Traditionally, the binary interaction parameters such as the ka, kb, k, ki in the Trebble-Bishnoi EoS have been estimated from the regression of binary vapor-liquid equilibrium (VLE) data. 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 variables are the temperature (T), pressure (P), liquid (x) and vapor (y) phase mole fractions of one of the components. The measurements of these variables are related to the "true" but unknown values of the state variables by the equations given next [Pg.231]

If the estimated best set of interaction parameters is found to be different for each type of data then it is rather meaningless to correlate the entire database simultaneously. One may proceed, however, to find the parameter set that correlates the maximum number of data types. [Pg.257]

Given an EoS, the objective of the parameter estimation problem is to compute optimal values for the interaction parameter vector, k, in a statistically correct and computationally efficient manner. Those values are expected to enhance the correlational ability of the EoS without compromising its ability to predict the correct phase behavior. [Pg.229]

Consider each type of data separately and estimate the best set of interaction parameters by Least Squares. [Pg.257]

Given a set of N binary VLE (T-P-x-y) data and an EoS, an efficient method to estimate the EoS interaction parameters subject to the liquid phase stability criterion is accomplished by solving the following problem [Pg.237]

Experimental data are available for this system at three temperatures by Campbell et al. (1986). Interaction parameters were estimated by Englezos et al. (1993). It was found that the Trebble-Bishnoi EoS is able to represent the correct phase behavior with an accuracy that does not warrant subsequent use of ML [Pg.244]

Figure 14.10 The stability function calculated with interaction parameters from constrained LS estimation [reprinted from Computers Chemical Engineering with permission from Elsevier Science], |

Because PC and TMPC form miscible blends in all proportions at all temperatures, there are problems in determining their mutual interaction parameters but a value for B of -7.5 J cm was estimated [86]. Kim and Paul [87] examined ternary blends of PCL with PC and TMPC to provide an indirect method of estimating interaction parameters. Samples of the ternary blends of PCL (PCL- [Pg.161]

When the fit is judged to be excellent the statistically best interaction parameters can be efficiently obtained by performing implicit ML estimation. This was found to be the case with the methane-methanol and the nitrogen-ethane systems presented later in this chapter. [Pg.243]

Data at two temperatures were obtained from Zeck and Knapp (1986) for the nitrogen-ethane system. The implicit LS estimates of the binary interaction parameters are ka=0, kb=0, kc=0 and kd=0.0460. The standard deviation of kd was found to be equai to 0.0040. The vapor liquid phase equilibrium was computed and the fit was found to be excellent (Englezos et al. 1993). Subsequently, implicit ML calculations were performed and a parameter value of kd=0.0493 with a standard deviation equal to 0.0070 was computed. Figure 14.2 shows the experimental phase diagram as well as the calculated one using the implicit ML parameter estimate. [Pg.246]

Figure 14.3 Vapor-liquid equilibrium data and calculated values for the carbon dioxide-n-hexane system. Calculations were done using interaction parameters from implicit and constrained least squares (LS) estimation, x and y are the mote fractions in the liquid and vapor phase respectively [reprinted from the Canadian Journal of Chemical Engineering with permission] |

The methane-methanol binary is another system where the EoS is also capable of matching the experimental data very well and hence, use of ML estimation to obtain the statistically best estimates of the parameters is justified. Data for this system are available from Hong et al. (1987). Using these data, the binary interaction parameters were estimated and together with their standard deviations are shown in Table 14.1. The values of the parameters not shown in the table (i.e., ka, kb, kc) are zero. [Pg.246]

Numerous assessments of the rehabiUty of UNIFAC for various appHcations can be found in the Hterature. Extrapolating a confidence level for some new problem is ill-advised because accuracy is estimated by comparing UNIFAC predictions to experimental data. In some cases, the data are the same as that used to generate the UNIFAC interaction parameters in the first place. Extrapolating a confidence level for a new problem requires an assumption that the nature of the new problem is similar to that of the UNIEAC test systems previously considered. With no more than stmctural information, such an assumption may not be vaHd. [Pg.252]

Based on the above, we can develop an "adaptive" Gauss-Newton method for parameter estimation with equality constraints whereby the set of active constraints (which are all equalities) is updated at each iteration. An example is provided in Chapter 14 where we examine the estimation of binary interactions parameters in cubic equations of state subject to predicting the correct phase behavior (i.e., avoiding erroneous two-phase split predictions under certain conditions). [Pg.166]

In addition to these faciUties for supply of data in an expHcit form for direct use by the system, there also are options designed for the calculation of the parameters used by the system s point generation routines. Two obvious categories of this type can be identified and are included at the top left of Figure 5. The first of these appHes to the correlation of raw data and is most commonly appHed to the estimation of binary interaction parameters. [Pg.76]

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