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Application of a maximum likelihood method using implicit constraints to determine equation of state parameters from binary phase behavior data

Application of a maximum likelihood method using implicit constraints to determine equation of state parameters from binary phase behavior data

Fluid Phase Equilibria, 50 (1989) 249-266 249 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands Application Equation of a ...

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Fluid Phase Equilibria, 50 (1989) 249-266

249

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

Application Equation

of a

Maximum

of State

Likelihood

Parameters

from

Method Binary

Using

Phase

Implicit

Behavior

Constraints

to Determine

Data

Vicki G. Niesenl and Victor F. Yesavage Department Colorado

of Chemical

Engineering

and Petroleum

Refining

School of Mines

Golden, CO 80401 (Received

December

30,

1988;

accepted

in final

May

form

20,

1989)

ABSTRACT The maximum

likelihood

in the determination uncertainties.

are complex

new form of the maximum systems

were obtained equation

m-cresol

functions

data and the associated

greatly facilitates

of pressure,

temperature,

likelihood

method

is illustrated

tetralin

+ quinoline,

mixing

to allow the use of implicit

from experimental

constraints

+ quinoline,

for six different

of state.

rules involving

has been modified

of model parameters

The use of the implicit

models typically binary

method

using recently

since

obtained

The

data for the

+ tetralin.

Parameters

with the Soave-Redlich-Kwong

The mixing rules ranged from a simple interaction

temperature

modeling

and composition.

and m-cresol

rules used in conjuction

experimental

phase behavior volume,

constraints

parameter

to more complex

and volume dependencies.

INTRODUCTION The estimation from very simple, statistical

accuracy

of parameters

of the

that is, the best estimates squares of the deviations variables

in mathematical

to very sophisticated fit. of the

methods.

Typically,

variables

a least squares

are those which minimize and dependent

are true values.

data

the weighted

values.

variables.

This

criterion

0 1989 Elsevier Science Publishers B.V.

is used;

sum of the

In this method A basic assumption

assumption

the is

does not take into

&-rent address: Thermophysics Division, National Institute of Standards and Technology (formerly National Bureau of Standards), Boulder, CO 80303.

0378-3812/89/$03.50

can range

chosen is based upon the desired

values from their calculated

are divided into two groups, independent

that the values of the independent

from experimental

in a simple method,

parameters

of the observed

models

The method

250

account

the experimental

uncertainties

cannot give the best possible When model,

using a least squares

the temperature

variables

criterion

difference

used

between

with the independent

estimation

criterion

for fitting

phase

the measured

case the following function

for estimating

and calculated

equilibria

data

and inherently

to a mathematical

(x) may be chosen

(p) and the vapor composition

as the basis

variables

of the parameters.

(T) and the liquid composition

while the pressure

typical

associated

statistical

the best bubble

as the independent

(y) are the dependent fit parameters

point

pressures,

variables.

One

is to minimize

Holder

(1986).

the

In this

is minimized:

N

s =

(pcalc

c

pmeas)a/(pmeas)z

_

(1)

i

where

N is the number

experimental

of experimental

errors associated

not use the measured A much experimental

vapor composition

more

sophisticated

(1973).

parameters

This method

likelihood”

neglects

the fact that there are

(T) and liquid composition

(x) and does

(y) at all.

method

utilizes

errors to obtain the best statistical

known as the “maximum Luecke

data points.

with both the temperature

method

In the case of evaluating

are chosen so as to minimize

all experimental

estimates

as described parameters

data

of the parameters. by Anderson

and the

associated

One such method

is

et al. (1978) and Britt and

from vapor-liquid

equilibrium

data,

the

variables

and

the following function:

N

s=

c

UP

cak

_

pmeas)~

/gzi

+

(plc

-

pms):l~‘.

Y’

i

+ (TCalC - Tmeas)q /&

where

c2

,

cr2,

cr2 T’

cr2

+

are the statistical

N is thePnumYber of datz points. within prescribed The experimental errors.

variances,

maximum

Essentially,

thereby

likelihood

measurements

In addition,

(xcalc

models

maximum information. methods. experimental

cannot

likelihood

/cti}

variances

associated

the method

be known

adequately principle

requires

and that

that

the

values to float

the physical

to an empirical

such as the maximum uncertainties,

distribution

are subject

properties

for weighting

likelihood

likelihood

within

(1983) states

of pure or mixed

procedure

method

method,

should be a superior

of the

only to random

the measurements

Skjold-Jorgensen

is true not only for the maximum

a method

statistical

the measurements

uncertainties.

represent reduces

data and experimental

with the measured

allows all experimental

the model must be capable of representing

This certainly However,

xme$

giving the best possible fit of the VLE data. method

less than the order of the experimental many

-

the

an order that since

fluids,

the

available

but also for other

which uses all available

empirical

method.

251

One inherent Anderson

problem

with

et al. (1983) is that

terms of the independent

the

development

it requires

variables

that

of the

maximum

the dependent

and parameters.

likelihood

variables

be written

In the case of phase equilibria,

method

by

explicitly

in

the constraints

are:

f; = ff When

an equation

functions

of state

of pressure,

functions.

To determine

example,

if fugacity

may be considered be assumed

(EOS)

coefficients

derivation between

use the

presents

DEVELOPMENT IMPLICIT

explicitly

the equations

cannot

requires

are complex

be reduced

certain

and vapor composition.

to explicit

approximations.

is fairly independent

of the pressure.

original

implicit

relationships

of the maximum

likelihood

coefficient

These approximations

between

For

In the liquid phase it can

upon the fluids and the region of interest.

Obviously,

the

variables.

method

may or

a more accurate The

for implicit

following

relationships

for two constraints.

OF THE MAXIMUM

It is assumed

the remaining

LIKELIHOOD

that there are N experimental variables,

variables

independent

variables,

which

METHOD

FOR

EQUATIONS

of a set of measured

chosen

of pressure

a development

the variables

compositions

are used in each phase, then the vapor phase fugacity

a weak function

that the fugacity

would

and

two of the variables

may not be valid depending scheme

is used to solve the constraints,

temperature,

Y and

M.

K variables

being dependent.

variables

observations

X is a vector

and is of length

with each observation

are chosen to be the independent

NK.

that

consists

consisting

variables

with

of the true values of the

In the case where

Z are each a vector of the true values of a dependent

there

are 2 dependent

variable

and both are of

length N. Since there are two dependent relate the dependent

F(X,Y,

Z,O)

variables

variables,

to the independent

two equations variables

(or constraints)

and parameters.

will be required These are written

= 0

G (X,Y, Z,19) = 0 where

(5)

0 is a vector of L undetermined

equation

to as:

parameters,

and

F and

G are vectors

of length N (one

for each data point). The criterion

S = (X -

for convergence

XmjT X (X-X")

is chosen as the minimization

+ (Y - Ym)T7 (Y-y”)

+ (z _

pjT

6 (z

of the following function:

_

p)

(6)

252

Superscript

m indicates

whose elements

a measured

are the squared

6 are diagonal

matrices

The symbol

variables

equation

on the constraints

Y

and

X is a diagonal

of the variances

of size (N,N) whose elements

of the dependent 6 based

value.

reciprocals Z

are the squared

respectively.

matrix

for the independent

If

Y

of size (NK,NK) variables.

reciprocals

and

Z

4 and 5, then the above function

7 and

of the variances

can be eliminated can be minimized

from in the

following manner.

as/ax= aslao=

0

(7)

0

(8)

To eliminate accomplish

Y

and

Z from equation

this substitution,

expansion



GY

the constraint

F’ + Fx (x - XT) +

variables

are linearized

of the parameters

which provides

in terms

of

X

0 . To

and

by a first order Taylor series

and the most recent estimates

a Newton-Raphson

of the

aspect.

(9)

F, (Y - Y’) + F, (Z _ Zr) + F,(O _ 0r)

(10)

G’+G,(X-X’)+G,(Y-Y’)+G,(Z-Z’)+G~(~_~’)

The superscript

r indicates

should approach

zero as convergence

and

equations

about the most recent estimates

true values of the independent

F

6, they must be written

G equal zero.

previous

iteration.

variables

in vector

derivatives matrices

of F

z--z=

=

By definition

F,

of F and

X

and

G,

are the derivatives

and are sparse G with respect F,

and

G,

8 and are matrices

These two linear equations and the previous

The error associated N containing

and

in vector

iteration.

is approached.

Fr and Gr are vectors of length

of size (N,N).

parameter

the previous

values of F and

matrices

G with respect F, , F, ,

to the dependent

variables of

F

Y and

and G

9 and 10

equations,

the values of F and

of size (N,NK).

are the derivatives

with equations

of the constraint

F

G at the

to the independent G, , and

G,

are

Z and are diagonal with respect

to each

of size (N,L).

(9 and 10) can be solved for Y-Yr and Z-Zr in terms of X , 0 , G

-(G,-G,F;‘F,)-‘((G’-G,F,‘F’) + (G, - G, F;’ F,)( X - Xr) + (G, - G, Fj’ F,)(O - 0’))

y - y’

=

(11)

- (G, - G, F,’ Fy)-l ((G’ - G, F,’ F’) +

(G, - G, F,’ F,)(X

+

(G, - G, F,’ Fe)(6’ -Or))

-x’)

(12)

253

These equations

can be substituted

into equation

of Fr, Gr, 6’ , Yr, Ym, Zr, Zm, and

s =

6 to give the function

to be minimized

in terms

X

(X-Xm)TX(X-Xm) (Yr-Ym-(Gy-GzF;‘Fy)-l ((G’-

G, F,’ F’) + (G, - G, F,’ F,)(X

7 (Y’-

-X’)

+ (G, - G, F,’ F&I

-Er)))T

Ym -(Gy - G, F,’ F,)-’

((G’ - G, F,’ F’) + (G, - G, F,’ Fx)(X - X’) + (G, - G, F,’ F,)( B - /I=))) (Z’ - Zm - (G, - G, Fj’ FJ-’ (( G”GyF;’

F’) + (G, - G, FJ’ F,)(X

- X’) + (G, - G, Fjl F&(0 - fP)))T

S(Zr-Zm-(Gz-GyFylFz)-l ((G’ - G, Fj’ F’) + (G, G, F;’ F,)(X

- Xr)+( G, - G, Fjl F,)(O -P)))

(13)

V The following

quantities

are redefined:

fr = Y’ - (G, - G, F,’ FY) -’ (G’ - G, F,’ Fr)

(144

f, =

- (G, - G, F,’ Fy)-l (G, - G, F,’ F,)

(14b)

fs =

- (G, - G, F,’ F,)-’ (G, - G, F,’ F,)

(14c)

gr =

Z, - (G, - G, FY’ F,)-’ (G, - G, F;’ F’)

(14d)

gx =

- (G, - G, F;’ F,)-’ (G, - G, Fj’ F,)

(14e)

ge =

- (G, - G, FJ1 F,)-’ (G, - G, FJ1 Fe)

(14f)

So that,

s =

(~-xm)~~(x-xm)+[~-~m+f,(x-xr)+f~(e-er)]~7 * [F - ym + f, (x - xr) + f, (0 - or)] + [gr - zm + g, (x - xr) + ge

(0 - eqT 6

* Lgr - zm + g, (x - xr) + ge (0 - or)] By redefining equation developed

variables

11 in the paper here

is exactly

in equations

by Anderson the

same

(15)

14a-14f the resulting et al.

as the

Thus

explicit

equation

from this method

point

15 has the same form as on, the implicit

as described

by Anderson

method et al.

254

Following equation

their

development

the two minimizing

conditions

(equations

7 and 8) are applied

to

15 to obtain:

A0 = #+l -or

= -ITT

- RTD-‘R]-l

[U - RTDV1 Q]

(16)

and AX

= X=+‘-X=

= -D-l[Q+RAO]

where the following

matrices

(17)

are defined as:

D

= x + f: ,y f, + g’x s g,

(184

R

= f: r f, + g: 6 ga

(18b)

TT

= f; -/fe + g; 6ge

(18~)

U

= f; .y AYm + g; S AZm

(18d)

Q

=

X AXm + f: ,y AY” + g: 6 AZm

UW

AXm=

X=-Xm

(W

AYm=

Y=-Ym

(1%)

AZm=

Z=-Zm

(18h)

Equations equations

16 and 17 are the basis for the maximum

12 and 13 in the paper by Anderson

and Xr+l are the new estimates from equation

Computationally,

be inverted

is (NK,NK);

experimental

the computational

et al. (1978) also provided

very useful for comparing

different

variables.

Most

can be easily written.

data points and the number of independent

Anderson

variables.

Equations

11

The new value of S(SN) is

to the last value of S

is not difficult.

algorithms

therefore,

to

Convergence

is achieved

level.

the algorithm

for which inversion

and correspond

Or+1are the new guesses for the parameters

of the dependent

15 and compared

when (SN - S)/Ss is below the criterion symmetric

method

for the “true values” of the independent

and 12 are solved to give the new estimates then calculated

et al.

likelihood

equations

a statistical

matrices

are either

diagonal

or

matrix

to

to the number

of

The largest symmetric

time involved

is related

variables. test to compare

of state or mixing rules.

models that proves to be

A value of s2 is calculated

for each model.

s2 = S/(N-L)

(19)

255

S = minimizing function as defined by equation 15 N = number of data points L = number of parameters The ratio of the distribution,

s2

values for two models is calculated

and will be distributed

assuming errors due to lack of fit are normally distributed

as the F

(Anderson et al. 1978).

F = s; /s; where subscript 1 refers to model 1 and subscript 2 to model 2. At a contidence level a , the value of F is compared to tabulated and

F(a:z+ v2) values.

vi

v2 are the degrees of freedom for the two models and are calculated as the number of data

points minus the number of parameters, models are considered statistically

(N-L).

If F is greater than F(a:vr, vs) then the two

different at a confidence level of (x . It is also possible to use

the same statistical test to compare how well a single model represents two different sets of VLE data.

MAXIMUM LIKELIHOOD METHOD FOR PHASE EQUILIBRIA

MEASUREMENTS

The preceding derivation can be used to fit almost any type of thermodynamic as many other types of data.

When the maximum

likelihood equation

data as well

is applied to phase

equilibria data, the following equations are used. For binary phase behavior, the Gibb’s phase rule indicates that there are two degrees of freedom.

Therefore, if the variables are temperature

(T), pressure (P), liquid composition

(x),

and vapor composition (y) then fixing any two of the variables will determine the other two, based upon two equilibria

requirements

which require that the component

fugacities (f) in the two

phases be equal:

f: = ft

and fi = fi

These equations can be written in terms of the compositions and the fugacity coefficients

(22) where y1+

Yz

=

1

and

xi + xi = 1 .

(23)

256

The two constraint

equations

F(X,Y,Z,0)

= $;yr-$;x,

GV,Y,Z,Q

=

for the maximum

are given as:

(24)

4;~~-4& = 0

(25)

If T and x are chosen as the independent and vector

vector containing

method

= 0

sets of T and x and will be 2N long. variable

likelihood

variables,

Vector

then the vector

Z will consist of N values of y as the other dependent L interaction

parameters

X will consist of N

Y will consist of N values of P as a dependent

for the mixing rule.

The function

variable.

0 is the

to be minimized

is:

N

c

{(P CHIC-

S=

pmeas)f

jnii

+ cycalc

- peas)~

/gfi

i

+ (Tcalc -Tme+

where

gi , f

variances

/&

, g2 T'

a2 are the variances x

are estimated

as the experimental

with the experimental

apparatus

The derivatives either

be calculated

were found numerically,

likelihood

difference

likelihood

best possible

rules.

P , T , x , and

y . These

and are based on experience

GB as given in equation

routine, + tetralin

expressions

of state.

To change to a

needs to be changed.

METHOD

TO EXPERIMENTAL

interaction

parameters

system

(SRK) equation acentric

as reported of state, factors

of state to the pure component

of the geometric

the derivatives

et al., 1972).

of the equation coefficients

13 can

for the derivatives

was to calculate

(Abramowitz

fugacity

Pure component

fit of the SRK equation

mixing rules are modifications

formula

LIKELIHOOD

Each model used the Soave-Redlich-Kwong

method

is independent

fitting

data for the m-cresol mixing

Fa and

GY'

The analytical

that calculates

OF THE MAXIMUM

one of six different

Y'

A much easier

algorithm

EOS, only the subroutine

fit to phase behavior

values

errors in the variables

or numerically.

cumbersome.

Using the maximum

(29)

of the measured

Fx , Gx , F, , G, , F

using a five point central

APPLICATION

/gzi}

used to obtain the data.

analytically

to be quite

The maximum different

+ (xcalc - xm=)f

DATA

for six models were

by Niesen, et al. (1988).

(Soave, 1972) in addition were optimized vapor pressures.

mean mixing rule for the attractive

to obtain

to the

All of the term

a as

shown below for binary mixtures:

b=

c

xi bi i

(29)

251

a=

Xi Xj aij

(30)

Model 1:

a = x: a1 + xi a2 + 2x, x2 ( a1 az)Oe5 (1 - k,,)

(31)

Model 2:

a = xf a1 + xi a2 + 2x1 x2 ( a1 a2)o’5 (1 - A - B/T)

(32)

Model 3:

a = x~a1+x~az+2x1xz(a1az)0*5(1-A-B/~)

(33)

Model 4:

a = x: a1 + $ a2 + 2x1 x2 ( a1 a2)Oe5 (1 - A + B/vRT)

(34)

Model 5:

a = xf a1 + xi a2 + 2x, x2 ( a1 az)Oe5 (1 - A)

cc i

j

(~/VW (x1 cl2

+ 23 x2

Model 3 (equation Luedecke,

(35)

33) was proposed

(36)

by Holder, (1986) and Model 5 (equation

35) was proposed

by

et al. (1985).

The standard experience standard

aT

czl)

a = Xi a1 + X; a2 + 2x, x2 (a1 az)Oa5 (1 - A - B/T - C/v)

Model 6:

ffP

+ x2

deviations

were estimated

with the experimental deviations

apparatus

were estimated

=

* 0.7 kPa below 689.5 kPa

=

* 3.5 kPa between

=

* 13.79 kPa above 3500 kPa

=

*O.lK

from the experimental

used to obtain

the data.

errors

and were based on

For all of the VLE fits, the

to be:

689.5 - 3500 kPa

ux, cy = * 0.002 mole fraction The fitting in less than

routine

was found to work very well and convergence Also any reasonable

10 iterations.

initial

was possible

for all mixing rules

guess could be used without

changing

the

final value of the parameters. Results

of the phase behavior

and the parameters parameters

modeling

attempted

in this research

are listed in Table 2. Model 6 was only studied

are not given.

In

oij is the level of confidence

Table 1, the results

of the

to which it can be stated

F that

are tabulated

in Table 1

briefly and the corresponding

test were used to compare models

i

and

j

models.

are statistically

258

TABLE

1

Comparison Model

of thermodynamic s2

ail

models:

m-cresol

ai

ai

4

21.8 19.6

-

: 5 6

21.0 17.3 a.7 10.0

$0 ns 9ngp9 9?9 99.0 99.0

+ tetralin

Qi4

-<90 99.5 99.0

ai

99.9 99.0

ns

at 523-598 K

AP (%)

AY (%)

0.90 1.00

1.29 1.42

1.06 1.02 0.46 -

1.19 1.27 1.75 -

ns = significantly less than a 907.confidence level

TABLE

2

Interaction

parameters:

m-cresol

+ tetrabn

PARAMETERS A = 3.833-2

1

2

A = -1.993-2

B = 31.973

3

A = 8.593-2

B = 18.475

A = 5.363-2

B = 27259270

4 5

A = -5.173-2

6

A = -2.1233-3

Cl2 = -5.71E-2

c2, = -5.29E-2

B = 29.556

C = 64.19

The values of Cl2 and C21 are normalized by ai(TJ2

different.

In addition,

variables

temperature

a more (T)

pressure

(p) and vapor composition

between

measured

Ap

values of p and

Using the results

from the

but model 5 showed superior Table average

measure

composition

(y) were calculated. y and predicted

of comparison

(x) were The

held

is also presented.

constant

and the

average absolute

values were calculated

percent

The

variables deviations

and are presented

as

Ay .

and

better.

conventional

and liquid

When comparing 1)

F test, there is little statistical

performance

over all other

the choice of the best model is much

absolute

percent

models

the models using the more conventional deviation

in pressure

more obscure.

difference

between

at a confidence means

models

14

level of 99% or

(Ap or Ay, as shown in

Model 5 has a significantly

than all of the other models;

however,

lower

the average

259

absolute

percent

deviation

of y is the largest

of all of the models.

to choose one of the other models which did not predict composition

better.

orders of magnitudes

Since the physical

values

and have different

experimental

the relative

values of one variable

deviations.

To resolve the discrepancies

Ay), graphs of models l-5 the

F

test obtained

between

represents

these experimental

These results

reasons

are three

F

it becomes

are presented

likelihood

of comparison

vapor

are different

quite difficult average

in Figures

method.

test is a superior

dependency

for the superior

major

composition

absolute

to weigh percent

(F test vs. Ap or

1 and 2. It can clearly

Graphs

method

of the models

of determining

by

at different

which model best

data.

superiority.

the case, then models 24 There

the

errors,

the two methods

can allow some conclusions

functional

none show statistical be primary

that

and vapor

the data is model 5 (shown in figure B) as predicted

from the maximum

also confirm

In this case, one might be led

quite as well, but predicted

when simply comparing

at the 523 K isotherm

isotherms

have a different

of pressure

over another

be seen that the model that best represents

pressure

temperature,

These same dependencies predictive

capabilities

should show increased

distinctions

to be drawn

(volume,

between

modeling

models

320 1

24

about

Model 5.

or volume

are also present of the mixing abilities

24

each

and temperature)

yet

in Model 5 and may

rule.

If this were truly

over model 1, but they do not.

and model 5:

(1) model 5 has a cubic

-K-----,.

__-_

80

MOLE

P&CENT

Models

rn%?ESOL

Figure 1. M-Cresol + Tetralin at 523 K Solid = Model 1, Long Dash = Model 2. Short Dash = Model 3

100

260

Ftgure 2

M-Cresol + Tetralin at 523 K Solid = Model 4, Dash = Model 5

5000”““‘1”1”U”““““““““‘I”‘I”I”‘llll 20

80

MOLE

P&CENT

TEkLIN

Ftgure 3. MGesol + Tetralin at 523 K SolId = Model 1. Dash = Model 5

100

compositional

dependency

whereas

models

14

second term in model 5 is not multiplied and (3) model 5 has one additional In an attempt briefly.

to determine

has a third

are multiplied

parameter.

Table I, when statistically

the

F

superior

the reason for the superiority

by aij

test

14

6 is the additional

rather

using these presented

binary

same models in Tables

all of the models

3-6.

volume,

different,

systems,

m-cresol

of model 5 is not as great. reasonably

significant

difference

well for this between

TABLE

1 2

model

A further

conclusion

data)

43.0 30.0 33.0 41.0 19.0 19.0

may be drawn

14

yet

that

since

of model 5 is the additional

and quinoline

likelihood Models

method

14

Model

+ tetralin,

(Niesen,

+ quinoline have little

In fact, graphical

system,

1987).

system,

were studied The results

the difference

statistical

observations

6 was not

the results

significantly

to be not statistically

models:

Qil

ai

<90 <90

-

;I 99

of models

difference

are

between and the

indicate

that

all models

since

there

was little

studied

models l-5.

+ quinoline

of thermodynamic

S2

6 was

the most significant

were very similar

better

different

ns <90 90-95 90-95

ai

<90 95 95

tetralin

from model

ai

+ quinoline

5, thus

ns = significantly less than a 90X confidence level

at 523 - 598 K

ai

AI’ (%)

AY (%)

-

1.35 1.11 1.20 1.37 0.49

1.38 2.37 1.62 1.23 2.26

-

to the m-cresol

than all other models.

3

Comparison Model

models,

dependencies

(for these

+ quinoline

system.

where model 5 performed

and again found

As shown in

level; model 5 and model 6 were

is that

In the case of the m-cresol

performed

tested

yet

binary.

6 to all other

the main improvement

and the maximum

superiority

system

model

and all

models 24,

form.

is much less clear cut.

For the tetralin

as models 24

+ tetralin

and temperature

parameter.

than the functional

Two other

the

of Model 5, Model 6 was studied dependency

a 99% confidence

A logical conclusion

models 5 and 6 are not statistically parameter

24,

dependency,

different.

parameter.

of model

models

temperature

this model closely resembles

on the m-cresol

at almost

Model 6 has the compositional, has an additional

compositional

Essentially,

was used to compare

to models

(2) unlike

which has a built-in

parameter.

Model 6 was tested

shown to be not statistically

tetralin

are only quadratic, aii

This mixing rule has the same quadratic

three parameters

aspect

by

+

Model 6 was

corroborating

the

262

TABLE 4 Interaction

parameters:

tetralin

+ quinoline PARAMETERS

MODEL

A = -1.523-2

1 2

A = -.1355

B = 67.791

3

A = 2.533-2

B = 14.657

A = 3.1333

B = 31083465

4

Cia = -7.183-2

5

A = -.1356

6

A = -9.7993-2

Cz, = -6.663-2

B = 15.242

c = 70.384

The values of CIz and Czl are normalized by ai

TABLE 5 Comparison

of thermodynamic s2

Qil

models:

ai

Qi3

1

21.5

; 4 5 6

21.4 18.3 ns - ns 21.2 ns 16.6 ZO GO ns not tested on this system.

ns :

m-cresol

Qi4

+ quinoline

Oi5

<90

-

at 523- 598 K

AP (%)

AY(%)

1.15

2.36

0.95 1.13 1.13 0.81

2.28 2.09 2.30 1.80

significantly less than a 907.confidence level

TABLE

6

Interaction

parameters:

m-cresol

+ quinoline

Model

Parameters A = -8.OOE-2

1

2

A = -5.583-2

B = -13.652

3

A = -.1389

B = -18.516

A = -9.683-2

B = -24613426

4 5

A = -7.573-2

The values of Cl2 and C2l are normalized by ai(TJ2

Cl2 = 7.09E-3

C2i = -2.163-3

263

conclusion

drawn from the m-cresol

Figure 6 is a plot of the tetralin was the simple correlating

interaction 4%.

+ quinoline

parameter

the data except

high by almost

+ tetralin

prediction

by putting

must be taken obtained

better

indicate

by model 1 which

a quinoline

job of

vapor pressure

that is

the quinoline

vapor pressure,

it should

Figure 6 shows that indeed for the high quinoline

a bend in the model to bring it back in line with the data. mixing rule which better

the data to correct

for an erroneous

when using multiparameter that the pure component

of state before meaningful

support.

due to the error in vapor pressure.

than model 1. However,

mixing

are not the result of simply curve fitting

results

has additional

1 does a reasonable

Notice that model 5 quickly corrects

not the result of using an improved result of “fitting”

Model

of state predicts

of state for predicting

not be able to fit the data significantly

pressure

and Model 5.

the entire curve is shifted

Since model 5 uses the same equation model 5 fits the data very well.

This conclusion

data at 598 K and the prediction

model

that the equation

Consequently,

system.

comparisions

represents

Certainly,

that

the phase boundary

properties

the phase behavior,

vapor pressure.

rules to ensure

care

that

In addition,

modeled

but a

extreme

the parameters

curves.

must be adequately

can be made between

vapor This is

are these

by the equation

mixing rules.

CONCLUSIONS The parameters

maximum

likelihood

from experimental

also the experimental information

errors

likelihood

method

dependent

variables

cannot

maximum

for implicit

and tetralin

the

maximum

method

method

explicit

for explicitly

for obtaining

In addition,

constraints. without

the method

yields

A major limitation

of the

In many applications,

simplifying

developed

in this paper eliminates

modified

maximum

model

values but

the The

assumptions.

this limitation

since it is

data for the binary systems The data were modeled

mixing

rules.

Besides

method

likelihood

m-cresol

method,

+ quinoline,

parameters m-cresol

using the Soave-Redlich-Kwong

providing

also produced

estimates statistical

were

+ tetralin equation

for the model parameters, information

which

of the

was used to

and analyze the models studied.

component

systems,

vapor pressures

provided

reasonable

523 and 600 K, temperature affect

with the measurements.

be solved

likelihood

For the binary

method

useful

not only uses all measured

analysis of the model and the data.

use of the

+ quinoline.

state and six different compare

The method

constraints.

from experimental

modified

is an extremely

was that it required

likelihood

To illustrate obtained

associated

useful in the statistical

maximum modified

method

measurements.

the

capabilities rule with

performance

the SRK equation

within

experimental

of state was not capable accuracy.

fits for the mixing rules. or volume

of the mixing

functional rule.

For these systems, dependencies

The major

of a mixing rule was found to be the addition three

parameters

was capable

Nevertheless,

at temperatures

the pure likelihood between

were found not to significantly

contributing of a third fitting

of not only modeling

of modeling

the maximum

factor

to the

parameter.

the experimental

data

modeling A mixing but also

264

attempted the

to correct

simple

pressure

for poor vapor pressure

interaction

predictions

parameter

it was also concluded properties

model

by the equation that

predictions

by the equation

performed

well

of state prevented

the equation

of state

of state.

qualitatively;

more quantitative

must

however, results.

be able to model

well before the mixing rules can be more definitively

In several cases

evaluated.

A, B, C

model parameters

D

defined by equation

F

a vector of length (N) of a constraint

equation

as in equation

4

G

a vector of length

equation

as in equation

5

18a

Fx 2Gx

the derivatives

of F and

G with respect

to the vector X

FY ’ GY Fz , Gz

the derivatives

of F and

G with respect

to the vector Y

the derivatives

of F and

G with respect

to the vector Z

F-0) Ge

the derivatives

of F and

G with respect

to the vector 0

K

number

of independent

variables

L

number

of parameters

N

number

of experimental

R

defined by equation

s

the function

T

temperature

TT

defined by equation

18~

Q

defined by equation

18e

U

defined by equation

18d

X

vector consisting

of the independent

Y

vector consisting

of one of the dependent

variables

of length

Z

vector consisting

of one of the dependent

variables

of length (N)

observations

18b

to be minimized

as in equation

2

variables

of length (NK) (N)

a

attractive

term of the Soave-Redlich-Kwong

EOS

ai

attractive

term of the Soave-Redlich-Kwong

EOS for component

b

repulsive

term of the Soave-Redlich-Kwong

EOS

bi talc

repulsive

term of the Soave-Redlich-Kwong

EOS for component

f!

fugacity

f;

fugacity

the superscript

fr, f, , fs, g* , g, , gs kij

indicates

a calculated

of component

i in the liquid phase

of component

i in the vapor phase

defined by equations

interaction

property

parameter

(14a - 14f)

vapor

the pure component

LIST OF SYMBOLS

(N) of a constraint

poor

Consequently,

using the model

i i

265

meas

the superscript

P r

pressure

indicates

a measured

the superscript

r indicates

T

the superscript

T indicates

V

volume

Xi

liquid composition

of component

i

of component

i

experimental

the value at the previous iteration the transpose

of the matrix

Yi

vapor composition

Qij

confidence

A0

defined by equation

16

AX

defined by equation

17

AX”’

defined by equation

18f

AYm

defined by equation

18g

AZrn

defined by equation

18h

6

(N,N) diagonal

matrix of variances

for vector Z

-r x

(N,N) diagonal

matrix

for vector Y

(NK,NK)

u

degree of freedom,

4:

liquid phase fugacity

$1 ll

fugacity

coefficient

standard

deviation

c2

variance

of the experimental

0

vector of parameters

level of model i compared

of variances

diagonal matrix

property

to model j

of the variances

for vector X

N-L coefficient

of component

of component

i

i in the vapor phase

measurement

of length

(L)

ACKNOWLEDGEMENTS The authors Department and Grant

gratefully

of Energy.,

acknowledge

the financial

Office of Fossil Energy through

support

of the United

States

Grant #DE_FG22_80PC30230

#DGFG22_84PC70006.

REFERENCES Abramowitz, M. and Stegun, IA., 1972. Publications, Inc., New York, 9th ed.

Handbook

Of

Mathematical

Anderson, T.F., Abrams, D.S., and Grens, E.A., 1978. “Evaluation Thermodynamic Models,” AIChE Journal, 24: 20-29. Britt, H.I. and Luecke, R.H., 1973. “The Estimation Technometrics, 5: 233.

of Parameters

Functions.

of Parameters in Nonlinear,

Dover

for Nonlinear Implicit

Models,

266

Holder,

G.D., and Mohamed, R.S., 1986. “High Pressure Phase Behavior in Systems COr and Heavier Compounds with Similar Vapor Pressures,” Presentation Meeting, New Orleans, La.

Containing at AIChE

Luedecke, D., Prausnitz, J.M.,, 1985. “Phase Equilibria for Strongly Nonideal Mixtures From and Equation of State with Density Dependent Mixing Rules,” Fluid Phase Equilibria, 22: 1-19. Niesen,

V.G., and Yesavage, V.F., 1988. “Vapor-Liquid Equlibria for m-Cresol/Quinoline Temperatures between 523 and 598 K,” J. of Chem. & Eng. Data, 33: 138-143.

Niesen,

V.G., and Yesavage, V.G., 1988. “Vapor-Liquid Equilibria for m-Cresol/Tetralin and Tetralin/Quinoline at Temperatures between 523 and 598 K,” J. of Chem. & Eng. Data, 33: 253-258.

Skjold-Jorgensen, S., 1983. “On Statistical Fluid Phase Equilibria, 14: 273-288. Soave, G.S., 1972. “Equilibrium Constants Chem. Eng. Sci, 27: 1197-1203.

Principles

in Reduction

from a Modified

of Thermodynamic

Redlich-Kwong

Equation

at

Data, of State,”