Thermodynamic consistency test for binary VLE data

Fluid Phase Equilibria, 27 (1986) 405-425 Elsevier Science Publishers B.V., Amsterdam

Thermodynamic

consistency

test

Department

2.

Department

3. 4.

binary

VLE data

of Mathematics of

Technical

Chemical

University

(author

for

K.Heberger3,P.Angya14,f.Thury2

K.Kollar-Hunekl,S.KemenyT 1.

405 in The Netherlands

- Printed

Engineering

to whom correspondance

Central

Research

Academy

of

Research

I H-1521

of Budapest, Institut

Sciences,

should for

H-1525

Chemistry

of the

Budapest,

Hungary

Hungarian

Hungary

Development Company for the H-1428 Budapest, Hungary

Industry,

Budapest,

be addressed)

Organic

Chemical

ABSTRACT A generalized to

binary

data,

- isothermal

based

on the

The essential equation for

specific equidistant

complete on a strict

quadratic

limited

and

solve

is

possibly

proposed.

the

equation)

conditions

coexistence

considering

the

at step

the

heat

the

of mixing

two edges.

sizes

and was applied

-thus

without

to systems

with

as well.

to be applied

mathematical

equilibrium

Van Ness to

test

To calculate the t(x) or P(x) a cubic spline-polynomial is used

miscibility

criteria

is

non-equidistant

- was developed

and

The test

method

points

using

fitting

of

Gibbs-Ouhem phase

consistency

- vapor-liquid

method

system.

Van Ness’

An algorithm curve

isobaric

the

vapor

at

thermodynamic

of the

from

the

rejecting

or

step of the

function

for

isothermal

(deduced

non-ideality data

method

to

statistical

residuals

are

established

basis.

INTRODUCTION During made in

the

measurement ~~ apply

data

are

last

predicting data these

required.

0378-3812/86/$03.50

twenty

years

successful

multicomponent using methods

semiempirical both

Measurement

basis

liquid

models.

good models data

attempts

VLE on the and

of good

@ 1986 Elsevier Science Publishers B.V.

reliable

quality

have

been

of binary experimental must

contain

406 small

random

In

and

preferably

an earlier

to

identify

the

data.

errors

stage

certain

the

could Thus

error.

showed

of

recently-

checking

has

to

been

work

is

algorithmize of

the

the

1977)

and

of

correct

to

systematic to

consequences

prove

that

(picture

type

correction

we hoped

then

possible

a specific

of

of

of

systematic

experimental

or

refused.

-after and

experimental

al.,

types

was not

certain of

also

this

to

consistency

it

et

errors

that

as

possibility

data

The aim

errors.

(Kemeny

systematic

be attributed

the

literature

work

of

be identified

consistency

residuals)

our

types

The experience

cannot

using

no systematic

of

refining

the

establish

data

in

the

methods

applied

decision

on the

detect

systematic

order

to

method

for

checking

differential

of

errors. LITERATURE SURVEY The basis

of

consistency Gibbs

of

GE 1s the is

HE From

the

a binary

total

the the

excess

system:

molar

liquid

excess mole

Gibbs

(la)

of

is

the

molar

change

1s the

molar

heat

of mixing

the

Gibbs-Ouhem

la

coefficient

follows

Tl+

x2 dln

The consistency contradiction

r2

of

= ;

itself with

energy

fraction

activity

xldln

VE

for volume

i-th

the

component

i-th

component

on mixing equation:

HE RT2 dT

dP-

means

the

the

that

the

Gibbs-Duhem

(lb) data

are

not

in

equation.

tests

To isothermal 1967)

data

sets

may be applied.

small, and

the

the 1s .

Eq.

Area

proposed is

-- 8 dT + 2 dP+ 5 ln yidx. 1 RT RT’ i=l

d

;

ever

data

for

energy;

where

any

VLE

the

it

also

second remains

the

As the

therm zero

of if

Redlich-Kister term

Eqn. the

test

VE/RT is (la)

is

equation

usually

practically is

integrated.

(HBla

et

al.,

negligibly equal

to

zero,

407 1

Tl

In The

allowed

zero

depends

on the

method

1967)

has

for

numerical

a limiting

points

of

the

they

are

however

mild. not the

A local

being

value

which

Method

of

of

from

among

value

of

measured

The

based

et

al.

based usually

for for

mole of

In

= g+x2

integrated

as

the

excess

been

which

is,

three the

compared

with

in

the

of

cases in

the

order most

vapor-liquid

equation directly

by Sayegh

first The

numerically.

or

only,

zero.

form

of

the

errors

around the

energy

energy

sets only

(random

differential

reviewed

data using

errors

sets in

isothermal

Gibbs

has b),

residuals.

consistency

Gibbs

pressures.

(x,t,y,P)

and

scatter

is

excess for

variables

written

this

fractions,

that

is

the

solve

coefficients

functions

rl

to

vapor

isothermal

data

checking

is

coefficients, phase

which

total sensitive

better.

the

residuals

data,

on the

extremely

(1985

area

of

component

cystematic

consistent the

equation

activity

give

equation

data are

calculated

the

not

boiling

used

s Bme principle

four

the

consistency

recognition

may be

should

system.

measured

for

et

The

(2)

commonly

the

much

reflect

errors)

the

Fenclova

(1973)

(as

(Hala

using

the

pure

deviations For

data

of

the

measured one

the

method

that

on the

on the

data

sets.

by Eqn.

tests

and

from

by Herington VLE data

These for

integral

measured

calculated

the

all.

deficiences,

Gibbs-Duhem

equilibrium

as

these

data.

efficient

The

test

residuals

differential

vapor

at

taken

fourth

no systematic

and

by Dohnal

the

experimental

may be

addition,

values

value,

These

vity

In

is

the

the

binary

characterize

used

Van Ness

The method

of

the

19721,

(Van Ness et al. ,1973),

recently free

of

expressed

components

defect

proposed

value

empirically

isobaric

sufficiently

are to

extended

integral

pure

too

Stevenson,

the

As Van Ness stated do not

numerical (precision)

and

been

of

exceed

the

accuracy

checking

value

pressure

(2)

(T=const)

of

by Ulrichson

The

tests

= 0

deviation

discussed al.,

dxl

r2

0

and

for for Vera

actithe (1980).

may be expressed following

way:

(3a)

+$ 1

408

(3b) where

Expressing

the

-ideality is

in

total the

pressure

vapor

phase)

the

Eqn.(3)

(neglecting

following

the

differential

non-

equation

obtained: P=xlP;Q+x2P;12

where

py and

To solve has

been

tion

(5)

the

pi

= x1Py exp

denote

propoded has as

a function

is

reduced

Thus

fixed

polinomial

fitting is

coefficients

xl,

the

of

fitted and

component (5)

vapor

(1965).The

this

way the

total

pressure

not

P(x,)

the

curve

is

by the

method

of

of

g(x,>

then

the

vapor

phase

of

least

independent

by equal is

to

P

P may be

number

step

be calculated

concentrations.

required:

values

equa-

x1 and

measured

the

the

differential However

discretization

(5)

of discretization

variables:

For are

g-x,?)

pressures.

a method

temperature).

one.

which

+x.p!+v(

ax 1 )

al.

the

case

tabulated

et

of to

x 1 values

previous

Having

with

isothermal

g+x2 as

two independent

variables in

pure

by Mixon

expressed

certain

(

equation

apparently

pj’ is

sizes

the

differential

(because

at

using

a spline

squares.

function

mole

the

fractions

activity may be

calculated: yj=xj

The majority constant such

of

pressure.

cases

constant

isobaric

data

of

the

data

According

isobaric

Fabries Anderson

Thus

data

and et

in

the

to

way is

temperatures set.

Maximum likelihood and

the

a practicable

several

(6)

j=l,Z

TjPP/P

in

a locally

literature

a remark

is of

given

Van Ness

to make isothermal the

temperature

isothermal

for (1973)

in

treatment range

at

of

description

the is

made

set. methods

Renon al.

(19751, (1978)

Peneloux during

the

et

al.

parameter

(1975a,

1975b,

estimation

1976)

409

procedure

also

measured and

give

variables

calculated data

it

was pointed

and

also

existing

models)

the

estimates (this

also

the

deficiences

the

Redlich-Kisler

this

distortion.

the

effect

the

if will the

can

measured

model case

As used

with

is the

be misleading, and

model, 1973;

errors

Fenclova

hopefully

be avoided

Ness,

the

consistency

measurement

Dohnal

also (Van

the

variances.

the the

only for

the

error

drawn

model.

of

between

frequently

not

spline-polynomial

Kemeny,

is

equation

This

values

judging

a1.(1982)

conclusions of

for of

et

reflect

used suitable

to

residuals

true

The deviations

by Kemeny

adequate

the

the

may be used

lead out

completely

because

for

(x,t,y,P). variables

of not

an estimate

but

(1985

b)

avoiding

by fitting

Kollar-Hunek

a and

1978).

dr

AY

proposed

0,02

Wilson

OlO2 a a

Fig.1.

l

e-

The effect

of

(Simulated P=lOl.

l

0 _ l ‘I * -1.0

applying

a model

chloroform-methanol

data

in

with

data random

reduction. errors.

325 kPa).

DEVELOPMENTOF IMPROVED METHODS Extension In

to

the

isobaric

general,

derivative

data

sets

non-isothermal in Eqns (3a)

appearing

Without neglecting

&Q = * 1 instead

of

Eqn.(5)

_ 1 the

x1

and and

the right

d In

non-isobaric (3b) will

case the be partial

hand side of Eqn. (lb)

Pl _ x d ln Y2 2 dxl dxl following expression is obtained:

and

(7)

410

P = xlp;

+x2p;

In

the

exp

isothermal

conditions

This

led

cannot

- $-

HE + RT

+

*

(8) ,1

(

at

moderate

fol low ing

pressures

the

case

dP=O but

fulfilled:

dT -=O dxl

term

*

case

are

-HE RT2

E v RT

and

to Eqn.(5).

dPw0 dxl

In the

isobaric

the

HE

be neglected.

Instead case)

g-x1 i

of a P(x)

a t(x)

spline-polynomial

spline

is

fitted

(as

minimizing

in

the

the

isotermal

following

criterion:

~i-z(xli)l 5

where

the

summation

i

is

for

all

a t(xl,P) function is fitted 2 gp expresses the uncertainity resulting

variance

pragation

law as:

of

(9)

c2*tcxl,P)i

data

points.-As

where

P is

of this

nt(xl,P)

a matter

assumed

of fact,

to be constant,

“constant”

value.Thus

can be obtained

from

the

the error

(10) The

were

approximate

taken

equation) the

majority f12nt of

for

partial

from calculations using

measurement and

values

x,t,P

of the

total

of cases led

the

to the

Thus the concentration,

were

taken

spline

same spline

Then the

differential

Mixon

al.

et

made by some model

data.

liquid

pressure

(19651,

derivatives

regression

Eqn.(lO)

(e.g.

Wilson

errors

that account.

of the

of the

temperature

However

using

constant

is solved

by

in

the

variances

function.

equation which

into

random

in

(5)

involves

the

following

the

method

assumption:

411

ag

=A_2

ax,

which

is

allowed

Comparing above cannot

more

be

data

rather

one

well

or

P(x,)

because to

discover

et

a1.(1967)

polynomial This

originally

one

through

the

kind

the

of

best

is

too

it

is

small) not

function between derivative

two

describe

the

the

Ay effort

best

way of

fitting

polynomials of

as

dy

residuals was

power

a third

We rejected

this

illustrated

by Fig.2.

is

with

residuals.

made

(second) of

compared

measured

spline-polynomials.

reduced

instead

intervals. fit

of

of

intervals

of

by the

order condition. where

the

(Kollsr-Hunek

modified

et

intervals

the

spline-polynomial

measurement knots

(it

the too

and

are

enough If

a spline-polynomial

number

the

the

must

much

spline

contain at

however.

a1.,1982).

intervals

number). will

crude

temperature-

influences

fitting

flexible

sufficient

situation,

using

Therefore

of

determined If

a crude the

distortion

rest

spline

too

HE

details

spline

plot

Even

function

approximation

way of

intervals. is

the

experimental

equation,

is

assumption HE usually

improves

Wilson

used

in

the

procedure.

proposed

Number The

the

the

last

improved

the

Mixon

and as

because

literature.

spline

any

in

first

that

cases

function

parameters

by the

the

obvious however,

the

HE(xl) from

obtained

in

is

on mathematical

t(xl>

order

in

the

energy

Consideration

it isobaric

unfortunately

expressed

Van Ness

(7) in

scarce of

independent

data

and

neglected,

are

The

(1)

awkward

approximation that

if

Eqns

is

(11)

Ax1

too does

(the

not of

the the

of

fit

the

contain

changes attention

special

the

number

spline of

intervals

points

because

parameters

intervals

parameters

points; calls

length

long cannot

number many

the

of

is

and

it

too

of large,

will

the

wave

of

sign

of

to

this

phenomenon.

the

second

412

Van --

Fig.2.

Ness

Comparison

technique. Fig.3.

proposed

sPliW

of

the

modified

and

original

(Pyridine-tetrachloroethylene

shows

the

Ay

on

rather

residuals

spline

spline-fit

measured

with

different

data at 333.15 K).

numbers

of

spline

intervals. Experience measurement

should

data

sets

be contained

measurement

contain

6

showed in

points

three-six

extensive

that

an interval

does

not

points

if

exceed there

number

of

about if

three the

number

and

more

points of

an intervalshould

than

15 data

3

intervals

AY

and

measurement

total

fifteen are

simulated

points.

intervals

A l

0.02--a*

lm

X

0

0

l* 0 *

.o

0

l

.

b-X l

1:o

-0.02 -0 l

l

t Fig.).

The

-methanol

effect

of the number of spline-intervals.

data with random error.s.P=101.325

(Simulated kPa).

chloroform-

413

As the intervals

measured are not

consisted

in

As the

it

is

one

the

of

seen

the

fourth

in

types

of

residuals

investigate

one

The results the

is

worth

function but

in

to

a1.(1973)

OP

the

residuals error

it

is

not

data

calculate

Ot,

above,

same.

x,t,P

but

only any

residual also

for

may be

influences

sufficient

studies in

in

all

the

to

total

many cases pressure

background

The solution

of

at

P. the

ordinary

did

not

After

careful

following

differential

two points: g=O at

and that

starting

Xl’1

these

(12)

conditions

values any

a few cases our

et

mathematical

without

during

Ness

spline

only.

remarking with

be the

using

systematic

errors the

known

g(x)gO,

happened

Ax, of

simulation

x1=0

automatically, cases

them

of

g = 0 at is

be used

mentioned of of

(8)

can thus

was found.

equation

calculated

equidistantly the number of points

to

by Van

be

type

systematic

explanation

required

paper

one; any

reconsideration

It

the

variables

However

reflect

is

may

unused

plotted.

for

the

consideration

another

of

fulfilled

iteration in

“solution”

investigation

are

80 per

from cent

may be found

effect

of

the of the as

it

systematic

errors

proposed

method

P.

The results are

interval

fiy residuals

three

in

data are not situated of equal length if the

compared

obtained

chloroform-methanol

s Remarks

by the

on Fig.47

on the

simulated

the the

azeotropic

systems

shown

on

acetone-benzene

figures: and methyl-ethyl-

systems:

dx= tiy=lg4; Different

and of

system.

For chloroform-methanol, -ketone-water

original

on an example

values

eT=161K of

variances

; are

Cp=13,3

Pa

marked

in

captions.

414 without

with

constraint at

x =O

+25

mm Hg

9 -0 Ap

=

AY

and

constraint

x -1

AY

Fig.4.

Application

(Simulated

Statistical

constraints:

g(O)=0

the

to

for

systematic

consistency

observe

(deviation

Fredenslund

data

with

and

g(l)=0

random

errors.

kPa). tests

To judge proposed

the

chloroform-methanol

P=101.325

trend

of

errors

of

a data

qualitatively from

(1975)

random

proposed

set

the

Van Ness

residuals

scattering). a useful

et

if

al.

they

Christensen quantitative

(1973) show

any

and criterion

of

acceptance: JnylLdX where

dx and

dy are

measurements; the

value

volumes (1)

is

the 0.01.

allow

the right

This

by Gmehling not

Christensen proposed stating

+

known and

for that greater

sy uncertainties hand

proposal et

usually

al. and

Fredenslund

isothermal the method deviations

(13)

side has

of is

does

not

the

by

the

y

approximated accepted

heat

disappear

use

data sets for is not strict in

been

As the

(1975)

x and

practically

also

(1977).

the

same

isobaric in this residuals.

by

for

the

of

mixing

in

in

isobaric

method

008 Eqn. case,

as

data sets but case and one should They

also

415.

propose

that

function

in

criterion, making

the

in

the

“a”

is

and

E (

(in

the of

is

random For

test

where

HE(x) The

between an algorithm

on a strict

to check in

book,

if

our

for

The test

there

is

case)

has

Its

null

1962).

E stands

constant. to

the

a been

expected

value

statistics

is

1=1

mean of follow

Dy values

be equal is

to zero

statistics

is

null

the

(no

is

taken

errors shift

in this

deviations than

hypothesis

value

to check following

the

much smaller

systematic

appropriate

values.

some trend, are

The

without

t-test

Ayi

Rcrit (n)

critical

should

(14)

( Ayi- Ay12

the

data

to

Eqn. (14):

. .

scattering. a

consistency

Linnik’s

Ay)=a,

residuals

subsequent

exceeds

the

reduction.

AYi12 .&T 51 ( Oyi+l-

G the

data

us to distinguish

Ay residuals

in

according

1 -2 n-l

If

to use

for

Our aim was to give

sensitive

n-l t-

where

allow

a non-specified

be calculated

R =

used

basis.

variables

is:

be better

on the

by Abbe (cited

hypothesis

would model

not

decision

severe

proposed

it the

errors.

statistical

A rather

and

does

and systematic

mathematical trend

from

however

random for

principle

calculated

from the

in

accepted

the if

case

R

a table.

constant

residuals). null

between

that

“a”

above

The Student’s

hypothesis.

The

test

expression:

(15) The null Vapor

hypothesis

phase the

accepted

if

t
correction

Considering takes

is

the

following

non-ideality more complete

of the form:

vapor

phase

Eqn.

(5)

416

P = q

exp {g+x*

[q

-Wxl)]}+

1 Ig--

I

1

VE

qJ(x,)=m 2 z.=exp J The data

B. Jk or

2 (3

REAL

(16) I, i

dP ---

HE

dxl

dT

RT*

(17)

dxl

ykBjk-B)P-Bjjp;-V;(P-p;)

k=

(18)

RT

!

second from

virial

I

coefficients

a generalized

Neglecting significant

exp

Wx,)

s-x1

where

g

the changes

necessary in

are

taken

from

vapor

phase

Ay residuals,

VAPOR-PHASE

correction Bs it

is

CONSIDERING

TREATED

PVT

measured

correlation. one may face seen

from

Fig.5.

NON - IDEALITY

AS IDEA4

AY

AY

0.01

0

-0.01

Fig.5.

The influence -benzene P=101.325

of

simulated kPa)

neglecting data

real with

vapor-phase.

random

errors.

(Acetone-

417 Consideration In

of

isobaric

heat

cases

neglected.

Neglecting

introduce

a systematic

(see

Fig.6).

smaller

if

curve

is

Y(xl>

this

do not

tapic

in

during

Eqn.(lG)

cannot

the

reduction

distorting results

affect

remarkably kJ

of

of

further

our

higher

residual it

was the

can

/ mol

effect

data

the

numerical Hiax 41

The

the

term

term

error

(e.g.

1Yy~O.001).

values

mixing

the

Concerning

HE values

residual

of

and

be can

function found

shape

be well

smaller

that of

the

neglected heat

of

mixing

examination.

AY

0,Ol

X -0,Ol

with

without consideration

Fig.6.

Isoba,ric mixing with

data

reduction

enthalpy random

The

with

limited

method

miscible

As

systems.

allows

the

04x

41

showiig subintervals.

the

discretization

, we have

limited

miscibility

has

mathematics to

of

simulated

HE(max)=-2285

data kJ/moleb

s',.'&y=o,ool)

in

above

also tried

kPa;

kPa;

miscibility

described

application

(Ether-chloroform P=26.66

errors.

values

with/without

term.

%=O*lK; 6’p=0.13 Systems

of HE

for extend

dividing

the

been

liquid used

phase only

inherent part the the

of

in the

method [O;l]

for

completely

the

method

interval to

systems

interval

into

418 The

main

boiling

difficulty point

different

arised

curve

for

character

miscibility.

in mixtures

from

An

fitting

that

example

is

the

with for

t(xl)

a miscibility

mixtures

shown

on

curve,

as gap

with

the

is

of

unlimited

Fig.7.

Methyl-ethyl-ketone

-Water \

04+o02

Diethylamine

-Triethylamine

;_x

“I+--

Fig.7.

The

ll0

6.5

different

character

of

systems

with

limited/

unlimited

miscibility. Fitting

a spline

curves.

Polynomials

the

form

tried

but

inflection

two

cases,

flexible

function the

function

of

order

boiling

point

atfk

a/(bxl+c)

were

rational

order the

boiling

fitted

such

to

steep

functions

of

also

been

have

and extrema

was also

or

t

q

tk +

+ 1

is

the

boiling

is

that

for

= tf-tk

for

first

represent

type

hopeless rational

obtained

in

the

function

was

point

curve.

the

two edges

A

of

curve:

0(x;

tf

to

the

Dtfk

t = tk+

tk

points

another

is and

and

while

enough

of

where

third

alxl+bl/(a2xf+b2x1+c)

first not

polynomial

Atfk d(l-xlP

point the

pure

of

the

component

+l

heteroazeotropic

mixture

419 This 1982)

formula but

it

led

to much better

was also

rejected

results

because

et

al.,

a remarkable consonance as the vs x 1 residuals

was found between the At vs xl and ay inadequacies in the t(x1) fit were also residuals

(Kollar-Hunek

reflected

in

the

y

(Fig.8.).

I, At, OC

0

l

0

02 0, -633

1.0 : .+x

l

- 0.2--

a

4

-0.1

-0.;

’

l

. l

i

t Fig.8.

Application

of t=tk+

unmiscible data

systems.

with

random

Reconsidering it

is

not

the

necessary

discretization previous case, used

for

the

t(x,>

there

or P(x,)

from each

are method,

of the equal

curve

three but

points the

for

(in

If difficulties

is

being set

which data

(8 ).

and the

done. are are set

we found

during

equation

data whole

kPa).

sizes

fitting

fitting simulated

Mixon method

step

function.

other

topic

methyl-ethyl-ketone-water that

+l> function

P=101.325

may be omitted

work on this

proposed

tix!

differential

or P (xl)

much different further

details

of the t(x1)

errors.

to use

respectively)

nt,,/(

(Methyl-ethyl-ketone-water

that

the This

way the

isobaric

or isothermal

measured

values

the

are

step sizes are may arise.however,

Results

of a simulated

shown on Fig.9. detected would

very

to

It

is

be wrong

be qualified

seen hy

as

420 inconsistent

using

the

previous

curve

fitting

curve

fitting

method.

AY t

f

Fig.S.Comparison

of

systems

curve

with

fitting)

miscibility. method

(spline

limited

of

P,x,t

spline,

was also

used

methods

(Simulated

random for

errors.

methyl-ethyl-

P=101.325

with

obtained steps)

for

steps

systems

results

and

by the

original new method

steps)

by simulation,

kPa).

(without

unlimited

by the

fit+non-equidistant

are

adding

compared.

random

errors

data. scattering

obtained data)

of

method is

residuals

by new method

traditional thus

reduction

non-equidistant

was prepared

calculated

residuals the

with

spline

fitting

miscibility

data

using

set

curve

data

fit+equidistant

The random In

without

program

previous

The data the

o

On Fig.lO/a.

(without to

with

different

-ketone-water The computer

l

more

less

trend

more

(without

some part

or

a virtual

is

convincing

curve

of

scattering

smoothed

by the

may appear

in

from

fitting). (namely fitting

Ayi

that

of

values

a

being

overpronounced. Results as

obtained

random

-benzene

errors

from (real

a data vapor

system

are

shown

for

data

of

residuals subject

to

systematic

Similar

pictures

are

set

subject

phase

on Fig.lO/b

a system

error found

(in if

to

treated with

while

systematic

systematic ideal) Fig.11.

limited

measuring the

as

the

as well for

acetone-

shows

miscbility

the and

temperature). errors

were

made in

421 SIMULATED

4 AY

4

0.01

--

without

DATA,

t(x)

NO SYSTEMATIC

AY

sPline

A

ERROR

with

t (x) spline

0.01 -’

*

e

0

l

a

l

0-O

.

0

I

*

0

0

0

l

I 1.0

#

0

.*

)-

X

a

oo*

O-

0.

*.

l

‘0 0.’

l

I, *x 1.0

.

0

l

--O.Ol--

0.01

V

b) AY

REAL

df hout

VAPOR

t(x)

PHASE

AY

spline

Cl.03

Fig.10.

TREATED

AS

with

IDEAL

t (x 1 splint?

0.0

Comparison

of

systems

without

-benzene

data

different miscibility with

random

data

reduction gap

errors.

methods

(Simulated P=101.325

for

acetonekPa).

422 theliquid with

concentration.

limited

residuals the these Mixon

applying

the

grows

because

fit

can

data

to

not

steps

the

is

from

absolute

of test

the

slope

is

that

in

use

of

to

of the

of P or T values The

measurement

values

owing

and

necessary.

the

systems

be achieved

problem

points

far

high

case

curve-fitting,

points

of this

data

rapidly

of the

in

or T(x1)

equidistant

measurement

calculations

of

point

that

more practical

P(x,)

of measurement

The main

method

from

is

accuracy

number

curves.

it

a previous

a suitable

small

far

miscibility without

because

We concluded

error

data

of

points

(dP/dxl)

or

testing

are

(dT/dxl)

derivatives. The computer available

programs

from

the

used

for

consistency

authors

CONCLUSIONS Our aim was to consistency

construct

of binary

a reliable

VLE data

algorithm

on the

basis

for

of

the

checking

of

Gibbs-Ouhem

equation. It

was found

torted

by model

deficiencies

purpose

of testing

for

the

that

Spline-polynomials previously

to

differential giving

are the

miscible is

It

was found

in

x l,t,P should

this

and doesn’t

was developped

equation

at

-water

became

to

those For

into It

for

tractable.

isobaric

data that

sets however

in

sacrificing

points.

Thus

method the

the

is

comparison

even well

the

errors

Ay residuals. E?XCept

of

showing this

type

differential

the

system

applied were

to

rather

MEKdata similar

curve-fitting. in

of mixing with

points.

mixtures

sets

coexistence

HE should

heat

of

data

results

previous

on accuracy

case

the

but

of data

smoothes

For

data

of spline

effect

the

of data.

of mixtures

number

their

solving

by applying

cases

curve-fitting

in

appropriate

or T(xl)

number

the

dis-

coexistence

in

optimum

as

not

P(x,>

that

phase.

This

miscibility

consideration, was found

well liquid

non-equidistant

obtained

the

fit

considering

without

fit

are

consistency

of the

was found

previous

in

a method

limited

to

underemphasises

miscibility

without

used

phase

be neglected

the-curve limited

It

they

thermodynamic

solution

be chosen

that

data

methods are generally

therefore the

widely

liquid to

maximum-likelihood

numerical

equation.

intervals

This

the

the

principle data

are

measurement

be taken rather errors

scarce. HE, s

423. AY

At = - 0.5 K

h

0.02--

-0.02

0.

.0.*

**a*

At =+0.5

Fig.11.

The result

of data

miscibility.

Data

(Simulated

P, Instead of established

make easier

the

are

for

distorted

a system

the

Hkax doesn’t variances:

exceed

with

by systematic

metyl-ethyl-ketone-water

may be neglected if considering typical

well

reduction

K

limited errors

data;P=101.325

the

value

kPa).

of 1 kJ/mol

= Q y = lo-3

visual statistical

investigation of residuals mathematically tests were proposed in order to

algorithmization.

424

The

Acknowledgement: and

dr.P.Jedlovszky

authors for

are

very

helpful

grateful

to

dr.J.Manczinger

discussions.

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