Fluid Phase Equilibria, 50 (1989) 249266
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
mcresol
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 SoaveRedlichKwong
The mixing rules ranged from a simple interaction
temperature
modeling
and composition.
and mcresol
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.
03783812/89/$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 vaporliquid
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
SkjoldJorgensen
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 (XX")
is chosen as the minimization
+ (Y  Ym)T7 (Yy”)
+ (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 NewtonRaphson
of the
aspect.
(9)
F, (Y  Y’) + F, (Z _ Zr) + F,(O _ 0r)
(10)
G’+G,(XX’)+G,(YY’)+G,(ZZ’)+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
zz=
=
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 YYr and ZZr 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
(XXm)TX(XXm) (YrYm(GyGzF;‘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(ZrZm(GzGyFylFz)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)~~(xxm)+[~~m+f,(xxr)+f~(eer)]~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)
14a14f 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=
= Dl[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/(NL)
(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
(NL).
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 SoaveRedlichKwong
method
is independent
fitting
data for the mcresol 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(1AB/~)
(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:
mcresol
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 523598 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:
mcresol
+ tetrabn
PARAMETERS A = 3.8332
1
2
A = 1.9932
B = 31.973
3
A = 8.5932
B = 18.475
A = 5.3632
B = 27259270
4 5
A = 5.1732
6
A = 2.12333
Cl2 = 5.71E2
c2, = 5.29E2
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 l5 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. MCresol + Tetralin at 523 K Solid = Model 1, Long Dash = Model 2. Short Dash = Model 3
100
260
Ftgure 2
MCresol + 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
36.
volume,
different,
systems,
mcresol
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 l5.
+ quinoline
of thermodynamic
S2
6 was
the most significant
were very similar
better
different
ns <90 9095 9095
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 mcresol
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 mcresol
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 mcresol
at almost
Model 6 has the compositional, has an additional
compositional
Essentially,
was used to compare
to models
(2) unlike
which has a builtin
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 = 1.5232
1 2
A = .1355
B = 67.791
3
A = 2.5332
B = 14.657
A = 3.1333
B = 31083465
4
Cia = 7.1832
5
A = .1356
6
A = 9.79932
Cz, = 6.6632
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 :
mcresol
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:
mcresol
+ quinoline
Model
Parameters A = 8.OOE2
1
2
A = 5.5832
B = 13.652
3
A = .1389
B = 18.516
A = 9.6832
B = 24613426
4 5
A = 7.5732
The values of Cl2 and C2l are normalized by ai(TJ2
Cl2 = 7.09E3
C2i = 2.1633
263
conclusion
drawn from the mcresol
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
mcresol
method,
+ quinoline,
parameters mcresol
using the SoaveRedlichKwong
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
F0) 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 SoaveRedlichKwong
EOS
ai
attractive
term of the SoaveRedlichKwong
EOS for component
b
repulsive
term of the SoaveRedlichKwong
EOS
bi talc
repulsive
term of the SoaveRedlichKwong
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
NL 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: 2029. 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: 119. Niesen,
V.G., and Yesavage, V.F., 1988. “VaporLiquid Equlibria for mCresol/Quinoline Temperatures between 523 and 598 K,” J. of Chem. & Eng. Data, 33: 138143.
Niesen,
V.G., and Yesavage, V.G., 1988. “VaporLiquid Equilibria for mCresol/Tetralin and Tetralin/Quinoline at Temperatures between 523 and 598 K,” J. of Chem. & Eng. Data, 33: 253258.
SkjoldJorgensen, S., 1983. “On Statistical Fluid Phase Equilibria, 14: 273288. Soave, G.S., 1972. “Equilibrium Constants Chem. Eng. Sci, 27: 11971203.
Principles
in Reduction
from a Modified
of Thermodynamic
RedlichKwong
Equation
at
Data, of State,”
”