail. For the sake of simplicity, a single continuous ensemble is considered. There are no discrete components and the mixture may be characterized by a single extensive distribution function w(~'). Firstly, the equation of state (Eq. 6) is rewritten

nRT P= V-(nb)

(n2a) [V5-31(nb)][Vs-62(nb)]

Keeping V and T constant the pressure p may be considered to depend on functionals with respect to w(z). Therefore

Op ~ = Ow(z)

(B.1)

n, (nb) and (nZa) being

3p On 3p 3(nb) Op O(n2a) --5--5--On Ow(z) O(nb) Ow(~) O(n2a) 3w(7)

(B.2)

0where according to Eq. (B.1)

3p On

RT V - (nb)

Op O(nb)

(B.3a)

nRT 61 4[V-(nb)] 2 5- (n2a) [ V + 6,(nb)]2[Vs- 62(nb)] [V + 8l(nb)l[v + (~2(nb)] 2

}

(B.3b) Op 1 O(n2a------ff= - [V + 61(nb)][V + 62(nb)]

(B.3c)

Because

n = f w(T)dz'ofr[w(z)+t6w(z)]dr 1 ' Ot ,=0 = L3 w('r)d'r Eq. (17) results in

(B.4)

On/Ow(r)= 1. Furthermore, introducing the abbreviations

10(n2a)

O(nb)

(B.5)

U. Baer et al./ Fluid Phase Equilibria 127 (1997) 29-44

42

Eq. (B.2) reads Op

Op

Op _

Op

(B.6)

-na(r) Ow(r------)-- On + O--(~nb)b(r) + -O(nZa)

According to Eq. (16) and using Eqs. (B.3) and (B.6) the fugacity coefficient go(r) may be expressed by

V I nRT RTln~o(r)=RTln V-(nb) +-b(r) V-(nb)

+ naa(3,12, +

a2I~2)} -

na(~)I,l -

Rrln z

.,I

(B.7) where the quantities lik are integrals defined by dV Iik= + ~l(nb)]i[v + 82(nb)] k

(B.8)

fv Iv

In the interesting cases integration results in Ill

t

111

I2l= (a2--6l)nb

I12=

(B.9a)

~,62--61,nIn b V + 61(nb )

gq- 61(nb)

(62-61)nb V+62(nb )

I,~1

(B.9b)

Ill ]

(B.9c)

With the aid of Eqs. (B.9) and introducing molar quantities Eq. (B.7) reads [RTa Vm a 1 (Vm+ 61b)] Rrln~o(T)=b(r) Wm-------~_ _ b (Vm'-I-c~lb)(Vmqt-¢~2b) b 2 (62 _ _ 61) In V m . - }- ~2 b q- b(62 _ 61,1n gm q- 62b +RTIn

Vm7 b - R T l n Z

(B.10)

Appendix C

Here, Eqs. (19b) and (19d) are derived in more detail. Again, for the sake of simplicity, a single continuous ensemble characterized by the extensive distribution function w(¢) is considered. To calculate a(r) and b(r) as defined by Eq. (B.5), Eqs. (8) and (10)-(12) are rewritten

(C.1) ,F~ = f w('~) a(~). b-ff-i u

nA~m~ -S

(C.2)

U. Baer et a l . / Fluid Phase Equilibria 127 (1997) 29-44

.b=

(.2F,) (nF2) ; n2a = n

43

(C.3)

(nb)(nF2)

RT

To obtain b ( r ) = O(nb)/Ow(r) the quantity (nb) is considered as a function of n, (n2Fl), ( n F 2) being functionals with respect to w(z). Hence, in analogy to Eq. (B.2) and using On/Ow(z) = 1 (Eq. (B.4)) b ( r ) reads

1 -b(r)=

1-

[10(n2F,) F2 I n

Ow(r)

b O(nF2) b + R T Ow(r)

(C.4)

RT

Applying Eq. (17) to ~ = n2F ~ and to ~ = nF2 one obtains

=2

Ow(r)

w(r' JT'

a(r)

b(r,r'

dr'

RT

(C.5a)

m"

Ow(r----~ = b(r)

A

(C.5b)

lny(r)

Combining Eqs. (C.4) and (C.5) and introducing molar quantities one obtains b(r) =

1-

F2 '

2

,W(r') b ( r o " )

a(z,r')

RT

dr'+ --

RT b ( r )

RTlny(r)A - RT]

RT (C.6) To calculate ~(r) = 1/nO(n2a)/Ow(r) a similar procedure is applied to n2a = (nb)(nF 2) resulting

in a(r) = -,,

=-b(r)F2+b~

aw(r)

(C.7)

Ow(r)

With the aid of Eqs. (C.5b) and (C.6) and introducing molar quantities it is easy to find a(r) =

/:2 1-

2F 2

W(r') b(r,r') '

a(r,r') RT

dr'+ b

lny(r)-F b(r)

2

a

RT

(c.8)

References

[1] D.S.H. Wong and S.I. Sandier, AIChE J., 38 (1992) 671-680 [2] R.L Cotterman and J.M. Prausnitz, in G. Astarita and S.I. Sandier (Eds), Kinetic and Thermodynamic Lumping of Multicomponent Mixtures, Elsevier, Amsterdam, 1991, pp. 229-275

44

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[31]

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