UREA-NH3-CO2-H2O: VLE Calculations using an extended uniquac equation

Fluid Phase Equilibria, 53 ( 1989) 207-2 18 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

UREA-NH,-COz-H.0:

M. BERNHRDIS M.

SRNTINI,

ULE

and

CRLCULATIONS

G. CRRVOLI,

Rgrimont,

Porta

USING

AN EXTENDED

Istituto

Guido

Marghera,

Italy

207

UNIQUAC

Donegani,

EQURTION

Novara,

Italy

RBSTRRCT R

modifi ed

ideaiity

UNIQURC

of the

auaternary

equiii brium

Some

successfuLLy

equation system

equipments

simulatea

is

using

applied

to

caiculate

Liquio

phase

non

URER-NH3-C02-H20.

operating such

in different

an equation

for

ano VLE

strict

conditions

were

calcuiations

INTRODUCTION Correct

VLE

hard

to obtain.

very In

fact,

a

calcutations

unique

different

conditions;

licenced

by Montedison

2) Temperatures to 414

In

Fiqure

the

and

K and same

modei

referring

pressures

7 KPa

0378-3812/89/$03.50

shouid

(Pagani

in the

sections

1 - Partial

for

URER

URER

be abte to

and go

ID9

Mariani,

from

Last

473

simuiation

to treat

and

phase

equilibrium

(Isobaric

Doubte

19841,

Figure

K and

see

optimization

Recycie)

of an

goes

plant

1:

20 MPa,

ciose

from

to

IDR piant

0 1989 Elsevier Science Publishers B.V.

35%

is

in very

to the

react or

flash.

concentration

flowsheeting

plant

100%

by weight

I

208 b)

The URER,

high

In order

section

recovery

ammonia

vacuum

ano

Low

to describe

the

liquid

choose

conditions

we

atready

applied

with

tow concentration

of

temperatures. phase

ternary

ideatity

non LINIQUAC

extended

an

to the

streams

regards

system

in

ecuation

NH,-CO,-H,O

a

Large

(Sande;

(Bernardis

range

et at., al.,

et

of

1988) 1988,

1989).

CHEMICAL The

REACTIONS four

generating The

ions,

comptete

NHs

+ H20

CO2

+ Hz0

H.0

+ OH+ H*

_

HCO;

-

co;

F=

NH,COO-

+ (Nti2)*CO=

We negtect

the

ion

is atways

We

atso

needed

very

is the

foliowing: (1) (II) (IIb) (III) (IV) (V)

+ CO? ion dissociation

is very

(IIb)

so that

low,

the

because

concentration

the of

Last

reaction,

the equitibrium

time

in most

equipments

URER

is thus

treated

because

its

conditions

rate

is much

is so

Longer

tow than

of a IJRER plant.

as an inert

component

of

the

system.

T~IBLE 7. Finat

system

describing

In KX = In ak; tn KXr

In KX,

= In a$+

q

mNu,+

tlB2

q

mCol+ mHcoi+ mNH:=

liquid

In a:H,

aH,P

In a:,--

-In

In auto

mrrw:+ mNH,coc-

muco;+

= l./(l.+

phase

- In aH,P

+ ln a",+- Ln a"cP, - In au,,

= In a9Nulcoo-+Ln

MB,

xw

the

+ In a&--

= In akO;

In KXII

m,++

thermodynamic the carbonate

smait. the

to achieve

reactions

+ HzO

bicarbonate

themselves

react.

+ OH-

2NH,

neglect

dissociate

+ tl'

constant

equilibrium

can

sotution

in

equilibrium

NH+,

G== H*

He0

PHRSE present

in turn

of chemicat

ti

+ HCO;

LIQUID

species

which

set

HCO; NH3

IN THE

motecutar

mNH,coamNu,coo-+

,018 Cm,)

m,,-

aEroj

-In a:, il

that the

the

time

residence

209 Each

remaining

temperature at.,

reaction

dependent.

has Their

a thermodynamic vatues

are

equilibrium

reported

constant

elsewhere

K,,

only

(Pawtikovski

et

1982).

Introducing

the

mass

electroneutraiity fraction The

xw, we get

concentration

moiatity

balances

balance the

final

of

all

system

ammonia

and

equation

to

calculate

the

Liquid

describing

species

but

carbon

water

dioxide, the

phase

are

the

water

(Table

expressed

mote

I).

on

the

scale.

The

corresponding

activities

the

and

moiatities

as pure

The

In this

expressions

appear

paper

by

basea

on the

asymmetric

coefficients

Sander term.

Besides,

the

simultaneous

of activity

in Literature,

we aod

residual

activity

are

activity

convention.

in iabie

1 are

referrer

to

FRHMEWORK

explicit

NH,-CO,-H,O

the

sotvent.

THERMODYNAMIC

done

the

the

Both water

of

and

some

et

concentration

at.

(1986),

expressions presence

of the

of two

coefficients

and

in the

three

of

for

ternary

6,,,

system

as

of activity

2( are

Therefore

expressions

the

(1988).

parameters

calculation

terms

for

Carvoti

dependent

soLvents.

explicit

coefficients

Bernardis

atready

coefficient

modified

now

it's

convenient

the

quaternary

by

to report system

the tSe (see

Rppendix). This

extension

refer

to a mixture

constants the

also

refer

and

(a:,,,,),

=

: ,)1

-

a:,_.U being

and

dilution

VAPOR

(Ra

the

and

that with

activities

water),

URER

and water

respect

of

white

to pure

the components

the

form

equilibrium

an ideal

water

mixture

(a:,,)

or

to

following:

+ 21

(Ra

( I)

+ +&\

(I) an approximation.

XT,,,, are,

in pure

UREA

respectively, and

pure

R, is the moiar the

activity

ratio

water/URER.

coefficients

at

infinite

water

PHASE

URER Only

Supposing activities is the

problem:

CURER

(R,

equation

&'T,_

between

URER

s a,,,

another

solvents

to water.

relationship

water

rises

of

the

volatility three

vapor-liquid

is neglected,

other

motecutar

equitibrium,

since

so it does components ions

not

NH.,

cannot

appear CO2

Leave

and the

in the Hz0

vapor

are

Liquid

phase.

involved

phase.

in the

210 The

relating

phase,

are

Py, 'pz where

expressions, the

taking

= (ti/P!,,.R2.+ H:P),,) m,

i : NH3

into

account

the non-ideatity

of

the

vapor

fotlowing: ,S / (R,il.)

15y:

(2)

or CO?;

for water: Pywy, RLL

= PLY:

fugacity

(Nakamura Henry's

a, exp

:Vw

coefficients

et at.,

(P - Pi) are

/ RTI

catcutated

(3) using

Nakamura's

equation

of

state

19761.

constants

are pressure aependent: cp"z, = in Hi,w + Vi * (P - P:) / RT CP,'I = in Hi,u + U, m fP - P:, / RT

in H:P)w (P> Ln H,,U

(4) (5)

wnere: oiw :: F'$i

1 f:::

The whote

set

TRBLE

!n T + BS~W

T + B4iw

(6)

1 : : ,",,, Ln T + BSLY

T + BLiU

(7)

of B constants

appears

in Table

2.

Henry's

constants

as a function

Electrolyte

B,

of

temperature

NH,

-681.531

28.67202

co,

-8901.728

-22.198746

NH,

-1018.619

28.68392

co1

-3326.2777

-22.27357

NH3

mNH,,

and

COn

moiatities

range

-.0541721

-151.3689

273-423

.OOll3684

156.8756

273-423

(K)

Water

-151.41204

433-453

155.34374

433-453

.0117394 URER

activity

coefficients

are

referred

to

the

of sotvents: PMw

+ xu PM")

(xw PMw

+ xu PM~)

5 = ~,a,>/ (xv,

mco,,a

and

Rpptic.

B,

-.05250941

Sotvent:

mixture

(K).

B.

Ba

Solvent:

The

2.

= xco2/

(9) (3) (IO) (11)

ESTIMFlTION The

(Bernardis 373.15K

OF PRRRMETERS

first

set

et at.,

only.

of

parameters

1998,

In order

1999)

regarding was

to simulate

obtained URER

ptant

the

ternary

reducing equipments

system

experimental the

NH.-COS-HzO data

equitibrium

at must

211 be

retiabty

described

catcuiations

using

temperature To

do

that

new

complete

TRBLE

more

we

had

between

of

data

373.15K

to introduce

set

For

to 473.15K.

experimentat

now

range,

The

up

ternary

this

reason

(Mutter

and

we

revised

our

1968)

et at.,

ternary a

in

wioer

473.15K.

the concentration parameters

dependent

is reported

parameters

in Table

6,,.

3.

3

Complete

set

of ternary

parameters

and

experimental

measurements

used.

Experimental Source: Binary VLE Data i

k

NH.

e.0

CO,

-256~2.10 '107.7931T

P=O.-9.0i?Pa

+.OlEt?211T*

-.108b8171i

m..,, =O.-152.7

H.0

-:272.667

654.7

N&COO-

H.0

-569.5

815.2

HCO;

H,O

-753.5

615.b

NH*

NH:

2500.0

2500.0

NH,COO-

2500

co,

NH,COO-

-1094.0

HCOi

0

-835.3

co,

tico; NH:

NH:

NH,EOO-

NH:

NH.COO-

NH:

uco;

root

and

vapor

The

2500.0

232.6

-1168.2

2500.0

2500.0

mole

from

for

the quaternary

TRBLE

are

between

experimentai

110 data

points

are

between

URER

and

data shown

available

in

TabLe

and

caLcutated

respectively the

other

in Literature.

15.2%

reported

determinations

and

16.3%. were

The

4.

interaction

k

al.

used

parameters

akI

to optimize concerning

Exp.

the URER.

remarks

NH2

URER

-717.3

573.3

COP

UREFl

-957.7

2500.0

H.0

URER

.2982+T

Kawasumi

NH:

UREA

-634.1

-112.9

Durisch

NHnCOO-

URER

-676.9

1293.7

(1979)

(1952,

53,

pressures

components

I.

Experimental

i

o.-9.77

at

6.*

parameters

to be significant

m,,,=

et

(1966)

373887.0

deviations

fractions

O.-Z&.6

-9lEZ.E

LE11.6

squared

m.,,=

16.5

-28661k.0

interaction

optimized found

mean

a:.

2k58.1

6,.

The

nu11er,

2500.0

186.0

co.

et

mcox=O.-1.5

-702.8

NH,

Hough:on, (19571

P=or3.75nPa

at

Data

H,O

NH:

T-273.15.373.15K

-334031.43/T

fxperimental Source. Ternary VLC

References Guiltev~c et (1965) Pawtikowswci et at (:5&i:

1=373.15-503.15K

2282.919

+183ilL.L5/1

NH:

Remarks

a..

a*. 4969.77 -20.832351

SL)

-.53Sl*T et at.

parameters

we

212 We

have

to

comptetely phase

Finally

range

we

solvents,

even

(see

maintained

at every

experimental

are well

In xx

= AH,/R

(I/T

mean

vapor

2).

The

1979).Moreover

K and

constants

keeping

of

453

and

NH3

these

CO,

in

parameters

the mutuat

This

reporting and

vaoor

data

are

URER

as

K).

interaction

in mind

temperature.

comes

awnrER,uREn

ideality

from

the

the

two

solubitity

of

of

of LJRER in water

(Kirk

K and

deviations

and

and

Othmer,

(12)

fractions

8.3%

data

assuming:

Tr = 408

squared

main

Mariani,

Letting

between 22

for

x

= 1.

experimenta:

and

experimental

caLcuLated

quaternary

data

pressures points

are

(Pagani

and

is directly

fed

12.2%.

features

1984)

the

gases

are

the

respectivety

of

can

actually

latter

in the ammonia

Instead

in the

leaving

tne

say

a water/LlRER solution

fixed

or CO=

VLE

D'Rrminio In the

exactly

through

typical

ratio

Like

Monforte, stripper

1987)

were

the stripping

the

from

heart

of

the plant.

carbamate;

missing

and

the

the

in

two phases

flashes.

PATHS

isotherms

azeotropic ternary

to the

calcutated

behaviour

system

goes

due

NH.-C02-HsO.

stoichiometric

is minimum;

pressure

inserted

are

free

five

STRIPPING

boiling

in the

is atso

is set

or

are

stripper.

strippers

four

dioxide

at

20

MPa

and

formation

of

R, = 1.764.

pressure

in solution,

was

plant

is close

component

first

equipments

a characteristic

Nhs/COp

model

two

solution

FIND SIMULATED

motar

shows

temperature,

motecutar

Last

IDR process

of ammonia

carbon

second

the

shown

a part

the

the

the ureic

passed

of an

and

removed.

for

ratio

these

and

traditional

The

molar

that

first

Loop

only

1: part

is quantitatively

2 are

carbamate,

synthesis

in Figure

the

ISOTHERMS

In Figure

the

remaining

former

reactor

EQUILIBRIUM

of

shown

reactor;

In fact

The

633

data

Liquid

IDR PROCESS The

NH3

et at.,

(between

literature of both

- l/T,)

mole

respectiveiy

We

Durisch

sotubitity

reproduced

being Ati, =12050J/mol,

to

avaitabte

compositions

in water. the

root

few

and

Table

calcuiated

1983)

and

found

Henry's

were

URER

In fact,

THE

we

temperature

parameters

auREa,waTEn

solid

of

considered

adjustabLe

The

that

temperature

(Kawasumi,1952,1953,195~;

in a narrow

and

emphasize

pressure,

value

increasing

up because

to

In fact

when

in carbamate,

the

quantities

the concentration

the

at

a

of either of the

free

increasing. in

a

in order

temperature

simulation

computer

to calcuiate slightly

rises

stripping from

program

(Pagani

and

paths. top

to

bottom;

the

213

Fiqufe

2 _

stand

for

- Calculated top

and

isotherms

bottom

of

the

and

concentration

of ammonia

is almost

through

stripper,

removed

excess

the

of ammonia,

reduction In

Table

5 real

comparison TABLE

the

of carbamate

with

stripping

path

of

2nd

stripper.

T and B

equipment.

constant,

by gaseous

reduction

of CO*

while NH..

carbon

dioxide

As there

accompanied

is

decreases

is a stoichiometric by

a

nearly

equal

in solution.

operating

conditions

calculations

of such

an equipment

are

reported:

the

is satisfactory.

5.

Working

conditions

and

outcoming

countercurrent

stripper,

funning

equipment

be quantified

could

Exchanged

heat

Number

theor.

of

No UREA

liquid at

streams

20.

comparison

MPa.

The

gas

phase

for

the

leaving

as difference.

6279. plates

MJ/h

3

hydrolysis

F ""Em Liquid

feed

Gaseous

feed

13355. 0.

Fwas.. 7139. 5.

FNH,

F sn.rtr

FCO,

T

13663.

4306.

0.

455.

2495.

0.

50.

473.

Plant

outlet

13355.

7079.

13754.

3107.

0.

C76.

Calc.

outlet

13355.

7139.

13435.

3205.

0.

476.

lc the

214 The

simulation

reported) at the This

shows

bottom

is due

2nd

of

to both

the the

cold

concentration

has

value

of 4.75

% by weight.

Rgain

the

we

flashes

report

coming

Tables the

agreement

7,

presented

TRBLE

CO*

between

feed

2,

where

stripping with

behaviour,

plant

a plant for

comparisons

where all

are

hydrolysis residence

assumed

and

comparisons

9 that

and

because

We

low.

some

from

6 and

temperature

point

got

Figure

path

the

is also

lowest

value

equipment.

that

at

Moreover

(see

a characteristic

(B) of

especialty

stripper

from

running

calculated

between

are

ptant

data

and

high a

is good

catculated

are missing.

equipments

taking

UREA,

conditions

plant

the strippers

these

of times

It can

(Tabte for

conditions

countercurrent

Exchanged

heat theor.

of

Rssumed

URER

Liquid

feed

Gaseous

feed

and

stripper,

Number

be seen

emerging

in

from

small.

outcoming running

liquid

at 20

streams

comparison

for

the

6120. plates

M3/h

6 1.75%

hydrolysis

F U"EEl

Fwe,,a

13355.

7079.

0.

F

by weight

N*)

FCOt

13754.

0.

0.

F iorrt.

T

3107.

0.

L76.

9245.

65.

383.

Plant

outlet

12720.

6354.

3679.

2912.

0.

653.

outlet

12720.

5684.

4067.

3029.

0.

(62.

TABLE

7.

Working

conditions

and

outcoming

liquid

streams

comparison

for

a flash

running

HPa

Exchanged

heat

Feed

2@

MPa.

Calc.

7

6). some

6.

Working

at

N!i.

hydrotisis

data

the differences

place

and

86671.

F ""LP

FLAX4y.r

65116.

41643.

60171.

F

N*3

tlJ/h

F sn.rt.

T

33487.

318.

b67.

FCD,

Plant

outlet

65116.

35560.

26819.

8645.

0.

b56.

Catc.

outlet

65116.

36732.

28609.

8030.

0.

465. '.'

215 TRBLE

8.

Working at

conditions

and

outcoming

liquid

streams

comparison

for

a flash

running

1.3 MPa

Exchanged

heat

354b3.

Feed

HJ/h

F”llCf3

F WlTL"

F*u,

F,,,

65116.

35560.

26819.

6645.

0.

456.

F lnlrt.

T

Plant

outlet

65116.

30262.

6731.

1676.

0.

b23.

Calc.

outlet

65116.

29250.

8035.

19&O.

0.

42&.

TABLE

9.

Working at

conditions

and

outcoming

liquid

streams

comparison

for

a flash

running

.33 MPa

Exchanged

heat

10664.

Feed

f”“,Cl

FlAWIT.”

65116.

30262.

F

HJ/h

NH)

FCDx

6731.

1676.

F in.rtr 0.

T &23.

Plant

outlet

65116.

25619

2527.

653.

0.

396.

Calc.

outlet

65116.

26530.

2956.

490.

0.

402.

NOTHTION al

= activity

au1

= interaction

of component parameter

= parameters

B,,B,,B,,B, F

= Flow

Ha (V H1 tQ H%

= Henry's

AH,

= heat

KJ

= thermodynamic

m

= molality,

rate,

i between

in equations

component (6 and

k and

constant

7)

of component

i, MPa

kg/mol

constant

of component

i, at pressure

= Henry's

constant

of component

i, evaluated

MB,

= total

MB.

q

P

= pressure,

total

i, K

kg/h

= Henry's

of

component

fusion,

ammonia

MPa

pressure

J/mol

equilibrium

mot/kg

carbon

P at saturation

constant

of solvent

introduced dioxide

in solution,

introduced

mol/kg

in solution,

of solvent moL/kg

of

solvent

216

PL

= saturation

PM

= molecular

91

= surface

ri

= volume

R

= gas

pressure

of water

at system

temperature,

parameter area

parameter

constant, ratio

J/mot

R* T

= temperature,

Tr

= melting

point,

v _ u1

= partial

molar

volume,

= partial

moiar

volume

of molecular

solute

xi

= liquid

fraction

of component

i

mole

Yi

= vapor

2

= coordination

mole

water

K

= molar

Greek

MPa

weight

to URER

K K

fraction

cm3/moL

of component

at

infinite

dilution,

cm3/mot

i

number

letters

21

= activity

6

= concentration

'p

q

vapor

coefficient dependent

phase

fugacity

parameter

coefficient

Superscripts *

= asymmetric

S

= saturation

m

= infinite

V

=

+

= cation

-

= anion

convention

dilution

convention

based

on molality

scale

Subscripts b

= solute

i,k

= component

j

= reaction

m

= single

S

q

solvent

solvents

U

q

URER

W

q

water

REFERENCES Bernardis, extended a

large

M. and UNIPURC

CarvoCi,

G.,

equation:

composition

and

1983.

"Phase

application temperature

at

equilibria the

ternary

range,"

calculations system

Proceedings

using

NH.-LOa-Hz0 of

the

an in XIX

217 Conference Bernardis,

on

M.,

using

the Use

G. Carvoli

an extended

Durisch,

W.,

and

UNIQURC

A.

"Experimentat dioxide

of Computers P.

Delogu,

equation,"

Buck,

S.M.

investigation

ammonia-water

in Chemical

of

system

1989. RIChE

J.,

Cemkowitz

Engineering,

Gdteborg,

"NH,-CO,-H,O

VLE

35,

and

synthesis

calculation

- 317. Van

P.J.

the vapour-Liquid

at URER

314

79-64

den Berg

equilibrium

of

conditions,"

1979.

,

the

carbon

Chimia,

33,

293

- 299. Guiltevic, for

J.,

D.

the Binary

MPa,"

G.,

and water Chem.

Sci.,

temperature

and

temperature Kawasumi,

S.,

R.E.

and

D.F.

Othmer,

146

and

Systems zwischen

R.,

Properties

Butt.

in the

Equitibrium 503.1

K up

"Compressibility, region

O-36

atm

III,"

Data

to

7.0

fugacity

and

of

1963.

Sot.

Japan,

system 25,

the CO.-NH.-URER-H.0

BulL.

V," Bull.

CO*-NH.-UREA-HZ0

Chem.

of

0-lOO"C,"

Chem.

Sot.

Japan,

26,

Sot.

"Encyclopedia

Japan,

216

of Chemical

254

under

high

- 222.

system

27,

high

- 238.

system

the CO=-NH9-URER-Hz0 Chem.

227

under

under

high

- 259.

Technology,"

Wiley

549. G. Maurer,

1986.

uDas

Dampf-FLBssigkeitsgLeichgewicht

Rmmoniak-Kohtendioxid-Wasser 373

und

473

Pagani,

G.

the

Pagani,

and

Ber.

Kelvin,"

hohen

bei

Bunsen.Ges.

D'Arminio

and

L.

URER,"

Process

Wassergehalten

Phys.

E.M.,

Chem.,

92,

and

of

Rmmonia

Process

21,

764

B.,

Liquid Extended

A.

Dev.,

Fredenslund

Equilibria UNIQURC

in

"Some

Experiences

Proceedings

IDR

Nonpolar

- 564.

of the

Giardini

with

XVIII

Naxos,

399

an economical

process:

the

Congress - 406. way

of

4, 45 - 49.

J.M. and

"Phase

Prausnitz,

1982.

Carbon

Dioxide,"

Equilibria

Ind.

Eng.

for Chem.

- 770.

and

Rasmussen,

P.

Nitric

Equation,"

"The

Progr.,

Flqueous Solutions Des.

1967.

"Thermodynamic and

Polar

15, 557

Engineering,

1964.

Eng.

3. Newman

Dev.,

Simulator,"

in Chemical

1976.

Common

Des.

Monforte,

Mariani, Chem.

Prausnitz,

J.M.

Containing

Chem.

of Computers

G.

Pawlikovski,

Eng.

A.

and

Mixtures

Equation-Oriented

Use

producing

Gas

Ind.

Montedison

Breedvetd

G.J.F. of

Components,"

Sander,

and

- 160.

Nakamura,

on

pag.

1957.

of the

"Equilibrium

pressure.

ternaren

im Bereich

II,"

pressure.

E. Bender

Dioxide

"Equilibrium

1954.

453.1

-335.

Ritchie,

"Equilibrium

pressure.

23,

332

P.D.

"Vapour-Liquid

1995. at 403.1,

- 137.

and

Interscience, MLitler,G.,

132

1953. and

temperature

30,

and

of Carbon

6,

1952.

S.,

H. Renon,

Data,

McLean

solubility

S.,

Kawasumi,

Enq.

R.M.

Eng.

Kawasumi,

des

and

Water-Ammonia

J. Chem.

Houghton,

Kirk,

Richon System

1966.

Reid-Water-Nitrate

Chem.

Eng.

Sci.,

41,

"Calculation Salt

1165

Systems

- 1195.

of Vapour-

Using

an

218 RPPENDIX We

report

here

an extended For et For

the al.

the

expressions

of

the activity

coefficients

caiculated

using

UNIQUACequation.

solvents

there

are

no differences

with

respect

to the

paper

of

Sander

(1966).

the

solutes

you

have

to consider

that: (RI)

and So

now

we

have

a mixture

the Debye-Huckel

term

assumed

concentration

Instead

the

b) and

Besides

:

& = ea

- -

term

changed

at

being

all,

the dielectric

qb a surface number,

taken

area eoual

parameter

(both

referred

xbr./Dx*r*

(R3) (AL)

residual

term

In

L . = q,. 8'

-

expression

C*r Rk

= pc,,

Db

= ;p,2e,6,,,m

is:

Ln Sb - fib + CD, + E,

2B)/T

l

In

Qqwy:;w R,qw

g&f,,

B

FeiEs

(R6) (R7)

Cc*, (Cam

+ cm,)

(A9)

+ C,t¶)

(RIO) (Rll)

: are

referred

+ i

(R6)

Et2 = =

+ qu

qu+R,qwywu ]

= p,

F and

+ quy’Y;,u

(R5)

q”Y.w+%qw

ki Y = yJki 15%

to

to 10.

= x,q,/Ex,qla

being:

constant

is:

parameter,

z a coordination

The

5%

is not

sotvents.

-gzq,.

rb is a volume

solute

two

independent.

combinatorial

In -

where

of

summations

to the

Y *i=

exp

f brn =

e,-;f,,,,e,

(

solvents

-a*,/T)

over

ail

system

species,

expect

where

specified;

only; (R12) (R13)

IZ is _