True

To use Equation (13), it is first necessary to calculate the true fugacity coefficient (ft. This calculation is achieved by utilizing the Lewis fugacity rule... 

Next, and more difficult, is the calculation of the true mole fraction This calculation is achieved by simultaneous... 

By contrast, in the system propionic acid d) - methyl isobutyl ketone (2), (fi and are very much different when y 1, Propionic acid has a strong tendency to dimerize with itself and only a weak tendency to dimerize with ketone also,the ketone has only a weak tendency to dimerize with itself. At acid-rich compositions, therefore, many acid molecules have dimerized but most ketone molecules are monomers. Acid-acid dimerization lowers the fugacity of acid and thus is well below unity. Because of acid-acid dimerization, the true mole fraction of ketone is signi-... 

The estimated true values must satisfy the appropriate equilibrium constraints. For points 1 through L, there are two constraints given by Equation (2-4) one each for components 1 and 2. For points L+1 through M the same equilibrium relations apply however, now they apply to components 2 and 3. The constraints for the tie-line points, M+1 through N, are given by Equation (2-6), applied to each of the three components. 

There are two types of measurement errors, systematic and random. The former are due to an inherent bias in the measurement procedure, resulting in a consistent deviation of the experimental measurement from its true value. An experimenter s skill and experience provide the only means of consistently detecting and avoiding systematic errors. By contrast, random or statistical errors are assumed to result from a large number of small disturbances. Such errors tend to have simple distributions subject to statistical characterization. 

For each experiment, the true values of the measured variables are related by one or more constraints. Because the number of data points exceeds the number of parameters to be estimated, all constraint equations are not exactly satisfied for all experimental measurements. Exact agreement between theory and experiment is not achieved due to random and systematic errors in the data and to "lack of fit" of the model to the data. Optimum parameters and true values corresponding to the experimental measurements must be found by satisfaction of an appropriate statistical criterion. 

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

The algorithm employed in the estimation process linearizes the constraint equations at each iterative step at current estimates of the true values for the variables and parameters. 

Table 1 gives the measured data, estimates of the true values corresponding to the measurements, and deviations of the measured values from model predictions. Figure 1 shows the phase diagram corresponding to these parameters, together with the measured data. 

Measured Variables and Estimates of Their True Values for Acetone(1)/Methanol(2) System (Othmer, 1928)... 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

In the highly nonlinear equilibrium situations characteristic of liquid separations, the use of priori initial estimates of phase compositions that are not very close to the true compositions of these phases can lead to divergence of iterative computations or to spurious convergence upon feed composition. 

However, when carboxylic acids are present in a mixture, fugacity coefficients must be calculated using the chemical theory. Chemical theory leads to a fugacity coefficient dependent on true equilibrium concentrations, as shown by Equation (3-13). ... 

The Lewis fugacity rule is used for calculating the fugacity coefficients of the true species, and (2) the second virial co-... 

Convergence of this iteration is influenced by initial estimates for the true mole fractions, zThe following rules have been found to lead to rapid convergence in all cases. 

These initial estimates are used in the iteration function. Equation (37), to obtain values of the 2 s that do not change significantly from one iteration to the next. These true mole fractions, with Equation (3-13), yield the desired fugacity... 

There are two enthalpy corrections for strongly associating vapors. The dominant term is due to the combined enthalpies of reaction of the stoichiometric species, Ah, to form the true... 

The total enthalpy correction due to chemical reactions is the sum of all the enthalpies of dimerization for each i-j pair multiplied by the mole fraction of dimer i-j. Since this gives the enthalpy correction for one mole of true species, we multiply this quantity by the ratio of the true number of moles to the stoichiometric number of moles. This gives... 

This subroutine also prints all the experimentally measured points, the estimated true values corresponding to each measured point, and the deviations between experimental and calculated points. Finally, root-mean-squared deviations are printed for the P-T-x-y measurements. 

PRINTS THE results OF THE REGRESSION-OP VARRIANCE-COVARPIANCE MATRIX, CORRELAT AND THE VARRIANCE OF THE FIT. ALSO, PP DATA, THEIR ESTIMATED TRUE VALUES, AND ALL NN DATA POINTS. FINALLY, THF ROOT-DEVIATIONS ARE GIVEN FOR EACH 0 = THE... 

Y - VECTOP OF TRUE VALUES CF FIRST DEPENDENT VARIABLE... 

C 2 - VECTOR OF TRUE VALUES OF SECOND DEPENDENT VARIAL0E C... 

INITIAL ESTIMATES OF TRUE VALUES OF THE VARIABLES ARE SET EQUAL TO THE MEASURED VALUES. 

CALCULATES THE SUM OF THE SQUARES OF THE DEVIATIONS OF ALL MEASURED VARIABLES FROM THEIR TRUE VALUES FOR REGRES. 

APRAV OF TRUE VALUES CORRESPCNDING TC THE MEASURED VALUES IN THE XM ARRAY... 

VECTOR OF TRUE VALUES CORRESPCNDING TO THE MEASURED VALUES IN TH ZM VECTOR... 

CALCULATES THE CONSTRAINT FUNCTIONS FOP BINARY VAaOR-LIOUIO EQUILIBRIUM OATA PEOUCTION. THE CCNSTRAINT FUNCTIONS RELATING THE TRUE VALUES QP THE MEASURED VARIABLES ARE (1) PRESS = F(LI0 COMP,TEMP,PARAMETERS)... 

TRUE VAPOR-PHASE MOLE FRACTION. CALCULATED WHEN THE CHEMICAL THEORY IS USED. 

USE MULLER.S METK)0 TO SOLVE NONLINEAR EQUATIONS FOP THE TRUE VAPOR-PHASE COMPOSITION. 

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