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A study of simplified molecular models in phase equilibrium prediction
Fluid Phase Equilibria, 2 (1978) 181-198 0 Elsevier Scientific Publishing Company, Amsterdam -
A STUDY OF SIMPLIFIED MOLECULAR EQUILIBRIUM PREDICTION
JEAN PIERRE MONFORT
and MARiA
Znstituto Mexican0 de1 Petrbleo, City 14, D.F. (Mexico)
MODELS IN PHASE
DE LOURDES
Subdirecci6n
181 Printed in The Netherlands
ROJAS R
de Z.B.P., Apartado
Postal
14-805,
Mexico
(Received April 12th, 1977; in revised form February 27th, 1978)
ABSTRACT Monfort, J.P. and Rojas R., M.D.L., 19’78. A study of simplified molecular models in phase equilibrium prediction. Fluid Phase Equilibria, 2: 181-198. The accuracy of three molecular models of solution (Wilson, UNIQUAC and NRTL) is tested for 34 binary systems and 9 ternary systems. The aim of this paper is to attempt some evaluation of these models. Vapor-liquid calculations show a slight superiority of the UNIQUAC equation in the representation of experimental data, while liquid-liquid calculations using a method of determination of binary energetic parameters from mutual solubility data show a better prediction with the NRTL equation.
INTRODUCTION
In chemical process design, several models are used for the representation of phase equilibrium properties, in particular liquid-vapor and liquid-liquid equilibrium. Each model is best suited for different conditions, and good criteria are needed in order to attain the best compromise. The selection of these models is of primordial importance in view of the need for optimum design of separation equipment such as distillation or extractive systems. In this work the requirements that these models should fulfil for the correct prediction of equilibrium properties are presented. In order to compare these models, we calculated isothermal and isobaric equilibrium data for 34 binary and 5 ternary vapor-liquid mixtures and 4 ternary liquid-liquid mixtures. This study permits the evaluation of these models and gives selection criteria, along with new values of binary paran)eters necessary for the calculation of the multicomponent fluid phase equilibrium. SIMPLIFIED
MOLECULAR
MODELS
OF SOLUTIONS
Activity coefficients in liquid mixtures can be calculated from a model that expresses the excess Gibbs energy of the mixture as a function of com-
182
position and temperature. It is also possible to handle the liquid-phase in the same way as the vapor phase with fugacity coefficients. In this case, we require an equation of state or a partition function from which the fugacity coefficients can be obtained. Until now the results best fitted for the design of separation equipment have come from simple models based on solution theories. It is worth mentioning the growing number of rather sophisticated models developed within the framework of pure statistical mechanics as those of Gubbins (197 3), Uno (1975), and Barker and Henderson (1976), but their applicability still leaves much to be desired. The three models considered in this work belong to the so-called simple molecular models (Renon, 1978; Fredenslund, 1975), derived either from the concept of local composition as in Wilson’s and the NRTL model, or from a statistical mechanics basis as in the UNIQUAC model, which is an extension of Guggenheim’s quasi-chemical theory of liquid mixtures. These solution models involve a certain degree of empiricism since they contain parameters that are obtained by fitting experimental data of liquidvapor or liquid-liquid equilibria. Nevertheless, they are derived from consideration of molecular interactions between unlike species and have consequently some theoretical significance. Wilson molecular
model
Using the slightly modified expression of Orye and Prausnitz (1965) of the original Wilson’s equation (1964), the following expression for the logarithm of the activity coefficient 7h is obtained In Tk = 1 - ln(J?
X.jAf;j)
-
5
(3ci
i=l
j=l
*i/J2
Xi/Iii)
(la)
i=l
with Aij = (v,i/vi)
exP[-_(gij
(lb)
-gjj)/RT]
where Vi and Vi are the molar volumes and gij - gij is an adjustable energy parameter. Local composition
model
of Renon
and Prausnitz
The expression for the activity coefficient (Renon, 1966) :
(NRTL)
yJz in the NRTL equation is
@a) 2
i=
1
i-
1
Gi.,x,
183
where Gij = exp(--ocij&);
gij = (gij -gjj)lRT);
t2b)
with aij = eji. Eq. (2a) allows the description of complex phase behavior. This equation contains two energy parameters gij and tji as used in Wilson’s equation and the non-random parameter eij. This parameter can be determined a priori, based exclusively on the nature of the system under study, but, in practice, it may serve as an additional adjustable parameter. Model of A bmms and Prausnitz (UNIQ UAC) This model is an extension of Guggenheim’s quasi-chemical model and provides the following equation for the activity coefficient +yk(Abrams and Frausnitz, 1975): ln Yk = ln(%JJch) + (z/2)qk
ln(&J(a,)
+ 1, -
(34 where the energetic parameter Tij is defined as: rii = exP [-(gij
- gjj )IRTl
(=I
and the average area fraction ei and the average segment fraction ai are given by:
ei=(SiNi)l(x
q_iNj) = (qixi)l(E
@pi= (riNi)/( C rjNj) = (rixi)/(C
qjxj) rjxi)
(3c) (3d)
where Xi is the mole fraction and lj = $(rj - qj) - (rj - 1) and z is the coordination number which in this work is set equal to 10 as in the original paper. For a binary solution this model uses in addition two more parameters per compound. These two parameters r and q are related to the structure of the pure components. Our calculation of r and q follows that proposed in the original work. Like the NRTL equation, eq. (3a) can predict phase separation. In conclusion, these models use two adjustable energetic parameters per binary, and the NRTL and UNIQUAC models employ an extra adjustable empirical parameter or two easily obtainable structural parameters, respectively.
184 REPRESENTATION BINARY
AND
AND
TERNARY
PREDICTION
OF VAPOR-LIQUID
EQUILIBRIUM
OF
SYSTEMS
The reliability of the older models has been confirmed by a considerable amount of work. (Schreiber and Eckert, 1971; Holmes and Van Winkle, 1970; Nagara et al., 1976); the more recent model has not been so extensively tested. In the present work we analyze a total of 448 binary and 66 ternary points of vapor-liquid equilibrium (VLE). 1. Data reduction, selection equilibrium calculations
criteria
for binary parameters
and vapor-liquid
The first step in this comparative study is the determinaton of optimum values for the energetic parameters of the various models. The main difficulty in the data reduction of VLE data is the presence of random and systematic error in the data. Here we shall investigate the results of least squares fitting of exact data (absence of systematic error) and the only check for correctness of VLE data will be via the so-called thermodynamic consistency test. We use here the integral area test of Herington (1951). When isothermal data satisfy the GibbsDuhem relation, this test leads to the condition: 1
s
ln(yj/“lk)dx
= 0
(4)
0
In general, the integral is equal to some value I. For isobaric conditions, eqn. 4 is replaced by: x=1 j 0
W'Yjhlt)d~
hEIRT2 dT
=_f
(5)
x=0
Due to the absence of the heat of mixing data necessary to compute the integral of the left hand side of eq. (5), it is convenient to use Herington’s empirical technique to calculate it. If we denote the absolute value of the algebraic sum of the areas delimited by the curve ln(rj/yk) as Z, we can write the percentage deviation D as follows: D = 100 III/C
(6a)
and define the quantity J as: J = 150 I AT,,,/Z’mi,
I%
(6b)
where TMin is the lowest boiling temperature in the isobaric system within the composition range 0 Q x 4 1 and ATM,, is the maximum difference of boiling points in the total composition range of the isobaric system. Herington assumes that with D - J < lo%, a given set of VLE data is probably consistent.
185
To perform the integrals of eqs. (4) and (5) we plot, using a computational graph subroutine, values of ln(ri/Tk) versus x for each data point and the areas defined by the x axis are calculated by the weighting method. We obtain for isothermal VLE data, values of I between 0.75% and 5.9% and for isobaric VLE data, values of D -J between 0.23% and 10%. Systems for which I and D - J do not fall within the superior limits of these intervals are not considered. The energy parameters are obtained by correlating VLE data using eqs. (la), (2a) and (3a) to calculate activity coefficients. The optimization procedure used here is based on a direct numerical method without the use of gradient search. The algorithm proposed consists of a steepest descend method combined with a golden ratio search technique. This minimizes the sum of squares of relative deviations of the total pressure and composition which define the objective function F as follows: F
=
&
2
[(
Pcalc
-
P eXP
2 pexp +
i (
Ycalc
-Yap Yexp
2
where ND is the total number of experimental For a binary mixture j, k:
P talc =
(7)
13
points considered.
(~jTj?j3l(Pj) + (1-~j)(YkE/(Pk)
(8)
and
(9) where y is calculated with either eqs. (2), (3) or (4). We calculate fugacity coefficients cpvia the Redlich-Kwong equation. The vapor pressure Ps of pure component at equilibrium temperature T is obtained from API data or from the Antoine equation. For all mixtures studied the Poynting term is set equal to 1. The energy parameters gij - gjj and gii -gii are given for each binary studied in Table 1. It is well known, however (Desplanches et al., 1975; Brinkman et al., 1974), that estimated values of the energy parameters depend upon the objective function used to adjust them, and that they are strongly correlated. It is worthwhile to mention that, as a consequence of it, other sets of binary parameters can give a fitting of experimental data with the same accuracy. 2. Results from the comparative
study of binary mixtures
Bubble pressure and temperature calculations are performed tures mentioned above by means of the equilibrium equation. (PkYkP
=
Ykxkti
for the mix-
(10)
186 TABLE
1
Solution model parameters for binary mixtures. tion, second and third lo UNIQUAC and NRTL Mixture
T or P
Cyclohexanen-Heptane
278.15
K
First line corresponds to Wilson’s * equations, respeclively Number of data points
equa-
Data Ref.
g12 -g22 (cal/mol)
g21 -g11 (cal/mol)
255.11 8.07 98.23
-235.23 33.90 -232.90
10
9
-519.03 -1.56 415.08
16
4
12
1
Methanol-water
1 atm
-120.97 -68.03 144.12
Methanol-ethanol
1 atm
31.82 40.81 -109.53
Methyl acetatemethanol
1 atm
62.26 -3791.04 147.58
-774.81 4076.98 530.37
14
14
Methanolethyl acetate
1 atm
-1030.78 135.59 362.04
242.13 -758.67 380.49
17
13
Acetonemethanol
1 atm
-249.60 -274.41 332.06
-228.55 6.23 94.78
12
1
Acetone-ethanol
1 atm
-433.53 -153.35 290.55
-21.12 -26.52 105.84
9
1
-170.64 -1261.37 1314.41
9
Methanolbenzene
308.15
-1794.06
K
99.16 348.20 1 atm
0.9539
atm
17.27 -5.19 9.66
17
1
-1745.63 27.67 576.50
-306.70 -57.88
-1616.50 -55.34 698.71
-477.34 -1037.69 1042.31
15
-2336.58 246.74 371.26
-230.60 -1418.18 1708.74
10
1
1129.93
Ethanoln-Heptane
303.15
Ethanoln-octane
338.15
-2068.47 235.21 599.56
-435.83 -1321.33 1376.67
18
3
Ethanol-methylcyclohexane
338.15
-2305.99 182.40 481.95
-230.60 -1146.07 1342.08
8
10
-1254.46 117.61 258.27
-175.26 -721.77 1028.47
14
11
Ethanolbenzene
1 atm
K
187 TABLE
1 (continued)
Mixture
T or P
g12 --22
g21 -a1
(cal/mol)
Ethanol-toluene
(cal/mol)
Number of data points
Data Ref.
308.15
K
-1690.82 215.15 1086.12
-145.18 -1063.06 1289.05
10
10
318.15
K
-1602.66 235.21 156.81
-172.95 -1093.04 1406.65
15
24
-1282.13 156.81 412.77
-159.11 -807.09 860.13
16
11
-62.26 1033.08 1381.29
20
16
0.9947
atm
i-Propanol-ioctane
318.15
K
-1953.17 276.72 283.64
n-Heptanebutanol
323.15
K
-13835.92 -777.12 1284.434 -63.09 -336.67 1023.86
-811.71 242.13 189.09 -1067.67 65.72 15.63
12
8
17
12
Toluenebutanol
1 atm
n-Pentanol-2methylpentane
298.15
K
-1598.23 196.47 118.30
-177.33 -671.04 1233.70
9
22
n-Pentanol-3methylpentane
298.15
K
-10777.72 208.69 86.24
-309.46 -689.49 1261.37
9
22
n-Pentanol-2,3dimethylbutane
298.15
K
-1533.60 156.34 104.46
-166.49 -604.17 1210.64
9
22
n-Pentanol-2,2dimethylbutane
298.15
K
-1580.04 205.46 104.92
-158.54 -680.27 1233.70
9
22
nPentanoln-Hexane
298.15
K
-1538.25 156.11 188.63
33.64 -583.41 1074.59
9
22
Acetonitrilewater
1 atm
-14610.04 -502.70 523.46
-730.05 -96.62 1053.84
11
2
Acetone-water
1 atm
-10537.67 -477.34 191.86
-737.69 24.44 1508.12
13
17
-53.11 -11.74 2.67
4.29 -0.58 0.0
10
5
Acetoneacetonitrile
318.15
K
188 TABLE
1 (continued)
T or P
Mixture
g12-g22
R21-fill
(cal/mol)
(cal/mol)
Number of data
Data Ref.
points
Dibutyl ketonedimethylsulfoxide
333.15
1-Hexenedibutyl ketone
333.15
Acetonitrilen-Heptane
318.15
K
-2304.40 -25.60 1395.12
Benzeneacetonitrile
318.15
K
-885.82 130.52 119.91
Octane-nitrobenzene
364.15
K
-801.88 101.83 -172.58
i-octane-nitrobenzene
364.15
K
Octanebutyronitrile
364.15
K
i-Octanebutyronitrile
364.15
K
K
13
21
8
21
9
18
121.94 -539.60 604.17
11
18
-242.13 -309.00 1072.28
8
23
6
23
4027.36
14818.2 313.61 412.77
-322.054 -702.03 762.52
-451.97 281.33 156.81
8
23
-3507.96 -1466.03 2461.43
-1464.3 385.10 242.13
8
23
-652.59 860.13 K
105.84 1035.39
-16.55 -611.1
-13259.4 -1166.97
35.97 1713.3 -1158.00 -1217.56 1212.95
Chloroformmethanol
1 atm
378.43 -1289.046 1353.61
-1799.90 272.11 -114.84
20
15
Chloroformethyl acetate
1 atm
418.19 148.04 -214.92
22.21 40.81 -281.12
13
15
l
value Of
“ij
=
0.30.
where the known variables are the liquid composition xk and the pressure P or the temperature T, the remaining variables have been specified in the above section. These mixtures have been sorted into 8 groups according to their components. Each group presents different degrees of chemical or physical interaction. We present in Table 2 the .relative mean deviations of pressure and temperature and the absolute deviations of the vapor composition, calculated as
189
follows: hp/P = (l/ADPI
(P,,P - P&)/P,,P
AT/T’ = UINDPI
We,, -
AY; = (l/'ND)CIYiexp
I
(114 Wb)
TcalcM’expl
-YicalcI
WC)
First, we observe that Wilson’s equation gives better results for the two following types of mixtures: alcohol-hydrocarbon, chloroform-polar component or alcohol. For the remaining mixtures we observe that the results obtained from the UNIQUAC equation are in better general agreement with experimental data than those from the other models. A priori, it is not possible to correlate the predictive behavior of a solution model with the chemical nature of the mixture; nevertheless one can select the most appropriate model for a certain type of mixture from a table such as Table 2.
TABLE
2
Binary mixtures. Standard deviations on predicted values of temperature, pressure and vapor molar composition. First line corresponds to Wilson’s equation, second and third to the UNIQUAC and NRTL equations, respectively Mixture
ATIT
Hydrocarbon-hydrocarbon
APIP
*Yi
No. of data points
0.050 0.051 0.059
0.015 0.015 0.015
10
Alcohol-water
0.006 0.007 0.006
0.027 0.023 0.027
16
Alcohol-alcohol
0.008 0.008 0.008
0.012 0.012 0.013
12
0.007 0.005 0.008
0.014 0.012 0.014
52
0.018 0.030 0.030
226
0.093 0.016 0.086
57
0.014 0.012 0.014
71
0.023 0.023 0.024
24
Alcohol-polar
solvent
Alcohol-Hydrocarbon
Polar solvent-water polar solvent
0.022 0.035 0.033 or
0.090 0.018 0.087
Polar solventhydrocarbon Chloroform-polar or alcohol
0.021 0.016 0.021 solvent
0.008 0.008 0.008
190 TABLE
3
Ternary mixtures. Standard deviations on predicted values of temperature, pressure and vapor molar composition, according to Wilson’s UNIQUAC and NRTL equations Mixture
T or P
Ap/p
Acetonitrilebenzene-n-heptane
318.15
K
or
Number of data points
Data Ref.
0.015 0.006 0.013
15
18
AYI
AY2
0.029 0.020 0.030
0.015 0.010 0.013
AT/T
Acetone-methanolethanol
1 atm
0.015 0.015 0.015
0.007 0.007 0.007
0.055 0.056 0.055
14
9
Acetone-acetonitriiewater
1 atm
0.014 0.003 0.003
0.039 0.043 0.016
0.080 0.025 0.023
13
5
Acetone-methanolwater
1 atm
0.036 0.009 0.023
0.038 0.022 0.023
0.030 0.010 0.025
13
7
Chloroform-methanolethyl acetate
1 atm
0.014 0.003 0.003
0.039 0.043 0.016
0.012 0.014 0.014
11
14
3. Results from the comparative
study of ternary mixtures
The advantages of these models are that reliable estimates of vapor-liquid equilibrium data can be made for multicomponent systems using only binary pair parameters; no ternary or higher pair parameters are required. From the binary pair parameters listed in Table 1, we study 5 ternary mixtures that present strong polar effects. The results obtained are in Table 3 and confirm the trends observed in the study of the binary mixtures. The UNIQUAC equation is slightly superior in the prediction of the vapor composition than the remaining models; the NRTL equation seems to be, in the case of ternary mixtures, somewhat superior to Wilson’s equation. The same behavior is observed for the temperature or pressure predictions. 4. Conclusions The UNIQUAC equation seems to be the best on the basis of the systems studied. It gives slightly better results than the Wilson’s or NRTL equations. It also seems that for binary systems containing an alcohol, Wilson’s equation gives the best predictive results, this trend having been discussed in a previous report (Schreiber, 1971).
191 PREDICTION
OF LIQUID-LIQUID
EQUILIBRIUM
In this section we examine the capabilities of the NRTL and UNIQUAC equations for predicting the complex phase behavior of liquid-liquid systems. We present results for 2 systems with a single pair of immiscible components (mixtures 4 and 2, type I) and 2 systems with two pairs of immiscible components (mixtures 3 and 4, type II), which are showed in Table 5. As it is well known, of the 3 solution models used in this work, only the UNIQUAC and NRTL models can predict phase instability (Orye, 1965; Renon, 1966; Abrams and Prausnitz, 1975). For the systems examined here, the binary UNIQUAC and NRTL parameters are selected to match vapor-liquid data for the miscible pairs and mutual solubility data for the partially miscible components. This technique, which is not as yet generally accepted (Abrams and Prausnitz, 1975; Joy and Kyle, 1969) because of the poor agreement obtained between experimental and calculated tie lines, can be notably improved by choosing .better binary parameters. Another alternative is, of course, the fitting of the binary parameters on ternary tie line data. This force-fitting procedure is very successful but cannot be adequately extended to the case of multicomponent liquid-liquid equilibrium. The first choice is adopted and the improvement is achieved by setting physically meaningful pairs of parameters for the partially miscible components. 1. Reduction
of binary miscibility data
We start from a recent correlation scheme (Mattelin of binary miscibility data that uses a special searching suitable pairs of binary parameters for the UNIQUAC The binary mixture is characterized in terms of the such that u = (a + b)/[ln(xt/x;)]
-
and Verhoeye, 1975) technique to obtain and NRTL models. quantities LJand w
2ab/[ln(x~/x~)]
(124
and w = 2/[ln(xk/x;)]
-
(a + b)/[ln(x$/xi)]
Wb)
where a =x:/x;;
b =x:/x;
In these relations (eqs. (9b), (9c) of Mattelin and Verhoeye’s
(12c)
paper) X: and 3t: are the mutual solubilities. They correspond to the end points of the binary diagram. Following these authors the different steps necessary to compute the parameters are: (1) Assign a class to each type of mixture, depending of the values taken by -v/w, either larger or smaller than one. From this it is possible to know, a priori, the interval of variations for the parameters eij, Gij and Gji. (2) Solve for (Yij, Gjj and Gii via a set of non-linear equations which
192
depend on the objective function chosen to minimize the differences between calculated activities corresponding to the equilibrium compositions. (3) Eliminate solutions provided by step 2 that have no thermodynamic significance by introducing a constraint conditions: calculate the free energy of mixing AGM and select the previous values for which AGM is negative. In practice, step 2 leads to a large number of numerical solutions, characterized by an oscillatory behaviour which makes them difficult to handle when applying the liquid phase instability condition (step 3). Therefore, we choose to abandon this approach. We replace step 2 and 3 by an unique selection criterium: we fit the abscissae of the points of contact of common tangents to the curve AGM versus x directly to the mutual solubility data (XT and x;), using the condition AGM < 0 with 0 d x < 1. The algorithm used in this work is based on the calculation of the Gibbs energy of mixing. First we initialize with a set of parameters taken from an interval of variation defined in step 1. They are then incremented by simultaneous positive values and we calculate new AG”. From the points XT and xf, the mole fractions corresponding to the contacts of the tangent to the curve are calculated and the parameters are chosen to minimize the quantity F defined as: (13) where x; talc and xi _ic are the calculated abscissae of the tangent to the curve AGM versus x. The computing time and the screening of physically acceptable solutions depend both on the type of mixture studied and the experimental solubility data. 2. Results
obtained
for binary mixtures
(i) NRTI, equation The proposed method may lead to several possible sets of binary parameters, depending of the values of (Yij, as in the original method of Mattelin and Verhoeye. The advantage is that the number of pairs of binary parameters (and the computation time) is greatly reduced. This is probably due to the fact that the condition of minimum free energy of the system, along with our force fitting technique of the mutual solubility data, is a sufficient condition to obtain the physical meaningful set of parameters. (2) UNIQUAC equation We extend the proposed method to the screening of the binary parameters of eqn. (3b). In this case the trend above mentioned is not observed; we arrive at a single pair of physically meaningful binary parameters. This seems, a priori, an understandable result: the asjusted parameter (~lijis replaced by the two fixed structural parameters r and q. As an illustrative example of the results obtained with our method of ob-
323.15
293.15
283.16
273.16 0
0.1
0.2
03
44
0.6
0.6
a7
0.8
0.9
I 1.0
Fig. 1. Experimental and calculated mutual solubility data for the n-Heptane-n-methylpyrrolidone mixture. The solid curve is the result of the present correlation of binary miscibility data with NRTL equation, the dashed line is the calculated curve using binary parameters of the literature (Data Ref. 6), (0) present work, calculated with UNIQUAC equation. Data source: (o), (X) Data Ref. 6. TABLE Solution
4 model
Mixture
parameters T W
from binary miscibility g12-a2 (cal/mol) NRTL
data 821 -a1 (cal/mol)
UNIQUAC -6.4
NRTL
UNIQUAC
1337.4
-1158.3
01 NRTL
Data Ref.
0.377
18
0.396
19
Acetonitrilen-heptane iso-octanefurfural
318.15
1577.7
303.15
1591.1
-287.9
n-Hexanefurfural n-Heptanefurfural Cyclohexanefurfural 1 -HexeneDMSO
303.15
1519.4
-338.5
1400.2
-250.3
0.383
19
303.15
1545.0
-288.0
1588.8
-268.9
0.398
19
303.15
1414.0
-725.2
895.3
0.322
19
333.15
1835.51
-275.06
0.389
21
/
1683.4
1559.00
232.3
-6.1 -295.4
194 taining the NRTL and UNJQUAC binary parameters we generate the solubility curve of the n-Heptane-n-Methylpyrrolidone mixture, the calculated and experimental solubility data appear in Fig. 1. It can be seen that the calculated solubilities are in good agreement with the experimental results. The parameters obtained are shown in Table 4. The pure component parameters r and q are calculated in the same way as in section 2.1. The Olii values are adjusted to the binary miscibility data. 3. Tie line calculations
of ternary
mixtures
The phase equilibrium calculation is based on the condition that the activity of each component in both phases is the same. The tie line curve is obtained by solving the following equation: [ln(yjl’x!‘))
-
ln(yf2)~$2J)]”
= ()
(14)
where i = 1, 2, 3. At first we solved eq. (14) for the case of the NRTL equation using a Newton-Raphson procedure that has been discussed elsewhere (Renon et al. 1971). However, it was not possible to solve eq. (14) with the UNIQUAC equation because of convergence problems. Therefore the tie line computations were performed with a direct search technique - here, the so called simplex method (Murray, 1972). This technique has proved to be extremely effective in practice, particularly when the objective function is not differentiable or the derivatives are difficult to obtain. When we applied this method to the resolution of eqn. (14) with the NRTL equation, the results obtained are similar to or better than those obtained with the Newton-Raphson procedure. We compared experimental and calculated tie line composition in terms of the quantity Ax defined as
AX = (l/ND)
CC j
i
lXicalc
-xi
exp
I
(15)
wherei=1,2,3andj=l... ND. The results are shown in Table 5 for both solution models. Although only a modest number of ternary systems have been studied, we can observe several trends. First, the NRTL model gives better results in the ternary liquid calculations for systems of type II with plait points than for systems of type I, without plait points. The differences observed between values of AX are small and do not correspond to the general assessment about the difficulties of predicting ternary diagrams that present one plait point. This is due, perhaps, to the fact that our ternary liquid calculations are based on the “best” binary parameters for the partially miscible components. The improvement of the ternary system equilibrium prediction, in this case, is quite significant. Regarding the UNIQUAC model, the deviations are larger than those obtained with the NRTL model, especially for systems of
195 TABLE
5
Liquid-liquid equilibria. Mean deviations between experimental and calculated liquid composition with UNIQUAC and NRTL equations Mixture
-:
“2.“‘
Acetonitrile(l)-benzene(Z)n-Heptane( 3) n-Hexane( l)-5-nonanone(Z)Dimethylsulfoxide( 3) i-Octane( 1 )-n-hexane( 2)furfural( 3) n-Heptane(l)-cyclohexane( furfural( 3)
T
2)-
Mean Deviation
AX
Number of data points
Data Ref.
NRTL
UNIQUAC
0.0030
0.0031
48
18
0.0010
0.0200
36
21
0.0037
0.0388
36
19
0.0063
0.0125
36
19
type II. This could indicate that the UNIQUAC model is not suitable for these systems. This analysis confirms the good predictive qualities of the NRTL model for the representation of ternary liquid equilibrium composition. We may conclude that this solution model is better adapted to calculations of liquidliquid systems than of vapor-liquid equilibria. CONCLUSIONS
Our study on the applicability of three simple molecular models to the estimation of equilibrium properties has confirmed some known facts but also pointed out some unexpected trends. Over the 8 types of mixtures studied, the UNIQUAC equation has shown a slight superiority over the NRTL and Wilson’s equations in the representation of vapor-liquid properties for binary systems. Nevertheless, two kinds of mixtures are better represented with Wilson’s equation: viz. alcohol-hydrocarbon and chloroform-polar component or alcohol mixtures. In the case of ternary mixtures, the differences of prediction of vapor--liquid equilibria between the three models are still smaller. It is difficult, if not impossible, to establish the absolute predictive accuracy of each equation used here and we should emphasize that one must be very careful in the selection of the solution model when requiring very precise quantitative estimates of fluid phase equilibria. ACKNOWLEDGMENT
It is a pleasure to thank Dr. J. Tejeda for many helpful discussions and for his contribution to the optimization program of binary parameters.
196 NOTATION
a, b D F .!?ij_gij
Gij h J 1 N ND P Q I; T V v, w x, Y 2
ratio of the mole fractions of both components in one phase, in eqn. (12~) percentage deviation as defined in eqn. (6a) objective function energetic parameters, Wilson’s eq. (lb), cal/mol-’ NRTL parameters, eqn. (2b) enthalpy defined by eq. (6b) defined after eq. (3a) number of components number of data points pressure, atm pure component area parameter pure component volume parameter gas constant, cal K-l mol-’ equilibrium temperature, K molar volume of liquid, cm3 mole-r defined by eqs. (12a) and (12b) liquid and vapor phase mole fraction lattice coordination number, set equal to 10
Greek symbols a:ij Yh ki Pi @i
ei tij rij
non-randomness parameter of the NRTL equation activity coefficient of component k Wilson parameters, eq. (lb) vapor phase fugacity coefficient of component i segment fraction of component i area fraction of component i defined by eqn. (2b) defined by eqn. (3b)
Subscripts talc refer to a calculated value of a quantity refer to an experimental value of a quantity exp i, j, k, 1 component i, component j, component k, component
1
Supercripts (Q) (b)
refers to the liquid phase on the left side of the concentration diagram refers to the liquid phase on the right side of the concentration diagram
197
S E
saturated condition excess property
REFERENCES Abrams, D. and Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: a new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J., 21: 116-128. Barker, J.A. and Henderson, D., 1976. What is “liquid”? Understanding the states of matter. Rev. Modern Phys., 48: 587-671. Brinkman, N.D., Luh, CT. and Weber, J.H., 1974. A study of the effect of optimization function on calculated parameters of Wilson equation. Can. J. Chem. Eng., 52: 397404. Desplanches, H., Chevalier, J.L. and Llinas, R., 1975. Calcul des parametres de l’kquation de Wilson. Rev. Inst. Franc. Petrole., 30: 951-967. Fredenslund, A., 1975. Vapor-liquid equilibria in strongly non-ideal mixtures, 5th International Congress CHISA, Prague. Gubbins, K.E., 1973. Perturbation methods for calculating properties of liquid mixtures. AIChE J., 19: 684-698. Herington, E.F.G., 1951. Tests for the consistency of experimental isobaric vapor-liquid equilibrium data. J. Inst. Petrol., 37: 457-470. Holmes, M.J. and Van Winkle, M., 1970. Prediction of ternary vapor-liquid equilibria from binary data. Ind. Eng. Chem., 62: 21-30. Joy, D.S. and Kyle, B.G., 1969. Calculation of ternary liquid-liquid equilibria, AIChE J., 15: 298-300. Mattelin, A.C. and Verhoeye, L.A., 1975. The correlation of binary miscibility data by means of the NRTL equation. J. Chem. Eng. Sci., 30: 1933200. Murray, W., 1972. Numerical Methods for Unconstrained Optimization. Academic Press, London. Nagata, I., Ohta, T. and Nakagawa, S., 1976. Excess Gibbs free energies and heats of mixing for binary alcoholic liquid mixtures. J. Chem. Eng. Japan, 9: 276-281. Orye, R.V. and Prausnitz, J.M., 1965. Multicomponent equilibria with the Wilson equation. Ind. Eng. Chem., 57: 18-26. Renon, H., 1966. Thermodynamic properties of non-ideal mixtures. Ph.D. Dissertation, University of California, Berkeley. Renon, H., Asselineau, L., Cohen, C. and Raimbault, C., 1971. Calcul sur ordinateur des equilibres liquide-vapeur et liquideliquide. Ed. Technip, Paris. Renon, H., 1978. Qualities of models for the evaluation, representation and prediction of fluid phase equilibrium data. Fluid Phase equilibria, 2: 101-118. Schreiber L.B. and Eckert, C.A., 1971. Use of infinite dilution activity coefficients with Wilson’s equation. Ind. Eng. Chem. Process Des. Develop., 10: 572-575. Uno, K., Sarashina, E., Arai, Y. and Saito S., 1975. Application of the perturbation theory to the calculation of Henry’s constants for normal fluids. J. Chem. Eng. Japan., 8: 201-206. Wilson, G.M., 1964. A new expression for the excess free energy of mixing. J. Am. Chem. Sot., 86: 127. EQUILIBRIUM
DATA
REFERENCES
1 Amer, H.H., Paxton, R.R. and Van Winkle, M., 1956. Methanol, ethanol, acetone. vapor-liquid equilibria. Ind. Eng. Chem., 48: 142-146. 2 Blackford, D.S. and York, R., 1965. Vapor-liquid equilibrium of the system acrylonitrile-acetonitrile-water. J. Chem. Eng. Data, 10: 313-318.
.
198 3 Boublika, L. and Lu, B.C.Y., 1969. Isothermal vapour - liquid equilibria for the ethanol-n-octane system. J. Appl. Chem. 19: 89-92. 4 Bredig, G. and Bayer. R., 1927. The vapor pressure of the system methanol-water. J. phys. Chem., 130-14. 5 Chu, H.C. et al., 1950. Distillation Equilibrium Data. Reinhold, New York. 6 Fabries, J.F., 1976. Contribution 1 l’etude des forts &arts $ I’idealitk en phase liquide. Thesis; University Paul Sabatier, Toulouse. 7 Griswold, J. and Buford, C.B., 1949. Separation of synthesis mixtures, vapor-liquid equilibrium of acetone-methanol-water. Ind. Eng. Chem., 41: 2397-2351. 8 Hala, E., Wichterle I., Polak, J. and Boublik, T., 1968. Vapoul-Liquid Equilibrium Data at Normal Pressure. Pergamon Press, London. 9 Katayama, T., Sung F.K. and Lightfoot, E.W., 1965. Isothermal activity coefficients for the system cyclohexane-heptane-toluene at 25’C. AIChE. J., 11: 924-929. 10 Kretschmer, C.B. and Wiehe R., 1949. Liquid-vapor equilibrium of ethanol-methylcyclohexane solution. J. Am. Chem. Sot., 71: 3176-3179. 11 Landwehr, J., Yerazunis, S. and Steinhauser, H., 1958. Vapor-liquid equilibria of toluene-ethylalcohol and benzene-ethylalcohol. Chem., Eng. Data Series, 3: 231234. 12 Mann, R.S. and Schemilt, L.W., 1963. VapoFliquid equilibrium at atmospheric pressure. J. Chcm. Eng. Data, 8: 189-190. 13 Murti, P.S. and Van Winkle, M., 1958. Vapor-liquid equilibria for binary systems of methanol-ethyl alcohol, l-propanol and 2-propanol with ethyl acetate and l-propanol-water. Chem. Eng. Data Series, 3: 72-81. 14 Nagata, I., 1969. Vapor-liquid equilibrium data for the binary systems methanolbenzene and methyl acetate-methanol. J. Chem. Eng. Data, 14: 418-420. 15 Nagata, I., 1962. Isobaric vapor-liquid equilibria for the ternary system chloroformMethanol-ethyl acetate. J. Chem. Eng. Data. 7: 367--m373. 16 Neau, E. and Peneloux, A., 1972. Calcul des fonctions thermodynamiques d’exces des binaires alcohol-alcane dans le modele chimique reticulaire. J. Chim. Phys., 9: 13221331. 17 Othmer, D.A., Chudgar, M.M. and Levy, S.L., 1952. Composition of vapors from boiling solutions. Binary and ternary systems of acetone, methyl ethyl ketone and water. Ind. Eng. Chem., 44: 1872-1881. 18 Palmer, D.A. and Smith B.D., 1972. Thermodynamic excess property measurements for acetonitrilebenzene-n-heptane system at 45OC. J. Chem. Eng. Data., 17: 71-76. 19 Pennington, E.W. and Marvil, S.J., 1953. Prediction of liquid-liquid phase equilibria. Ind. Eng. Chem., 45: 1371-1377. 20 Polak, J. and Lu, B.C. Y., 1975. Vapor-liquid equilibria in system ammonia-water at 14.69 and 65 psia. J. Chem. Eng. Data., 20: 182-183. 21 Renon, H. and Prausnitz J.M., 1968. Liquid-liquid and vapor-liquid equilibria for binary and ternary systems with dibutyl ketone, dimethylsulfoxide, n-hexane and lhexene. Ind. Eng. Chem., Process Design. Develop., 7: 220-225. 22 Sayegh, S.A. and Ratcliff, G.A., 1976. Excess Gibbs energies of binary systems of isopentanol and n-pentanol with hexane isomers at 25OC. Measurement and prediction by analytical group solution model. J. Chem. Eng. Data., 21: 71-74. 23 Spelyng, W. and Tassios, D., 1974. Solvent selectivity for hydrocarbons with close molar volumes. Ind. Eng. Chem., Process Design. Develop., 13: 328-332. 24 Van Ness H.C., Soczck C.A., Peloquin, G.L. and Machado, R.L., 1967. Thermodynamic excess properties of three alcohol-hydrocarbon systems. J. Chem. Eng. Data., 12: 217-224.
,
A STUDY OF SIMPLIFIED MOLECULAR EQUILIBRIUM PREDICTION
JEAN PIERRE MONFORT
and MARiA
Znstituto Mexican0 de1 Petrbleo, City 14, D.F. (Mexico)
MODELS IN PHASE
DE LOURDES
Subdirecci6n
181 Printed in The Netherlands
ROJAS R
de Z.B.P., Apartado
Postal
14-805,
Mexico
(Received April 12th, 1977; in revised form February 27th, 1978)
ABSTRACT Monfort, J.P. and Rojas R., M.D.L., 19’78. A study of simplified molecular models in phase equilibrium prediction. Fluid Phase Equilibria, 2: 181-198. The accuracy of three molecular models of solution (Wilson, UNIQUAC and NRTL) is tested for 34 binary systems and 9 ternary systems. The aim of this paper is to attempt some evaluation of these models. Vapor-liquid calculations show a slight superiority of the UNIQUAC equation in the representation of experimental data, while liquid-liquid calculations using a method of determination of binary energetic parameters from mutual solubility data show a better prediction with the NRTL equation.
INTRODUCTION
In chemical process design, several models are used for the representation of phase equilibrium properties, in particular liquid-vapor and liquid-liquid equilibrium. Each model is best suited for different conditions, and good criteria are needed in order to attain the best compromise. The selection of these models is of primordial importance in view of the need for optimum design of separation equipment such as distillation or extractive systems. In this work the requirements that these models should fulfil for the correct prediction of equilibrium properties are presented. In order to compare these models, we calculated isothermal and isobaric equilibrium data for 34 binary and 5 ternary vapor-liquid mixtures and 4 ternary liquid-liquid mixtures. This study permits the evaluation of these models and gives selection criteria, along with new values of binary paran)eters necessary for the calculation of the multicomponent fluid phase equilibrium. SIMPLIFIED
MOLECULAR
MODELS
OF SOLUTIONS
Activity coefficients in liquid mixtures can be calculated from a model that expresses the excess Gibbs energy of the mixture as a function of com-
182
position and temperature. It is also possible to handle the liquid-phase in the same way as the vapor phase with fugacity coefficients. In this case, we require an equation of state or a partition function from which the fugacity coefficients can be obtained. Until now the results best fitted for the design of separation equipment have come from simple models based on solution theories. It is worth mentioning the growing number of rather sophisticated models developed within the framework of pure statistical mechanics as those of Gubbins (197 3), Uno (1975), and Barker and Henderson (1976), but their applicability still leaves much to be desired. The three models considered in this work belong to the so-called simple molecular models (Renon, 1978; Fredenslund, 1975), derived either from the concept of local composition as in Wilson’s and the NRTL model, or from a statistical mechanics basis as in the UNIQUAC model, which is an extension of Guggenheim’s quasi-chemical theory of liquid mixtures. These solution models involve a certain degree of empiricism since they contain parameters that are obtained by fitting experimental data of liquidvapor or liquid-liquid equilibria. Nevertheless, they are derived from consideration of molecular interactions between unlike species and have consequently some theoretical significance. Wilson molecular
model
Using the slightly modified expression of Orye and Prausnitz (1965) of the original Wilson’s equation (1964), the following expression for the logarithm of the activity coefficient 7h is obtained In Tk = 1 - ln(J?
X.jAf;j)
-
5
(3ci
i=l
j=l
*i/J2
Xi/Iii)
(la)
i=l
with Aij = (v,i/vi)
exP[-_(gij
(lb)
-gjj)/RT]
where Vi and Vi are the molar volumes and gij - gij is an adjustable energy parameter. Local composition
model
of Renon
and Prausnitz
The expression for the activity coefficient (Renon, 1966) :
(NRTL)
yJz in the NRTL equation is
@a) 2
i=
1
i-
1
Gi.,x,
183
where Gij = exp(--ocij&);
gij = (gij -gjj)lRT);
t2b)
with aij = eji. Eq. (2a) allows the description of complex phase behavior. This equation contains two energy parameters gij and tji as used in Wilson’s equation and the non-random parameter eij. This parameter can be determined a priori, based exclusively on the nature of the system under study, but, in practice, it may serve as an additional adjustable parameter. Model of A bmms and Prausnitz (UNIQ UAC) This model is an extension of Guggenheim’s quasi-chemical model and provides the following equation for the activity coefficient +yk(Abrams and Frausnitz, 1975): ln Yk = ln(%JJch) + (z/2)qk
ln(&J(a,)
+ 1, -
(34 where the energetic parameter Tij is defined as: rii = exP [-(gij
- gjj )IRTl
(=I
and the average area fraction ei and the average segment fraction ai are given by:
ei=(SiNi)l(x
q_iNj) = (qixi)l(E
@pi= (riNi)/( C rjNj) = (rixi)/(C
qjxj) rjxi)
(3c) (3d)
where Xi is the mole fraction and lj = $(rj - qj) - (rj - 1) and z is the coordination number which in this work is set equal to 10 as in the original paper. For a binary solution this model uses in addition two more parameters per compound. These two parameters r and q are related to the structure of the pure components. Our calculation of r and q follows that proposed in the original work. Like the NRTL equation, eq. (3a) can predict phase separation. In conclusion, these models use two adjustable energetic parameters per binary, and the NRTL and UNIQUAC models employ an extra adjustable empirical parameter or two easily obtainable structural parameters, respectively.
184 REPRESENTATION BINARY
AND
AND
TERNARY
PREDICTION
OF VAPOR-LIQUID
EQUILIBRIUM
OF
SYSTEMS
The reliability of the older models has been confirmed by a considerable amount of work. (Schreiber and Eckert, 1971; Holmes and Van Winkle, 1970; Nagara et al., 1976); the more recent model has not been so extensively tested. In the present work we analyze a total of 448 binary and 66 ternary points of vapor-liquid equilibrium (VLE). 1. Data reduction, selection equilibrium calculations
criteria
for binary parameters
and vapor-liquid
The first step in this comparative study is the determinaton of optimum values for the energetic parameters of the various models. The main difficulty in the data reduction of VLE data is the presence of random and systematic error in the data. Here we shall investigate the results of least squares fitting of exact data (absence of systematic error) and the only check for correctness of VLE data will be via the so-called thermodynamic consistency test. We use here the integral area test of Herington (1951). When isothermal data satisfy the GibbsDuhem relation, this test leads to the condition: 1
s
ln(yj/“lk)dx
= 0
(4)
0
In general, the integral is equal to some value I. For isobaric conditions, eqn. 4 is replaced by: x=1 j 0
W'Yjhlt)d~
hEIRT2 dT
=_f
(5)
x=0
Due to the absence of the heat of mixing data necessary to compute the integral of the left hand side of eq. (5), it is convenient to use Herington’s empirical technique to calculate it. If we denote the absolute value of the algebraic sum of the areas delimited by the curve ln(rj/yk) as Z, we can write the percentage deviation D as follows: D = 100 III/C
(6a)
and define the quantity J as: J = 150 I AT,,,/Z’mi,
I%
(6b)
where TMin is the lowest boiling temperature in the isobaric system within the composition range 0 Q x 4 1 and ATM,, is the maximum difference of boiling points in the total composition range of the isobaric system. Herington assumes that with D - J < lo%, a given set of VLE data is probably consistent.
185
To perform the integrals of eqs. (4) and (5) we plot, using a computational graph subroutine, values of ln(ri/Tk) versus x for each data point and the areas defined by the x axis are calculated by the weighting method. We obtain for isothermal VLE data, values of I between 0.75% and 5.9% and for isobaric VLE data, values of D -J between 0.23% and 10%. Systems for which I and D - J do not fall within the superior limits of these intervals are not considered. The energy parameters are obtained by correlating VLE data using eqs. (la), (2a) and (3a) to calculate activity coefficients. The optimization procedure used here is based on a direct numerical method without the use of gradient search. The algorithm proposed consists of a steepest descend method combined with a golden ratio search technique. This minimizes the sum of squares of relative deviations of the total pressure and composition which define the objective function F as follows: F
=
&
2
[(
Pcalc
-
P eXP
2 pexp +
i (
Ycalc
-Yap Yexp
2
where ND is the total number of experimental For a binary mixture j, k:
P talc =
(7)
13
points considered.
(~jTj?j3l(Pj) + (1-~j)(YkE/(Pk)
(8)
and
(9) where y is calculated with either eqs. (2), (3) or (4). We calculate fugacity coefficients cpvia the Redlich-Kwong equation. The vapor pressure Ps of pure component at equilibrium temperature T is obtained from API data or from the Antoine equation. For all mixtures studied the Poynting term is set equal to 1. The energy parameters gij - gjj and gii -gii are given for each binary studied in Table 1. It is well known, however (Desplanches et al., 1975; Brinkman et al., 1974), that estimated values of the energy parameters depend upon the objective function used to adjust them, and that they are strongly correlated. It is worthwhile to mention that, as a consequence of it, other sets of binary parameters can give a fitting of experimental data with the same accuracy. 2. Results from the comparative
study of binary mixtures
Bubble pressure and temperature calculations are performed tures mentioned above by means of the equilibrium equation. (PkYkP
=
Ykxkti
for the mix-
(10)
186 TABLE
1
Solution model parameters for binary mixtures. tion, second and third lo UNIQUAC and NRTL Mixture
T or P
Cyclohexanen-Heptane
278.15
K
First line corresponds to Wilson’s * equations, respeclively Number of data points
equa-
Data Ref.
g12 -g22 (cal/mol)
g21 -g11 (cal/mol)
255.11 8.07 98.23
-235.23 33.90 -232.90
10
9
-519.03 -1.56 415.08
16
4
12
1
Methanol-water
1 atm
-120.97 -68.03 144.12
Methanol-ethanol
1 atm
31.82 40.81 -109.53
Methyl acetatemethanol
1 atm
62.26 -3791.04 147.58
-774.81 4076.98 530.37
14
14
Methanolethyl acetate
1 atm
-1030.78 135.59 362.04
242.13 -758.67 380.49
17
13
Acetonemethanol
1 atm
-249.60 -274.41 332.06
-228.55 6.23 94.78
12
1
Acetone-ethanol
1 atm
-433.53 -153.35 290.55
-21.12 -26.52 105.84
9
1
-170.64 -1261.37 1314.41
9
Methanolbenzene
308.15
-1794.06
K
99.16 348.20 1 atm
0.9539
atm
17.27 -5.19 9.66
17
1
-1745.63 27.67 576.50
-306.70 -57.88
-1616.50 -55.34 698.71
-477.34 -1037.69 1042.31
15
-2336.58 246.74 371.26
-230.60 -1418.18 1708.74
10
1
1129.93
Ethanoln-Heptane
303.15
Ethanoln-octane
338.15
-2068.47 235.21 599.56
-435.83 -1321.33 1376.67
18
3
Ethanol-methylcyclohexane
338.15
-2305.99 182.40 481.95
-230.60 -1146.07 1342.08
8
10
-1254.46 117.61 258.27
-175.26 -721.77 1028.47
14
11
Ethanolbenzene
1 atm
K
187 TABLE
1 (continued)
Mixture
T or P
g12 --22
g21 -a1
(cal/mol)
Ethanol-toluene
(cal/mol)
Number of data points
Data Ref.
308.15
K
-1690.82 215.15 1086.12
-145.18 -1063.06 1289.05
10
10
318.15
K
-1602.66 235.21 156.81
-172.95 -1093.04 1406.65
15
24
-1282.13 156.81 412.77
-159.11 -807.09 860.13
16
11
-62.26 1033.08 1381.29
20
16
0.9947
atm
i-Propanol-ioctane
318.15
K
-1953.17 276.72 283.64
n-Heptanebutanol
323.15
K
-13835.92 -777.12 1284.434 -63.09 -336.67 1023.86
-811.71 242.13 189.09 -1067.67 65.72 15.63
12
8
17
12
Toluenebutanol
1 atm
n-Pentanol-2methylpentane
298.15
K
-1598.23 196.47 118.30
-177.33 -671.04 1233.70
9
22
n-Pentanol-3methylpentane
298.15
K
-10777.72 208.69 86.24
-309.46 -689.49 1261.37
9
22
n-Pentanol-2,3dimethylbutane
298.15
K
-1533.60 156.34 104.46
-166.49 -604.17 1210.64
9
22
n-Pentanol-2,2dimethylbutane
298.15
K
-1580.04 205.46 104.92
-158.54 -680.27 1233.70
9
22
nPentanoln-Hexane
298.15
K
-1538.25 156.11 188.63
33.64 -583.41 1074.59
9
22
Acetonitrilewater
1 atm
-14610.04 -502.70 523.46
-730.05 -96.62 1053.84
11
2
Acetone-water
1 atm
-10537.67 -477.34 191.86
-737.69 24.44 1508.12
13
17
-53.11 -11.74 2.67
4.29 -0.58 0.0
10
5
Acetoneacetonitrile
318.15
K
188 TABLE
1 (continued)
T or P
Mixture
g12-g22
R21-fill
(cal/mol)
(cal/mol)
Number of data
Data Ref.
points
Dibutyl ketonedimethylsulfoxide
333.15
1-Hexenedibutyl ketone
333.15
Acetonitrilen-Heptane
318.15
K
-2304.40 -25.60 1395.12
Benzeneacetonitrile
318.15
K
-885.82 130.52 119.91
Octane-nitrobenzene
364.15
K
-801.88 101.83 -172.58
i-octane-nitrobenzene
364.15
K
Octanebutyronitrile
364.15
K
i-Octanebutyronitrile
364.15
K
K
13
21
8
21
9
18
121.94 -539.60 604.17
11
18
-242.13 -309.00 1072.28
8
23
6
23
4027.36
14818.2 313.61 412.77
-322.054 -702.03 762.52
-451.97 281.33 156.81
8
23
-3507.96 -1466.03 2461.43
-1464.3 385.10 242.13
8
23
-652.59 860.13 K
105.84 1035.39
-16.55 -611.1
-13259.4 -1166.97
35.97 1713.3 -1158.00 -1217.56 1212.95
Chloroformmethanol
1 atm
378.43 -1289.046 1353.61
-1799.90 272.11 -114.84
20
15
Chloroformethyl acetate
1 atm
418.19 148.04 -214.92
22.21 40.81 -281.12
13
15
l
value Of
“ij
=
0.30.
where the known variables are the liquid composition xk and the pressure P or the temperature T, the remaining variables have been specified in the above section. These mixtures have been sorted into 8 groups according to their components. Each group presents different degrees of chemical or physical interaction. We present in Table 2 the .relative mean deviations of pressure and temperature and the absolute deviations of the vapor composition, calculated as
189
follows: hp/P = (l/ADPI
(P,,P - P&)/P,,P
AT/T’ = UINDPI
We,, -
AY; = (l/'ND)CIYiexp
I
(114 Wb)
TcalcM’expl
-YicalcI
WC)
First, we observe that Wilson’s equation gives better results for the two following types of mixtures: alcohol-hydrocarbon, chloroform-polar component or alcohol. For the remaining mixtures we observe that the results obtained from the UNIQUAC equation are in better general agreement with experimental data than those from the other models. A priori, it is not possible to correlate the predictive behavior of a solution model with the chemical nature of the mixture; nevertheless one can select the most appropriate model for a certain type of mixture from a table such as Table 2.
TABLE
2
Binary mixtures. Standard deviations on predicted values of temperature, pressure and vapor molar composition. First line corresponds to Wilson’s equation, second and third to the UNIQUAC and NRTL equations, respectively Mixture
ATIT
Hydrocarbon-hydrocarbon
APIP
*Yi
No. of data points
0.050 0.051 0.059
0.015 0.015 0.015
10
Alcohol-water
0.006 0.007 0.006
0.027 0.023 0.027
16
Alcohol-alcohol
0.008 0.008 0.008
0.012 0.012 0.013
12
0.007 0.005 0.008
0.014 0.012 0.014
52
0.018 0.030 0.030
226
0.093 0.016 0.086
57
0.014 0.012 0.014
71
0.023 0.023 0.024
24
Alcohol-polar
solvent
Alcohol-Hydrocarbon
Polar solvent-water polar solvent
0.022 0.035 0.033 or
0.090 0.018 0.087
Polar solventhydrocarbon Chloroform-polar or alcohol
0.021 0.016 0.021 solvent
0.008 0.008 0.008
190 TABLE
3
Ternary mixtures. Standard deviations on predicted values of temperature, pressure and vapor molar composition, according to Wilson’s UNIQUAC and NRTL equations Mixture
T or P
Ap/p
Acetonitrilebenzene-n-heptane
318.15
K
or
Number of data points
Data Ref.
0.015 0.006 0.013
15
18
AYI
AY2
0.029 0.020 0.030
0.015 0.010 0.013
AT/T
Acetone-methanolethanol
1 atm
0.015 0.015 0.015
0.007 0.007 0.007
0.055 0.056 0.055
14
9
Acetone-acetonitriiewater
1 atm
0.014 0.003 0.003
0.039 0.043 0.016
0.080 0.025 0.023
13
5
Acetone-methanolwater
1 atm
0.036 0.009 0.023
0.038 0.022 0.023
0.030 0.010 0.025
13
7
Chloroform-methanolethyl acetate
1 atm
0.014 0.003 0.003
0.039 0.043 0.016
0.012 0.014 0.014
11
14
3. Results from the comparative
study of ternary mixtures
The advantages of these models are that reliable estimates of vapor-liquid equilibrium data can be made for multicomponent systems using only binary pair parameters; no ternary or higher pair parameters are required. From the binary pair parameters listed in Table 1, we study 5 ternary mixtures that present strong polar effects. The results obtained are in Table 3 and confirm the trends observed in the study of the binary mixtures. The UNIQUAC equation is slightly superior in the prediction of the vapor composition than the remaining models; the NRTL equation seems to be, in the case of ternary mixtures, somewhat superior to Wilson’s equation. The same behavior is observed for the temperature or pressure predictions. 4. Conclusions The UNIQUAC equation seems to be the best on the basis of the systems studied. It gives slightly better results than the Wilson’s or NRTL equations. It also seems that for binary systems containing an alcohol, Wilson’s equation gives the best predictive results, this trend having been discussed in a previous report (Schreiber, 1971).
191 PREDICTION
OF LIQUID-LIQUID
EQUILIBRIUM
In this section we examine the capabilities of the NRTL and UNIQUAC equations for predicting the complex phase behavior of liquid-liquid systems. We present results for 2 systems with a single pair of immiscible components (mixtures 4 and 2, type I) and 2 systems with two pairs of immiscible components (mixtures 3 and 4, type II), which are showed in Table 5. As it is well known, of the 3 solution models used in this work, only the UNIQUAC and NRTL models can predict phase instability (Orye, 1965; Renon, 1966; Abrams and Prausnitz, 1975). For the systems examined here, the binary UNIQUAC and NRTL parameters are selected to match vapor-liquid data for the miscible pairs and mutual solubility data for the partially miscible components. This technique, which is not as yet generally accepted (Abrams and Prausnitz, 1975; Joy and Kyle, 1969) because of the poor agreement obtained between experimental and calculated tie lines, can be notably improved by choosing .better binary parameters. Another alternative is, of course, the fitting of the binary parameters on ternary tie line data. This force-fitting procedure is very successful but cannot be adequately extended to the case of multicomponent liquid-liquid equilibrium. The first choice is adopted and the improvement is achieved by setting physically meaningful pairs of parameters for the partially miscible components. 1. Reduction
of binary miscibility data
We start from a recent correlation scheme (Mattelin of binary miscibility data that uses a special searching suitable pairs of binary parameters for the UNIQUAC The binary mixture is characterized in terms of the such that u = (a + b)/[ln(xt/x;)]
-
and Verhoeye, 1975) technique to obtain and NRTL models. quantities LJand w
2ab/[ln(x~/x~)]
(124
and w = 2/[ln(xk/x;)]
-
(a + b)/[ln(x$/xi)]
Wb)
where a =x:/x;;
b =x:/x;
In these relations (eqs. (9b), (9c) of Mattelin and Verhoeye’s
(12c)
paper) X: and 3t: are the mutual solubilities. They correspond to the end points of the binary diagram. Following these authors the different steps necessary to compute the parameters are: (1) Assign a class to each type of mixture, depending of the values taken by -v/w, either larger or smaller than one. From this it is possible to know, a priori, the interval of variations for the parameters eij, Gij and Gji. (2) Solve for (Yij, Gjj and Gii via a set of non-linear equations which
192
depend on the objective function chosen to minimize the differences between calculated activities corresponding to the equilibrium compositions. (3) Eliminate solutions provided by step 2 that have no thermodynamic significance by introducing a constraint conditions: calculate the free energy of mixing AGM and select the previous values for which AGM is negative. In practice, step 2 leads to a large number of numerical solutions, characterized by an oscillatory behaviour which makes them difficult to handle when applying the liquid phase instability condition (step 3). Therefore, we choose to abandon this approach. We replace step 2 and 3 by an unique selection criterium: we fit the abscissae of the points of contact of common tangents to the curve AGM versus x directly to the mutual solubility data (XT and x;), using the condition AGM < 0 with 0 d x < 1. The algorithm used in this work is based on the calculation of the Gibbs energy of mixing. First we initialize with a set of parameters taken from an interval of variation defined in step 1. They are then incremented by simultaneous positive values and we calculate new AG”. From the points XT and xf, the mole fractions corresponding to the contacts of the tangent to the curve are calculated and the parameters are chosen to minimize the quantity F defined as: (13) where x; talc and xi _ic are the calculated abscissae of the tangent to the curve AGM versus x. The computing time and the screening of physically acceptable solutions depend both on the type of mixture studied and the experimental solubility data. 2. Results
obtained
for binary mixtures
(i) NRTI, equation The proposed method may lead to several possible sets of binary parameters, depending of the values of (Yij, as in the original method of Mattelin and Verhoeye. The advantage is that the number of pairs of binary parameters (and the computation time) is greatly reduced. This is probably due to the fact that the condition of minimum free energy of the system, along with our force fitting technique of the mutual solubility data, is a sufficient condition to obtain the physical meaningful set of parameters. (2) UNIQUAC equation We extend the proposed method to the screening of the binary parameters of eqn. (3b). In this case the trend above mentioned is not observed; we arrive at a single pair of physically meaningful binary parameters. This seems, a priori, an understandable result: the asjusted parameter (~lijis replaced by the two fixed structural parameters r and q. As an illustrative example of the results obtained with our method of ob-
323.15
293.15
283.16
273.16 0
0.1
0.2
03
44
0.6
0.6
a7
0.8
0.9
I 1.0
Fig. 1. Experimental and calculated mutual solubility data for the n-Heptane-n-methylpyrrolidone mixture. The solid curve is the result of the present correlation of binary miscibility data with NRTL equation, the dashed line is the calculated curve using binary parameters of the literature (Data Ref. 6), (0) present work, calculated with UNIQUAC equation. Data source: (o), (X) Data Ref. 6. TABLE Solution
4 model
Mixture
parameters T W
from binary miscibility g12-a2 (cal/mol) NRTL
data 821 -a1 (cal/mol)
UNIQUAC -6.4
NRTL
UNIQUAC
1337.4
-1158.3
01 NRTL
Data Ref.
0.377
18
0.396
19
Acetonitrilen-heptane iso-octanefurfural
318.15
1577.7
303.15
1591.1
-287.9
n-Hexanefurfural n-Heptanefurfural Cyclohexanefurfural 1 -HexeneDMSO
303.15
1519.4
-338.5
1400.2
-250.3
0.383
19
303.15
1545.0
-288.0
1588.8
-268.9
0.398
19
303.15
1414.0
-725.2
895.3
0.322
19
333.15
1835.51
-275.06
0.389
21
/
1683.4
1559.00
232.3
-6.1 -295.4
194 taining the NRTL and UNJQUAC binary parameters we generate the solubility curve of the n-Heptane-n-Methylpyrrolidone mixture, the calculated and experimental solubility data appear in Fig. 1. It can be seen that the calculated solubilities are in good agreement with the experimental results. The parameters obtained are shown in Table 4. The pure component parameters r and q are calculated in the same way as in section 2.1. The Olii values are adjusted to the binary miscibility data. 3. Tie line calculations
of ternary
mixtures
The phase equilibrium calculation is based on the condition that the activity of each component in both phases is the same. The tie line curve is obtained by solving the following equation: [ln(yjl’x!‘))
-
ln(yf2)~$2J)]”
= ()
(14)
where i = 1, 2, 3. At first we solved eq. (14) for the case of the NRTL equation using a Newton-Raphson procedure that has been discussed elsewhere (Renon et al. 1971). However, it was not possible to solve eq. (14) with the UNIQUAC equation because of convergence problems. Therefore the tie line computations were performed with a direct search technique - here, the so called simplex method (Murray, 1972). This technique has proved to be extremely effective in practice, particularly when the objective function is not differentiable or the derivatives are difficult to obtain. When we applied this method to the resolution of eqn. (14) with the NRTL equation, the results obtained are similar to or better than those obtained with the Newton-Raphson procedure. We compared experimental and calculated tie line composition in terms of the quantity Ax defined as
AX = (l/ND)
CC j
i
lXicalc
-xi
exp
I
(15)
wherei=1,2,3andj=l... ND. The results are shown in Table 5 for both solution models. Although only a modest number of ternary systems have been studied, we can observe several trends. First, the NRTL model gives better results in the ternary liquid calculations for systems of type II with plait points than for systems of type I, without plait points. The differences observed between values of AX are small and do not correspond to the general assessment about the difficulties of predicting ternary diagrams that present one plait point. This is due, perhaps, to the fact that our ternary liquid calculations are based on the “best” binary parameters for the partially miscible components. The improvement of the ternary system equilibrium prediction, in this case, is quite significant. Regarding the UNIQUAC model, the deviations are larger than those obtained with the NRTL model, especially for systems of
195 TABLE
5
Liquid-liquid equilibria. Mean deviations between experimental and calculated liquid composition with UNIQUAC and NRTL equations Mixture
-:
“2.“‘
Acetonitrile(l)-benzene(Z)n-Heptane( 3) n-Hexane( l)-5-nonanone(Z)Dimethylsulfoxide( 3) i-Octane( 1 )-n-hexane( 2)furfural( 3) n-Heptane(l)-cyclohexane( furfural( 3)
T
2)-
Mean Deviation
AX
Number of data points
Data Ref.
NRTL
UNIQUAC
0.0030
0.0031
48
18
0.0010
0.0200
36
21
0.0037
0.0388
36
19
0.0063
0.0125
36
19
type II. This could indicate that the UNIQUAC model is not suitable for these systems. This analysis confirms the good predictive qualities of the NRTL model for the representation of ternary liquid equilibrium composition. We may conclude that this solution model is better adapted to calculations of liquidliquid systems than of vapor-liquid equilibria. CONCLUSIONS
Our study on the applicability of three simple molecular models to the estimation of equilibrium properties has confirmed some known facts but also pointed out some unexpected trends. Over the 8 types of mixtures studied, the UNIQUAC equation has shown a slight superiority over the NRTL and Wilson’s equations in the representation of vapor-liquid properties for binary systems. Nevertheless, two kinds of mixtures are better represented with Wilson’s equation: viz. alcohol-hydrocarbon and chloroform-polar component or alcohol mixtures. In the case of ternary mixtures, the differences of prediction of vapor--liquid equilibria between the three models are still smaller. It is difficult, if not impossible, to establish the absolute predictive accuracy of each equation used here and we should emphasize that one must be very careful in the selection of the solution model when requiring very precise quantitative estimates of fluid phase equilibria. ACKNOWLEDGMENT
It is a pleasure to thank Dr. J. Tejeda for many helpful discussions and for his contribution to the optimization program of binary parameters.
196 NOTATION
a, b D F .!?ij_gij
Gij h J 1 N ND P Q I; T V v, w x, Y 2
ratio of the mole fractions of both components in one phase, in eqn. (12~) percentage deviation as defined in eqn. (6a) objective function energetic parameters, Wilson’s eq. (lb), cal/mol-’ NRTL parameters, eqn. (2b) enthalpy defined by eq. (6b) defined after eq. (3a) number of components number of data points pressure, atm pure component area parameter pure component volume parameter gas constant, cal K-l mol-’ equilibrium temperature, K molar volume of liquid, cm3 mole-r defined by eqs. (12a) and (12b) liquid and vapor phase mole fraction lattice coordination number, set equal to 10
Greek symbols a:ij Yh ki Pi @i
ei tij rij
non-randomness parameter of the NRTL equation activity coefficient of component k Wilson parameters, eq. (lb) vapor phase fugacity coefficient of component i segment fraction of component i area fraction of component i defined by eqn. (2b) defined by eqn. (3b)
Subscripts talc refer to a calculated value of a quantity refer to an experimental value of a quantity exp i, j, k, 1 component i, component j, component k, component
1
Supercripts (Q) (b)
refers to the liquid phase on the left side of the concentration diagram refers to the liquid phase on the right side of the concentration diagram
197
S E
saturated condition excess property
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,