Binary Interaction Parameters

Fluid Phase Equilibria, 58 (1990) 117-132 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

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ESTIMATION OF MULTIPLE BINARY INTERACTION PARAMETERS IN EQUATIONS OF STATE USING VLE DATA. APPLICATION TO THE TREBBLE-BISHNOI EQUATION OF STATE *

P. ENGLEZOS, N. KALOGERAKIS * *, M.A. TREBBLE and P.R. BISHNOI

Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Drive N. W., Calgary, Alb., T2N IN4 (Canada)

(Received February 3,1989; accepted in final form October 31, 1989)

ABSTRACT

Englezos, P., Kalogerakis, N., Trebble, M.A. and Bishnoi, P.R., 1990. Estimation of multiple binary interaction parameters in equations of state using VLE data. Application to the Trebble-Bishnoi equation of state. Fluid Phase Equilibria, 58: 117-132.

A systematic approach for the estimation of binary interaction parameters for equations of state is presented. A least-squares procedure which is computationally very efficient is advocated for the calculation of the binary interaction parameters. Subsequently, if the calculated phase behavior represents the experimental data without a gross bias, the statisti- cally best parameters can be obtained by maximum likelihood (ML) estimation. For these cases, the use is advocated of an implicit ML estimation procedure which is computationally significantly more efficient than the error in variables method. The proposed approach is particularly suitable for equations of state which have more than one interaction parameter. In such cases, the best parameter set is chosen from among several combinations of interaction parameters present in the equation of state.

INTRODUCTION

Equations of state (EOS) are used extensively for the calculation of high pressure equilibria and/or properties of fluid mixtures. Empirical binary interaction parameters are normally utilized in these equations to increase their correlational flexibility, particularly in non-ideal fluid systems. Re- cently, Trebble and Bishnoi (1987, 1988) proposed a cubic EOS which can

* The material in this article was presented at the 4th International IUPAC Workshop, Thessaloniki, Greece, October 24-26,198s. * * Author to whom correspondence should be addressed.

037%3812/90/!03.50 0 1990 - Elsevier Science Publishers B.V.

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utilize up to four such parameters. The interaction parameters are usually obtained from the regression of binary vapor-liquid equilibrium (VLE) data. These data are the measurements of the vapor and liquid mole fractions of one of the components, the temperature and the pressure.

Traditionally, simple least-squares (LS) estimation has been used to obtain the parameters. From a statistical point of view, this is also the correct approach to use when the errors in the measurements are indepen- dently and identically (iid) normally distributed with zero mean and con- stant variance. However, the error variances are not the same for each of the measured variables, and may differ significantly from experiment to experi- ment. In this case, a statistically sound procedure that has been formulated for the estimation of parameters for non-linear functional relationships (Box, 1970; Britt and Luecke, 1973) can be employed. It is based on the maximum likelihood (ML) principle. One important underlying assumption in applying ML estimation is that the model is capable of representing the data without any systematic deviation. This assumption is reasonably satis- fied when correlating low pressure VLE or LLE data using excess Gibbs free energy models. Hence, the ML estimation method has been used extensively for the estimation of the parameters of the excess Gibbs free energy models (Fabries and Renon, 1975; Peneloux et al., 1976; Kemeny and Manczinger, 1978; Anderson et al., 1978; Sutton and MacGregor,,1977; van Ness et al., 1978; Neau and Peneloux, 1981; Patino-Leal and Reilly, 1982; Kemeny et al., 1982; Salazar-Sotelo et al., 1986). However, cubic EOS predict the properties of mixtures of polar components with a variable degree of success. The magnitude of the variation in the experimental error can be significantly less than the systematic deviation resulting from thermody- namic model inadequacies, and in such cases ML estimation should be used with caution.

Over the years, two ML estimation approaches have been followed: (a) parameter estimation (implicit formulation) and (b) parameter and state estimation (error in variables method). In the first approach, only the model parameters are estimated (Sutton and MacGregor, 1977). In the second, the true values of the state variables as well as the parameters are estimated. Neau and Peneloux (1981) and Kemeny et al. (1982) have suggested that both the methods yield the same parameters when they utilize the same amount of information. However, this is true for the implicit formulation only if the residuals in the objective function are normally distributed. Skjold-Jorgensen (1983) formulated the problem of parameter estimation for an EOS using ML-based estimation methods, but did not present computational results.

In the present work, we examine the problem of estimation of EOS interaction parameters in a systematic manner (Englezos, 1988). We present

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a computationally efficient approach which involves the use of a simple LS estimation procedure as well as an implicit ML estimation procedure to obtain the statistically best parameters. Conditions under which the simple LS procedure is adequate are exploited. The Trebble-Bishnoi EOS has been used throughout this work.

PROBLEM STATEMENT

It is assumed that a set of N experiments have been performed and that at each of these experiments, four state variables were measured. These mea- surements are the liquid 2 Fd vapor jJ mole fractions of one of the components, the temperature T, and the pressure p of the system, all taken at equilibrium. If x, y, T, P are the corresponding true but unknown values of the state variables, they are related to the measurements by the equations

Ji=xi+eXi; i=l ,**-, N 04 ji=yi+eyi- , i=l ,-*-, N (W

$=T+e,,; ,...,N i=l (14

Pi = Pi + e,,i; i= l,..., N (ld)

where e.,i are the corresponding errors in the measurements. The thermody- namic model (EOS) can be viewed as a functional relationship among the above true state variables, a set of unknown parameters k, and a set of precisely known parameters u, such as the pure component EOS parame- ters and critical properties. However, it should be noted that the assumption of precisely known pure component critical properties may not always be valid. The elements of the parameter vector k = (k,, k,, k,, k,)T are the binary interaction parameters. Given the above information and having a thermodynamic model, the objective is to determine the parameter vector k by matching the data with the model predictions by satisfying some optimal- ity criterion.

THEORETICAL BACKGROUND

ML parameter and state estimation (error in variables) method

According to the phase rule for a binary two-phase system, at each experimental point only two of the above four state variables are indepen- dent. Arbitrarily one can select two variables as independent and use the thermodynamic model and the equilibrium equations to solve for the other

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two, which become the dependent ones. For experiment i, let nij( j = 1, 2) be the independent variables (e.g. qil = xi and qiZ = T). The dependent variables tii can be obtained from the equilibrium relationships using the thermodynamic model, and in principle can be written as

tij=hj(nii, Iliz; k; u>; j=l, 2 (2)

In this case the ML estimate of k is obtained by minimizing the objective function (Anderson et al., 1978; Salazar-Sotelo et al., 1986)

E i [7)ij-Gij12 + [ hj(qil, qi2; k; U) - ii] ix1 j=l 4, 'tfj

The above quadratic optima&y criterion can be derived from the ML function by assuming that: (a) the experiments are independent; (b) the errors in the measurements of the state variables are normally distributed with zero mean and variance-covariance Cj = diag(&, (I;,~, U~i, u:,~); (c) the elements of the covariance matrix Ci are known a priori; and (d) the thermodynamic model is capable of representing the data without any systematic deviation.

In the case where the elements of the covariance matrix are not known, eqn. (3) should be modified by replacing the u terms with their estimates 6, and by adding an extra term in the optimality criterion.

Unless very few experimental data are available, the dimensionality of this problem is extremely large and hence difficult to treat with standard non-linear least-squares iterative procedures. Schwetlick and Tiller (1985), Salazar-Sotelo et al, (1986) and Valko and Vajda (1987) have examined and exploited the structure of this problem from a computational viewpoint and proposed efficient algorithms for its solution.

ML parameter estimation (implicit formulation) method

If we wish to avoid the computationally demanding state estimation, we have to change the optimality criterion and impose additional distributional assumptions. In this case the residuals in the optimality criterion, instead of representing the errors in the state variables, are suitable implicit functions of the four state variables dictated by the isofugacity criterion.

The errors in the measurements of all four state variables are taken into account by the error propagation law (Fabries and Renon, 1975; Sutton and MacGregor, 1977). In this case the necessary assumptions are that: (a) the experiments are independent; (b) the variance-covariance matrix C of the errors in the measurement of the state variables is known a priori; (c) the residuals employed in the optima&y criterion are normally distributed; and

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(d) the thermodynamic model is capable of representing the data without any systematic deviation.

PROPOSED IMPLICIT ML PARAMETER ESTIMATION PROCEDURE

At each experimental point i, and for each component j, the equilibrium constraints (isofugacity criterion) can be written as

gi(xi, y,, T, Pi; k; u)=O; i=l,..., N; j=1,2 (4)

If we replaced the true but unknown state variables with their measure- ments, then the above functions would not be equal to zero, owing to the measurement errors, even if the model were exact.

gi(&, ji, t, ?i; k; u) =cij; i=l,..., N; j=1,2 (5)

If the assumptions stated in the previous section hold, the parameter estimation problem reduces to the minimization of the quadratic objective function

where u; ( j = 1, 2) is computed using the error propagation law. If the errors in the measurements of the state variables are not correlated, using a first order variance approximation for IJ$ we obtain

and similarly for ail2 we obtain

+( a)( $)G,i

(7)

(8)

where all the derivatives are evaluated at hi, j$, $, Fi. The choice of the functional form of the residuals in the optimality

criterion is of extreme importance, and should be based on the following considerations.

(a) On statistical grounds, the selection of the residual functions should be such that the errors, cij are approximately normally distributed, since a quadratic objective function is to be used..

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(b) On computational grounds, the choice should be such that no iterative computations are needed for the calculation of the residuals. Otherwise there is no advantage in using an implicit formulation over the error in variables method, which on statistical grounds has more realistic assumptions.

(c) The choice of the residuals should be such that it avoids, if possible, numerical instabilities (e.g. exponent overflow) and has favorable conver- gence characteristics.

Based on the above considerations and numerical experimentation with several binary systems, we chose the residual function

gj(xi,yi,T, P,;k;u)=lnh:-lnf,;; i=l,..., N; j=l,2

The calculation of the residuals is straightforward and does not require any iterative computations. As a result, the overall computational require- ments are significantly less than for the error in variables method.

PROPOSED SIMPLE LS ESTIMATION PROCEDURE (CURVE FITTING)

As discussed, cubic EOS may not be capable of predicting the phase behavior of certain mixtures without systematic deviations. These deviations can be significantly larger in magnitude than the experimental error. In such cases one should not attempt to estimate the best parameters in a statistical sense, but should rather view the parameter estimation problem as a curve fitting exercise. In other words, the objective should be to determine a set of interaction parameters (with as little computational effort as possible) which yields an acceptable fit within the limitations of the model.

Based on numerical experimentation we chose the following LS estima- tion procedure, whereby the parameters are obtained by minimizing the objective function

S,,(k) = 5 i qij[ln fi, - ln A;] i=l j=1

00)

where qij are user-selected weighting factors. In general, the choice of qi j should be such that equal attention is paid to all data points. If the ln fij are within the same order of magnitude, qij can simply be set equal to 1. Otherwise, qij can be chosen proportional to (ln fiy)-2, which corresponds to complete normalization of the residuals. It is noted that Paunovic et al. (1981) have proposed a similar LS procedure by formulating the objective function as the non-weighted summation of squares of normalized fugacity differences. Peters et al. (1987) have reported that use of the summation of squares of fugacity differences can easily lead to numerical instability. Skjold-Jorgensen (1983) has suggested the use of eqn. (10) with qij = 1.

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ALTERNATIVE LS ESTIMATION PROCEDURES

The error in variables method can be simplified to weighted LS estimation if the independent variables are assumed to be known precisely. On statisti- cal grounds this assumption is reasonable if the experimental error in the independent variables is much smaller than that in the dependent ones. On thermodynamic grounds, the selection of the independent variables should be based on the VLE behavior of the system. Systems with a sparingly soluble component, such as methane-methanol, are poor candidates for anything except isothermal-isobaric flash calculations because of the large derivatives of the pressure with respect to the liquid mole fraction. For systems with azeotropic behavior, bubble point pressure calculations should be performed at a specified liquid mole fraction and temperature.

If we further assume that the variances of the errors in the measurement of the dependent variables are known a priori, the following objective functions can be formulated:

( Mk)= i & I p.cc-~i)2 + (yidc-jTi)* 2 i=l 5.i I and

N (x;~c-4i)2 + (y;c-,)* &P(k) = c

i=l i .t i ti )

01)

02)

In the case of S,(k), the calculation of yFIC and Pyle is performed by bubble point pressure computations. In the case of S,,(k), ~7~ and yFIC are obtained by isobaric-isothermal flash computations. These flash calcula- tions have to be performed at each iteration during the minimization with the current parameter estimates, and hence the computational requirements increase substantially. Furthermore, when the parameters are away from their optimal values, the flash calculations may exhibit convergence and/or numerical difficulties which lead to convergence problems in the minimiza- tion procedure. Taking into account the above observations and the fact that important experimental information regarding the independent variables is ignored, we do not advocate use of these LS estimation procedures. Compu- tations with the above two objective functions were performed merely for comparison purposes.

SYSTEMATIC APPROACH TO THE PARAMETER ESTIMATION PROBLEM

When given a set of binary VLE data with no prior knowledge regarding the adequacy of the thermodynamic model (EOS), we advocate the following approach to the parameter estimation problem.

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(a) Use the proposed simple LS estimation procedure and determine the binary interaction parameters.

(b) Plot the data and the model predictions and judge whether the fit is acceptable. If the model predictions are deemed to be not acceptable, stop, and reconsider the applicability of the EOS/mixing rules. Otherwise, pro- ceed to the next step.

(c) Using the determined parameters as initial guesses, obtain the opti- mum parameter values by the proposed implicit ML estimation procedure.

At this point it is up to the user whether to proceed with further investigations, such as computation of the standard error of parameter estimates, computation of the standard error of estimate of the model prediction, detection of erroneous measurements (outliers), performance of model adequacy tests, etc.

Step (a) is very important when using EOS which utilize more than one binary interaction parameter. In such cases we have to select the best combination of interaction parameters to be used, besides determining their optimal values. For example, the Trebble-Bishnoi EOS can utilize up to four interaction parameters; hence the number of possible combinations that should be investigated is C:=i( 4) = 15.

In general, it is desirable to utilize as few interaction parameters as possible. Therefore, the investigation begins by considering all possible cases with one parameter. Of these, the best is used for the generation of the VLE predictions. Upon examination, we decide whether the fit is acceptable. If it is, we proceed to determine the statistically optimal parameter value by the implicit ML procedure. If the fit is not acceptable, we should proceed to utilize two binary interaction parameters. Again, all possible two-parameter combinations should be examined and the VLE predictions be generated with the optimal set of parameters. If necessary, we should continue with three or more parameters. It is noted that the decision as to whether the fit is acceptable is subjective and based on experience rather than a rigorous statistical test.

COMPUTATIONAL ASPlkTS OF THE MINIMIZATION ALGORITHM

In the applications that follow, the solution of the non-linear minimiza- tion problem was obtained using the Gauss-Newton method. To ensure convergence of the iterative procedure we used (a) Marquardts modifica- tion, and (b) singular value decomposition of the normal equation matrix to take the pseudo-inverse if the problem was ill-conditioned (Kalogerakis and Luus, 1983). When the initial guess of the parameters is not sufficiently close to the optimal values, several problems may be encountered. The minimiza- tion may proceed towards the wrong direction, or towards the proper

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direction but with excessive overstepping. To remedy the latter, an optimal step-size policy was used (Kalogerakis and Luus, 1983). The computations were performed on a CDC Cyber 860 computer with the NOS/VE operat- ing system.

APPLICATIONS

Three binary systems were used in the present study, namely methane- methanol (Hong et al., 1987), carbon dioxide-methanol (Hong and Kobayashi, 1988) and propane-methanol (Galivel-Solastiuk et al., 1986). The quadratic mixing rules used were those proposed by Trebble and Bishnoi (1988). It is noted that the binary interaction parameters were considered to be temperature independent.

Table 1 shows the parameter estimates with their standard error of estimate for the methane-methanol system. Also shown is the type of the objective function that was used, and the CP seconds required per iteration. The values of the parameters which are not shown in the tables are zero. As can be seen from the table, the parameter estimates with all three estimation procedures do not vary significantly. For this system, use of only one parameter, kd, was found to be acceptable. The interaction parameter k, was selected over k,, k,, k, from the results obtained by the simple LS estimation procedure. Using the interaction parameter value for k, obtained by LS estimation (-0.1903), the phase equilibrium calculations were per- formed. The calculated values of the liquid and vapor mole fractions at constant temperature and pressure were found to agree very well with the experimental data. Subsequently, ML estimation was performed and the value -0.2317 was obtained for k,. Using the ML estimate for k,, iso- baric-isothermal flash computations were performed. Typical VLE predict- ions for two isotherms are shown in Figs. 1 and 2.

Similar calculations were performed for the carbon dioxide-methanol system. The parameter estimates are shown in Table 2. Two binary interac-

TABLE 1

Parameter estimates for the methane-methanol system

Parameter Standard Objective deviation ( W) function

CPU/iteration b (CP s)

k, = -0.1903 f 14.9 a &s(k) 0.004 NP k, = - 0.2317 f 0.03 S,,(k) 0.014 NP k, = - 0.2515 f 0.01 a S,,(k) 0.032 NP

a As computed under the null hypothesis that uij is constant and qii = 1. b NP is the number of experimental points.

T q 220 K

0.0 0.2 0.4 0.6 0.8 1.0

X, Y Methane

Fig. 1. Vapor-liquid equilibrium for the methane-methanol system.

tion parameters were used. For this system the minimization algorithms using ST, and S, converged with a Marquardts parameter not equal to zero. For the isothermal-isobaric flash calculations, the ML estimates were used. The results at 290 and 273 K, shown in Figs. 3 and 4, were found to be slightly better than those obtained using the LS estimates. Comparing the required computational times, it can be seen that the LS estimation proce- dure has the smallest requirements, whereas the estimation procedures which are based on flash calculations have significantly larger requirements. For example S, requires 20 times more CPU time per iteration compared with

5oT------

0 0 H@Q 6t al. 1987

- Predict/ops

0.0 0.2 0.4 0.6 0.6 1.0

X, Y kktkne Fig. 2. Vapor-liquid equilibrium for the methane-methanol system.

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TABLE 2

Parameter estimates for the carbon dioxide-methanol system

Parameter Standard Objective CPU/iteration b deviation (%) function (CP s)

k, = 0.0605 f 9.54 a S,(k) 0.0054 NP k, = -0.1137 f 15.86 a k, = 0.0504 f0.18 S,,(k) 0.015 NP k, = - 0.0631 f 0.60 k, = 0.0511 f3.15 Sr, (h) 0.103 NP k, = - 0.0967 f 7.63 k, = 0.0566 f 3.91 S,,(h) 0.445 NP k, = - 0.2238 f 1.14

a As computed under the null hypothesis that uij is constant and qi, = 1. b NP is the number of experimental points.

simple LS, whereas S, requires 80 times more CPU time for this particular problem.

For the majority of binary systems, the use of one or two binary interaction parameters has been found to be adequate for the representation of VLE data (Trebble, 1988). As can be seen from Figs. 1-4, the phase behavior of the binary systems of methane and carbon dioxide with methanol are represented by the EOS with an accuracy reasonable for engineering-type calculations. This was not the case for the propane-methanol system, for which grossly biased predictions were obtained using one, two or three parameters determined by LS estimation. The improvement of the fit by

- Predictions 0 I 1 , I I I I

0.00 0.25 0.50 0.75 1.00 x, Y CO2

Fig. 3. Vapor-liquid equilibrium for the carbon dioxide-methanol system.

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6- I I I I

0 0 Hong and Kabayashi 1988 - Predictions

5--

x, Y co;! Fig. 4. Vapor-liquid equilibrium for the carbon dioxide-methanol system.

increasing the number of parameters, especially from two to three, was minimal. It is evident that the EOS with classical quadratic mixing rules is not capable of representing the behavior of this system with an accuracy that would make ML estimation of the parameters meaningful. ML calcula- tions were nevertheless performed, for comparison purposes, and the param- eter estimates obtained are shown in Table 3. Again, the computational requirements per iteration of S, are four times more than for the implicit ML, and 33 times more compared with the simple LS procedure. The parameters obtained by LS estimation were used in this case to generate the phase equilibrium predictions by bubble point pressure calculations. The results for two -isotherms are shown in Figs. 5 and 6. As can be seen, the calculated bubble pressures differ significantly from the experimental values.

TABLE 3

Parameter estimates for the propane-methanol system

Parameter

k, = o.i53i k, = - 0.2994 k, = 0.1533 k, = - 0.3218 k, = 0.2384 k, = -0.3222

Standard deviation ( W)

f 7.35 a f 9.34 a f 0.27 f 0.40 f 0.58 f 0.82

Objective function

S,,(k)

S,,(k)

S,,(k)

CPU/iteration b

(CP s)

0.006 NP

0.049 NP

0.196 NP

a As computed under the null hypothesis that uij is constant and qii = 1. b NP is the number of experimental Points.

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0 0 Galivel- Salastiuk et al. 1988 -Predictions

Y , ~~ I 0.0 d.3_ .

x, Y Fr:pacne d.9

Fig. 5. Vapor-liquid equilibrium for the propane-methanol system.

Moreover, erroneous liquid phase splitting is suggested. We also used the ML estimated parameters in the bubble point pressure calculation al- gorithm, and the results were similarly bad. This is a typical case where one should not proceed to compute the ML estimates, but should rather recon- sider the applicability of the thermodynamic model (EOS/mixing rules).

Schwartzentruber et al. (1987) introduced an empirical three-parameter mixing rule with temperature-dependent interaction parameters, and were able to correlate these propane-methanol data. Recently, Englezos et al. (1989) have proposed a methodology for the estimation of binary interaction

2.8

2.1

2

3

E 1.4

2

0.7

0.0

-I 1 I 1 I I

q 0

8

/

0

T = 343 K /

4/ 30 Galivel- Solartiuk et al. 1988 - Predlctions I I I I I I I I

0.1 3 0.2 0.4 0.6 0.6 1.0

X, Y hopane

Fig. 6. Vapor-liquid equilibrium for the propane-methanol system.

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parameters which yield the best possible fit of the VLE data and guarantee prediction of the correct phase behavior.

Finally, as can be seen from the examples given above, the computational requirements of the LS and the ML estimation procedures are very small when compared with the simplified error in variables method (S,,., S,,). In addition, the latter were found to converge very slowly as compared with the ML or the LS estimation procedures, which implies that the overall compu- tational requirements of. the error in variables method are excessive by at least two orders of magnitude.

CONCLUSIONS

A systematic approach for the estimation of binary interaction parameters for an EOS has been formulated and evaluated using the Trebble-Bishnoi EOS. It has been shown that a very efficient LS estimation procedure should be used for the selection of the best combination of interaction parameters. Subsequently, the statistically best interaction parameters can be obtained using the proposed implicit ML estimation procedure, if the model is not grossly biased. The latter procedure is computationally much more efficient when compared with the error in variables method. If the EOS is not adequately predicting the VLE data, the estimates obtained by the simple LS procedure should suffice.

ACKNOWLEDGEMENTS

Financial support by the Natural Science and Engineering Research Council of Canada is greatly appreciated. P. Englezos expresses his gratitude to the Izaak Walton Killam Memorial Scholarship Trust and to the Alberta Research Council for their financial support. The computations were per- formed using the facilities of the University of Calgary Computer Center.

NOMENCLATURE

k P

: T U X

Y

error fugaci ty interaction parameter vector pressure weighting factor objective function temperature vector of precisely known parameters liquid mole fraction vapor mole fraction

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Greek letters

c residual Bl? 772 independent variables 513 62 dependent variables

; standard deviation variance-covariance matrix

Subscripts

i index; 1, . . . , N j index; 1, 2 LS least-squares ML maximum likelihood TX temperature, liquid mole fraction TP temperature, pressure

Superscripts

talc calculated L liquid V vapor A

experimental

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