Abstract
The equations of the method based on the maximum likelihood principle have been rewritten in a suitable generalized form to allow the use of any number of implicit constraints in the determination of model parameters from experimental data and from the associated experimental uncertainties. In addition to the use of any number of constraints, this method also allows data, with different numbers of constraints, to be reduced simultaneously. Application of the method is illustrated in the reduction of liquidliquid equilibrium data of binary, ternary and quaternary systems simultaneously
Maximum likelihood; parameter estimation; phase equilibrium
APPLICATION OF A GENERALIZED MAXIMUM LIKELIHOOD
METHOD IN THE REDUCTION OF MULTICOMPONENT
LIQUIDLIQUID EQUILIBRIUM DATA
L. STRAGEVITCH and S.G. dÁVILA
Laboratório de Propriedades Termodinâmicas, Faculdade de Engenharia Química,
Universidade Estadual de Campinas, Caixa Postal 6066, Campinas, SP, 13081970, Brazil
(Received: March 14, 1996; Accepted: January 22, 1997)
Abstract  The equations of the method based on the maximum likelihood principle have been rewritten in a suitable generalized form to allow the use of any number of implicit constraints in the determination of model parameters from experimental data and from the associated experimental uncertainties. In addition to the use of any number of constraints, this method also allows data, with different numbers of constraints, to be reduced simultaneously. Application of the method is illustrated in the reduction of liquidliquid equilibrium data of binary, ternary and quaternary systems simultaneously.
Keywords: Maximum likelihood, parameter estimation, phase equilibrium.
INTRODUCTION
The problem of estimating parameters in mathematical models is an important step in the process of interpretation of experimental data in many fields of engineering science. When the experimental errors in the measured variables are taken into account, algorithms based on the maximum likelihood (ML) principle are considered to offer the most appealing possibility for optimum parameter estimation if the statistical distribution of the observations is known and is subject exclusively to random errors (SkjoldJørgensen, 1983). Moreover, the model is not supposed to introduce systematic errors greater than the experimental random errors, as, in this case, the application of the ML principle is not justified (Péneloux et al., 1990).
In addition to the best values of the parameters, the estimation methods based on the ML principle can provide useful information to evaluate the model and the quality of the data in statistical terms. As suggested by Anderson et al. (1978), an error analysis can be used to identify the best model to represent a given set of experimental data. The analysis of the residuals, that is, the difference between the experimental values and the estimate of the true values of the variables, may provide useful guidance to assess the laboratory procedures during measurement. If the model is good, large residuals may be attributed to systematic errors that must be identified and eliminated, if possible.
In the last two decades, several algorithms based on the ML principle were developed to estimate parameters of phase equilibrium models. Good reviews and discussions on the subject are found in papers by SkjoldJørgensen (1983), Péneloux et al. (1990), Englezos et al. (1990b) and in the book by Novák et al. (1987). Nevertheless, most of the proposed methods involve only two constraints (implicit or explicit), limiting the treatment to vaporliquid equilibrium (VLE) binary systems. A computational methodology appropriate for the simultaneous regression of the VLE and the liquidliquid equilibrium (LLE) data was presented by Vo ka et al. (1989), and one appropriate for the simultaneous regression of the VLE and the vaporliquidliquid equilibrium (VLLE) data was presented by Englezos et al. (1990a); however, in both cases, only binary mixtures were considered. On the other hand, Novák et al. (1987) formulate the problem of parameter estimation from ternary LLE data but do not present computational results.
In this work, a method based on the ML principle to estimate parameters, which considers a generic number of implicit constraints, is presented. The method is also an extension of that developed by Niesen and Yesavage (1989). These authors applied the algorithm to the determination of parameters of cubic equations of state from binary VLE data.
The present extension is useful to simultaneously treat sets of data related to different numbers of constraints. Typically, the application of the new method is adequate in the reduction of phase equilibrium data sets involving
distinct numbers of components and/or diverse types of equilibrium, a practical problem to which the process engineer is faced frequently.
ADAPTATION OF THE EQUATIONS
Since the multiple constraint case is not conceptually different from the more common two constraint case, the generalization is meant to be in the sense of adapting the equations in order to facilitate their computational implementation. Basically, the equations are rewritten in a suitable general matrix form by extensively making use of the partitioning of matrices.
For convenience, the artifice of using dependent and independent variables, as proposed by Anderson et al. (1978), is adopted, although this distinction between the variables is not necessary, as stated by Britt and Luecke (1973). Furthermore, the nomenclature used is as similar as possible to that adopted previously by Anderson et al. (1978) and Niesen and Yesavage (1989).
Considering that N experimental observations are grouped in d different sets, each characterized by a certain number of constraints, a given set k (k = 1, 2, ..., d) contains N_{k} observations so that
(1)
For example, given sets (systems) of LLE (or VLE) observations, 1, 2 and 3 can contain N_{1} binary data, N_{2} ternary data and N_{3} quaternary data.
Each experimental observation, in a given set k, consists of M_{k} measured variables, among which K_{k} are considered to be the independent and the remaining M_{k}_{} K_{k}, the dependent variables. The true values of the independent variables in each experiment j are placed in X_{j} which becomes a K_{k}´ 1 matrix. Making use of the partitioning of matrices, for the N_{k} observations is defined as , an N_{k}K_{k}´ 1 matrix.
Similarly, the true values of the dependent variables in each experiment j are assigned to Y_{j}, which is an (M_{k}_{} K_{k}) ´ 1 matrix, and is defined as , an N_{k}(M_{k}_{} K_{k}) ´ 1 matrix.
Considering that an observation j in a given set k involves M_{k}_{} K_{k} dependent variables, M_{k}_{} K_{k} equations (constraints) are necessary to relate the dependent variables to the independent ones and to the parameters. For each experiment j, these equations can be written as
(2)
where q is an L ´ 1 matrix consisting of L undetermined parameters. In the general case, as considered here, the number of constraints can vary from one data set to another, and the constraints can be mathematically different as well. The constraint equations are assigned to f_{j} = , which is an (M_{k}_{} K_{k}) ´ 1 matrix, and then is defined as , an N_{k}(M_{k}_{} K_{k}) ´ 1 matrix.
To take all groupings (k = 1, 2, ..., d) into account, the true values of the independent variables are placed in X where X = , an h ´ 1 matrix, and where
(3)
Also, the true values of the dependent variables are assigned to Y where Y = , a m ´ 1 matrix, and where
(4)
Finally, F is used to represent the constraint equations where F = , an m ´ 1 matrix, meaning M_{k}_{} K_{k} constraints to each experimental point in each grouping k. As a result, the constraints equations are now represented in the reduced form
(5)
From this point on, the steps followed were exactly as described by Niesen and Yesavage (1989). The equations derived are also similar, apart from the dimensions of the matrices involved, which are obviously completely different, as well their definitions.
Provided that the experiments are independent and only random errors are involved in the measurements, the ML principle establishes that the best estimate of the parameters corresponds to the most likely set of the true values of the measured variables (Anderson et al., 1978). Following Anderson et al. (1978) and Niesen and Yesavage (1989), the optimum parameters are determined by solving the system of equations (5) by adopting the minimization of the equation (6) below as the convergence criterion,
(6)
where the superscript m denotes a measured variable and the raised dot denotes matrix multiplication.
As the variables X and X^{m} and Y and Y^{m} have dimensions h ´ 1 and m ´ 1, respectively, l and d are h ´ h and m ´ m diagonal matrices, correspondingly. The elements of l are the values of the inverse of the variances in the independent variables and d contains the values of the inverse of the variances in the dependent variables.
In principle, it is possible to eliminate Y from equation (6) making use of the constraints in equation (5). The conditions necessary to minimize the objective function (6) then become
(7)
(8)
To eliminate Y from equation (6), the constraints (5) are now linearized by a firstorder Taylor series expansion about the most recent estimates of the parameters and of the true values of the variables as follows:
(9)
where r indicates the previous iteration; F^{(r)} has dimensions m ´ 1 and consists of the values of F in the previous iteration; is a sparse m ´ h matrix and contains the partial derivatives of F with respect to the independent variables; is a sparse m ´ m matrix and contains the partial derivatives of F with respect to the dependent variables; and finally, is a m ´ L matrix and consists of the partial derivatives of F with respect to the parameters q . This linearization procedure is similar to that adopted by Niesen and Yesavage (1989).
Using the linearized equation (9), Y can now be eliminated from equation (6). By definition, F = 0 and the system formed by the m equations (9) can be solved for Y Y^{(r)} in terms of X, q and the previous values of F:
(10)
For simplification purposes, the subscript r has been omitted in the Jacobian matrices.
Substituting equation (10) into equation (6) the final form of the minimization function is achieved:
(11)
The minimization conditions (7) and (8) are applied to the objective function (11), resulting in the h + L system of equations (12):
(12)
Now the equations (12) are solved to obtain the increments q ^{(r}^{+} 1)^{}^{q}^{ (r)} and X^{(r}^{+} 1)^{}X^{(r)} :
(13)
(14)
where
(15)
(16)
(17)
(18)
(19)
Similarly, as described by Anderson et al. (1978) and Niesen and Yesavage (1989), the functions F^{(r)} and the Jacobians , and can be calculated using the estimates X^{(r)}, Y^{(r)}, q ^{(r)} from the previous iteration. Through equations (13) and (14), the corrections D q and D X are obtained, and consequently, X^{(r}^{+} 1) and q ^{(r}^{+} 1). New values for the dependent variables Y^{(r}^{+} 1) are directly calculated by solving equations (5). The new value of the objective function S, S_{N}, is then calculated from equation (6). Convergence is attained when the ratio (S_{N}_{} S)/S_{N} satisfies a given criterion.
Like the previous methods, the algorithm does not impose serious problems in its implementation and applications. The greatest computational effort is directly related to the number of experimental points and constraints involved and resides mostly in the solution of the constraint equations, in the calculation of the Jacobian matrices F_{X}, F_{Y} and Fq and in the inversion of the matrices F_{Y} (m ´ m ) and D (h ´ h ). However, F_{Y} and D are sparse matrices which, if taken into account, can considerably reduce the computer time in the inversion step. More precisely, these matrices are blockdiagonal and that characteristic is a consequence of the presumed independence of the experiments. As a result, use of sparse matrix handling routines is completely unnecessary. Instead, the diagonal blocks can be individually treated by any standard matrix inversion procedure. Moreover, to prevent oscillations and accelerate the convergence, the method of Law and Bailey (1963) was employed and showed to be satisfactory.
As a final step, to complete the parameter estimation scheme, the equations used to perform an error analysis are briefly discussed. For this multiple constraint case the number of degrees of freedom is n = m  L, so that the estimated variance s^{2} is given by
(20)
For the variancecovariance matrix of parameters S , the formula is
(21)
Apart from dimensions and definitions of the matrices involved, equation (21) is the same formula presented by Anderson et al. (1978).
APPLICATION TO LLE DATA
When two liquid phases (I and II) are present, each experimental tieline is identified by temperature (T) and phase mole fractions (, ,¼ , , , , ¼ , ) where C is the number of components, in a total of 2C 1 measured variables in each observation. In this case, the constraints are the equilibrium equations that can be expressed in the following form:
(22)
where g _{i} is the activity coefficient of component i in the mixture. The constraints are implicit in T and the mole fractions.
If d sets (systems), with different numbers of components, are considered in the estimation procedure, the elements of F (equation 5) are given by
(23)
According to equation (22), C constraints have to be satisfied at each experimental point. Therefore, since there are 2C  1 measured variables, C 1 independent variables must be chosen for each observation, e. g., T and , i = 2, ¼ , C 1. The remaining mole fractions in each experiment are considered to be the dependent variables, that is, M_{k}_{} K_{k} = C_{k}. As a result, for the reduction of LLE data, the objective function (6) is given by
(24)
where s is the observed standard deviation.
To illustrate the application of the algorithm, parameters of the UNIQUAC model (Abrams and Prausnitz, 1975) were estimated from the LLE data of binary, ternary and quaternary systems simultaneously, involving cyclohexane, toluene, benzene and acetonitrile, at 25° C. The standard deviations for the measured variables were adopted as s _{T} = 0.05° C for the temperatureand s _{x} = 0.002 for the mole fractions based on laboratory experiments where resistance thermometers are employed to measure temperature and the compositions are determined by gas chromatographic analysis.
When treating LLE data it is the usual practice to reduce the number of adjustable parameters by determining them for some of the miscible pairs from VLE data, the remaining parameters being estimated from data taken from the systems presenting partial miscibility. Using the present algorithm, the parameters shown in Table A1 of Appendix A were determined from the VLE binary data found in the given references and their values were kept constant when simultaneously processing the data of the binary, ternary and quaternary LLE systems listed in Table 1. The reduction of lowpressure VLE data using the generalized method is briefly described in the Appendix A.
The initial estimate of the parameters was generated minimizing the objective function
(25)
where the elements of F_{a}(X^{m},Y^{m},q ^{(0)}) are calculated by equation (23) with the experimental values of the measured variables. The minimization procedure of Nelder and Mead (1965) was employed in this preliminary step. The additional computational effort introduced is not significant because complete minimization of equation (25) is not needed. Only a few iterations were satisfactory to generate a good initial estimate of the parameters.
^{a}Nagata and Ohta (1983)
^{b}Nagata (1985)
The global convergence of the algorithm was attained in five iterations for a stated tolerance of 0.001 in the objective function S. The estimated standard deviations generated by the algorithm are shown in Table 1. Estimated parameters, their respective uncertainties and other statistics are shown in Table 2. In general, the estimated mean standard deviations in mole fractions roughly agree with those previously presumed, which is to be expected when using models such as the UNIQUAC and also considering that only six parameters were allowed to fluctuate in the simultaneous fit. However, as can be seen in Figure 1, most of the residuals are within the presumed boundaries. A higher discrepancy is observed in the binary experiment and can be attributed to the fact that it was determined by a different method, the turbidity method, whereas the compositions of the ternary and quaternary systems were measured directly by the gas chromatography technique. Another global view of the results can be attained in Figure 2 where experimental and calculated phase diagrams are compared.
Figure 1: Calculated local residuals and global assumed and estimated standard deviations.
78% of the local residuals are within the presumed standard deviation.
Cyclohexane (1)/Benzene (3)/Acetonitrile (4) at 25ºC
(b)
Cyclohexane (1)/Toluene (2)/Benzene (3)/Acetonitrile (4) at 25ºC
(d)
Figure 2: Liquidliquid equilibrium diagrams. In all cases full circles (· ) denote the binary cyclohexane (1)/acetonitrile (4) and crosses (+ ) the estimated critical points. The quaternary (c and d) is represented by three planes with different toluene/benzene ratios (molar basis) denoted by R. In this case, calculated tielines were not drawn to avoid confusion.CONCLUSIONS
Except for the work of Novák et al. (1987), the published methods based on the ML principle for estimating model parameters consider only the case of two constraints. However, a more general formulation of the equations is needed to develop efficient computational procedures, for instance, applicable in the reduction of phase equilibrium data of systems involving more than two components. More particularly, in the reduction of LLE data not only binary data sets are used but also, and on a larger scale, ternary data sets. Furthermore, in recent years more quaternary experimental data sets are found in the literature. Naturally, it would be advantageous to the process engineer if all these data could be reduced simultaneously and also the experimental uncertainties taken into account. With the generalization presented in this work all these aspects could be covered, as demonstrated.
As an example, the UNIQUAC parameters were simultaneously estimated from binary, ternary and quaternary LLE data sets. Although only the UNIQUAC activity coefficient equation was considered, other activity models could be employed so that they could also be compared in order to choose among them. However, this was beyond the scope of the example that was intended only to illustrate the generalization.
In general, use of the new algorithm for the correlation of many other data sets required no initial estimate of the true values of the variables, other than the corresponding measured values. On the other hand, the starting values for the parameters were more critical. The initial estimate generated by minimizing objective function (25) was, in most cases, satisfactory. The application of the steplimiting procedure of Law and Bailey (1963) was decisive. With these features the algorithm has rarely been unsuccessful.
In addition to the use of experimental data with any number of constraints and different numbers of constraints simultaneously, the generalized method presented in this work also places no restrictions on the mathematical form of the constraints. Although not exemplified in this paper, the method can also be applied, for instance, to estimate binary parameters in mixing rules of models based on equations of state. A typical practical situation of this nonbinary application is found in modeling supercritical extraction processes. Another practical example in this sense is the simultaneous reduction of VLE, LLE, and VLLE data. Application of the algorithm to all these problems is straightforward. Only procedures dealing with the constraints require changes.
ACKNOWLEDGMENT
The authors gratefully acknowledge the financial support of FAPESP  the Fundação de Amparo à Pesquisa do Estado de São Paulo  through Grant No. 93/10370.
NOMENCLATURE
A_{ij} UNIQUAC binary interaction energy parameter for the ij pair
C Number of components in the mixture
C_{k} Number of components in data set k
d Number of data sets
D , an h ´ h matrix
, ,¼ , Constraint equations
f_{j}, an (M_{k}_{}K_{k}) ´ 1 matrix
= , an N_{k}(M_{k}_{} K_{k}) ´ 1 matrix
Reference fugacity for pure liquid i
F_{ijk}ijkth element of F
F Constraint equations, a m ´ 1 matrix
F_{a}F calculated with measured values of X and Y
F_{X} Partial derivatives of F with respect to the independent variables, a m ´ h sparse matrix
F_{Y} Partial derivatives of F with respect to the dependent variables, a m ´ m sparse matrix
Fq Partial derivatives of F with respect to the parameters, a m ´ L matrix
K_{k} Number of measured variables considered to be independent in each experiment of data set k
L Number of parameters to be estimated
M_{k} Number of measured variables in each experiment of data set k
N Total number of experiments, defined by equation (1)
N_{k} Number of experiments in data set k
P Pressure
Vapor pressure of component i
Q
an h ´ 1 matrix
R Molar toluene/benzene ratio in Figure 2; Universal gas constant
R, an h ´ L matrix
s^{2} Estimated variance
S Function to be minimized
S_{a} Objective function to be minimized to generate initial estimates of the parameters, as in equations (25) and (A5)
S_{N} New estimate of function to be minimized
T Temperature
T, an L ´ L matrix
U , an L ´ 1 matrix
v_{i} Saturated liquid molar volume of component i
x_{i} Liquid phase mole fraction of component i
Measured liquid phase mole fractions in experiment j and data set k, a C_{k}´ 1 matrix
X True values of independent variables, an h ´ 1 matrix
X_{j} True values of independent variables in experiment j, a K_{k}´ 1 matrix
= , an N_{k}K_{k}´ 1 matrix
D X Changes of deviation in X, a m ´ 1 matrix
y_{i} Vapor phase mole fraction of component i
Y True values of dependent variables, a m ´ 1 matrix
Y_{j} True values of dependent variables in experiment j, an (M_{k}_{} K_{k}) ´ 1 matrix
= , an N_{k}(M_{k}_{} K_{k}) ´ 1 matrix
Greek letters
g _{i} Activity coefficient of component i in the mixture
d Inverse of the variances for dependent variables, a m ´ m diagonal matrix
h Total number of independent variables, defined by equation (3)
q Model parameters to be estimated, an L ´ 1 matrix
Dq Changes of deviation in q , an L ´ 1 matrix
l Inverse of the variances for independent variables, an h ´ h diagonal matrix
m Total number of dependent variables, defined by equation (4)
n Number of degrees of freedom
s Standard deviation
S Variancecovariance matrix of parameters, an L ´ L matrix
f_{i} Vapor phase fugacity coefficient of component i
Pure saturated vapor fugacity coefficient of component i
Subscripts
1, 2, 3, ... Components; experiments; data sets
i Component i
j Experiment j
k Data set k
P Pressure
T Temperature
x Liquid phase mole fraction
y Vapor phase mole fraction
Superscripts
0 Initial estimate
I, II Liquid phases in equilibrium
m Measured variable
r Previous iteration number
T Transpose of the matrix
REFERENCES
Abrams, D.S. and Prausnitz, J.M., Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J., 21, 116128 (1975).
Anderson, T.F.; Abrams, D.S. and Grens II, E.A., Evaluation of Parameters for Nonlinear Thermodynamic Models. AIChE J., 24(1), 2029 (1978).
Britt, H.I. and Luecke, R.H., The Estimation of Parameters in Nonlinear, Implicit Models. Technometrics, 15(2), 233247 (1973).
Englezos, P.; Kalogerakis, N. and Bishnoi, P.R., Simultaneous Regression of Binary VLE and VLLE Data. Fluid Phase Equilib., 61, 115 (1990a).
Englezos, P.; Kalogerakis, N.; Trebble, M.A. and Bishnoi, P.R., Estimation of Multiple Binary Interaction Parameters in Equations of State Using VLE Data. Application to the TrebbleBishnoi Equation of State. Fluid Phase Equilib., 58, 117132 (1990b).
Gmehling, J.; Onken, U. and Arlt, W., VaporLiquid Equilibrium Data Collection. Vol. 1. Part 6a. DECHEMA. Frankfurt/Main (1980a).
Gmehling, J.; Onken, U. and Arlt, W., VaporLiquid Equilibrium Data Collection. Vol. 1, Part 7. DECHEMA. Frankfurt/Main (1980b).
Law, V.J. and Bailey, R.V., A Method for the Determination of Approximate System Transfer Functions. Chem. Eng. Sci., 18, 189202 (1963).
Nagata, I., Quaternary LiquidLiquid Equilibrium for AcetonitrileCyclohexaneTolueneBenzene. Fluid Phase Equilib., 24, 259267 (1985).
Nagata, I. and Ohta, T., LiquidLiquid Equilibria for the Systems AcetonitrileBenzeneCyclohexane, AcetonitrileTolueneCyclohexane, and MethanolEthanolCyclohexane. J. Chem. Eng. Data, 28, 256259 (1983).
Nelder, J.A. and Mead, R., A Simplex Method for Function Minimization. Computer J., 7, 308313 (1965).
Niesen, V.G. and Yesavage, V.F., Application of a Maximum Likelihood Method Using Implicit Constraints to Determine Equation of State Parameters from Binary Phase Behavior Data. Fluid Phase Equilib., 50, 249266 (1989).
Novák, J.P.; Matou, J. and Pick, J., LiquidLiquid Equilibria, Elsevier, Amsterdam (1987).
Péneloux, A.; Neau, E. and Gramajo, A., Variance Analysis Fifteen Years Ago and Now. Fluid Phase Equilib., 56, 116 (1990).
SkjoldJørgensen, S., On Statistical Principles in Reduction of Thermodynamic Data. Fluid Phase Equilib., 14, 273288 (1983).
Vo ka, P.; Novák, J.P. and Matou, J., Application of the Maximum Likelihood Method to the Parameter Evaluation in Heterogeneous Systems. Collect. Czech. Chem. Commun., 54, 28232839 (1989).
APPENDIX A  VLE Data Reduction
Owing to the large number of adjustable parameters arising in the treatment of multicomponent LLE systems, reduction of VLE data is also demanded, as discussed in the text, and this subject is presented here as a complement. In order to preserve as much as possible the physical significance of the estimated parameters, the number of adjustable parameters is usually reduced by determining them for some of the miscible pairs from binary VLE data. Further actions can also be considered by taking into account the estimated intercorrelation coefficients among parameters.
The VLE experimental observations of a C component system are completely characterized by pressure (P), temperature (T), liquid phase mole fractions (x_{1}, x_{2}, ¼ , x_{C}_{} 1) and vapor phase mole fractions (y_{1}, y_{2}, ¼ , y_{C}_{} 1) so that 2C variables are measured for each experimental point. When the gammaphi approach is chosen to describe the VLE, C equilibrium equations, given in (A1), make up the constraints which are rigorously implicit in all variables:
(A1)
In equation (A1) f _{i} is the fugacity coefficient of component i in the vapor mixture, g _{i} is the liquid phase activity coefficient of component i and is the pure liquid i reference fugacity that is calculated using equation (A2):
(A2)
where is the vapor pressure of component i, is the pure saturated vapor fugacity coefficient, v_{i} is the saturated liquid molar volume and R is the universal gas constant.
When PTxy data are taken into consideration, temperature T and C 1 liquid mole fractions of the system k are usually taken as the C independent variables, and the pressure P and C 1 vapor mole fractions are the corresponding dependent variables. On the other hand, when total pressure (PTx) data are considered, the number of measured variables is only C+ 1 (P, T and x_{i}, i = 1, 2, ¼ , C 1), and, accordingly, there is only one measured dependent variable. This poses no problem as the C constraints of equation (A1) can be easily reduced to one expression as in equation (A3)
(A3)
According to the measured variables involved, the objective function (6) is given by
(A4)
It should be noted that, for total pressure experiments, the yterm does not contribute to objective function (A4).
Good initial estimates of the parameters can be obtained by minimizing the following objective function S_{a}:
(A5)
As in the LLE application, complete minimization is not required and it can also be carried out using the Nelder and Mead method.
Separate correlation of single binary lowpressure VLE data sets are summarized in Table A1. The simplifying assumption that the following standard deviations, s _{P} = 0.1 mmHg, s _{T} = 0.05 K, s _{x} = 0.001 and s _{y} = 0.003, are the same in all data sets, was adopted. The Antoine constants used to calculate the vapor pressures and the pure component UNIQUAC parameters were used as given in the data sources. Ideal vapor phase was used and the Poynting correction was neglected. The parameters estimated in this way were kept constant in the simultaneous LLE fit as presented in the text.
^{}Fitted simultaneously.
^{a}Gmehling etal. (1980a)
^{b}Gmehling etal. (1980b)
 Abrams, D.S. and Prausnitz, J.M., Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J., 21, 116128 (1975).
 Anderson, T.F.; Abrams, D.S. and Grens II, E.A., Evaluation of Parameters for Nonlinear Thermodynamic Models. AIChE J., 24(1), 2029 (1978).
 Britt, H.I. and Luecke, R.H., The Estimation of Parameters in Nonlinear, Implicit Models. Technometrics, 15(2), 233247 (1973).
 Englezos, P.; Kalogerakis, N. and Bishnoi, P.R., Simultaneous Regression of Binary VLE and VLLE Data. Fluid Phase Equilib., 61, 115 (1990a).
 Englezos, P.; Kalogerakis, N.; Trebble, M.A. and Bishnoi, P.R., Estimation of Multiple Binary Interaction Parameters in Equations of State Using VLE Data. Application to the TrebbleBishnoi Equation of State. Fluid Phase Equilib., 58, 117132 (1990b).
 Gmehling, J.; Onken, U. and Arlt, W., VaporLiquid Equilibrium Data Collection. Vol. 1. Part 6a. DECHEMA. Frankfurt/Main (1980a).
 Gmehling, J.; Onken, U. and Arlt, W., VaporLiquid Equilibrium Data Collection. Vol. 1, Part 7. DECHEMA. Frankfurt/Main (1980b).
 Law, V.J. and Bailey, R.V., A Method for the Determination of Approximate System Transfer Functions. Chem. Eng. Sci., 18, 189202 (1963).
 Nagata, I., Quaternary LiquidLiquid Equilibrium for AcetonitrileCyclohexaneTolueneBenzene. Fluid Phase Equilib., 24, 259267 (1985).
 Nagata, I. and Ohta, T., LiquidLiquid Equilibria for the Systems AcetonitrileBenzeneCyclohexane, AcetonitrileTolueneCyclohexane, and MethanolEthanolCyclohexane. J. Chem. Eng. Data, 28, 256259 (1983).
 Nelder, J.A. and Mead, R., A Simplex Method for Function Minimization. Computer J., 7, 308313 (1965).
 Niesen, V.G. and Yesavage, V.F., Application of a Maximum Likelihood Method Using Implicit Constraints to Determine Equation of State Parameters from Binary Phase Behavior Data. Fluid Phase Equilib., 50, 249266 (1989).
 Novák, J.P.; Matou, J. and Pick, J., LiquidLiquid Equilibria, Elsevier, Amsterdam (1987).
 Péneloux, A.; Neau, E. and Gramajo, A., Variance Analysis Fifteen Years Ago and Now. Fluid Phase Equilib., 56, 116 (1990).
 SkjoldJřrgensen, S., On Statistical Principles in Reduction of Thermodynamic Data. Fluid Phase Equilib., 14, 273288 (1983).
Publication Dates

Publication in this collection
06 Oct 1998 
Date of issue
Mar 1997
History

Accepted
22 Jan 1997 
Received
14 Mar 1996