Adjustable parameters estimation

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

Here, a, a 2, and fl2i are the binary adjustable parameters estimated from experimental vapor-liquid equilibrium data. The adjustable energy parameters, a 2 and 21. are independent of composition and temperature. However, when the parameters are temperature-dependent, prediction ability of the NRTL model enhances. The Wilson, NRTL, and UNIQUAC equations are readily generalized to multicomponent mixtures. 

Figure 1 Relationships among variables affecting 6-month MDI/PDI, as revealed through structural equation analyses. Covariate-adjusted parameter estimates (and standardized regression coefficients) are indicated. All relationships significant at p = 0.05 or less, one-tailed tests. After Dietrich et al (1986)...

LEAD EXPOSURE AND EARLY DEVELOPMENT IN URBAN INFANTS Table 1 Crude and adjusted parameter estimates for cord blood lead category... 

While many methods for parameter estimation have been proposed, experience has shown some to be more effective than others. Since most phenomenological models are nonlinear in their adjustable parameters, the best estimates of these parameters can be obtained from a formalized method which properly treats the statistical behavior of the errors associated with all experimental observations. For reliable process-design calculations, we require not only estimates of the parameters but also a measure of the errors in the parameters and an indication of the accuracy of the data. 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

Substitution of Equations (2) and (3) into the equilibrium relations dictated by Equation (2-l)[Pg.99]

An apparent systematic error may be due to an erroneous value of one or both of the pure-component vapor pressures as discussed by several authors (Van Ness et al., 1973 Fabries and Renon, 1975 Abbott and Van Ness, 1977). In some cases, highly inaccurate estimates of binary parameters may occur. Fabries and Renon recommend that when no pure-component vapor-pressure data are given, or if the given values appear to be of doubtful validity, then the unknown vapor pressure should be included as one of the adjustable parameters. If, after making these corrections, the residuals again display a nonrandom pattern, then it is likely that there is systematic error present in the measurements. ... 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

SemiempiricalRelationships. Exact thermodynamic relationships can be approximated, and the unknown parameters then adjusted or estimated empirically. The virial equation of state, tmncated after the second term, is an example of such a correlation (3). 

Equation (8-42) can be used in the FF calculation, assuming one knows the physical properties Cl and H. Of course, it is probable that the model will contain errors (e.g., unmeasured heat losses, incorrect Cl or H). Therefore, K can be designated as an adjustable parameter that can be timed. The use of aphysical model for FF control is desirable since it provides a physical basis for the control law and gives an a priori estimate of what the timing parameters are. Note that such a model could be nonlinear [e.g., in Eq. (8-42), F and T t. re multiplied]. 

The first two examples show that the interaction of the model parameters and database parameters can lead to inaccurate estimates of the model parameters. Any use of the model outside the operating conditions (temperature, pressures, compositions, etc.) upon which the estimates are based will lead to errors in the extrapolation. These model parameters are effec tively no more than adjustable parameters such as those obtained in linear regression analysis. More comphcated models mav have more subtle interactions. Despite the parameter ties to theoiy, tliey embody not only the uncertainties in the plant data but also the uncertainties in the database. 

Overview Reconciliation adjusts the measurements to close constraints subject to their uncertainty. The numerical methods for reconciliation are based on the restriction that the measurements are only subject to random errors. Since all measurements have some unknown bias, this restriction is violated. The resultant adjusted measurements propagate these biases. Since troubleshooting, model development, ana parameter estimation will ultimately be based on these adjusted measurements, the biases will be incorporated into the conclusions, models, and parameter estimates. This potentially leads to errors in operation, control, and design. 

Parameter Estimation Relational and physical models require adjustable parameters to match the predicted output (e.g., distillate composition, tower profiles, and reactor conversions) to the operating specifications (e.g., distillation material and energy balance) and the unit input, feed compositions, conditions, and flows. The physical-model adjustable parameters bear a loose tie to theory with the limitations discussed in previous sections. The relational models have no tie to theory or the internal equipment processes. The purpose of this interpretation procedure is to develop estimates for these parameters. It is these parameters hnked with the model that provide a mathematical representation of the unit that can be used in fault detection, control, and design. 

Gro.s.s-error-detection methods detect errors when they are not pre.sent and fail to detect the gro.s.s errors when they are. Couphng the aforementioned difficulties of reconciliation with the hmitations of gross-error-detection methods, it is hkely that the adjusted measurements contain unrecognized gross error, further weakening the foundation of the parameter estimation. 

U.se additional mea.surement. sets that were not included in the development of the parameter e.stimate.s to te.st their accuracy. A certain subset of the raw or adjusted measurements is used to adjust the parameter estimate. Once the optimal values are attained, the model is used to predict values to compare against other measurement sets or subsets. These additional measurements provide an independent check on the parameter estimates and the model vahdity. 

When estimates of k°, k, k", Ky, and K2 have been obtained, a calculated pH-rate curve is developed with Eq. (6-80). If the experimental points follow closely the calculated curve, it may be concluded that the data are consistent with the assumed rate equation. The constants may be considered adjustable parameters that are modified to achieve the best possible fit, and one approach is to use these initial parameter estimates in an iterative nonlinear regression program. The dissociation constants K and K2 derived from kinetic data should be in reasonable agreement with the dissociation constants obtained (under the same experimental conditions) by other means. 

Based on therapeutic drag monitoring ( TDM) and the individual parameter estimates, every drag can be made applicable to every patient by individual dose adjustment. 

The scope of this book deals primarily with the parameter estimation problem. Our focus will be on the estimation of adjustable parameters in nonlinear models described by algebraic or ordinary differential equations. The models describe processes and thus explain the behavior of the observed data. It is assumed that the structure of the model is known. The best parameters are estimated in order to be used in the model for predictive purposes at other conditions where the model is called to describe process behavior. 

Activity coefficient models offer an alternative approach to equations of state for the calculation of fugacities in liquid solutions (Prausnitz ct al. 1986 Tas-sios, 1993). These models are also mechanistic and contain adjustable parameters to enhance their correlational ability. The parameters are estimated by matching the thermodynamic model to available equilibrium data. In this chapter, vve consider the estimation of parameters in activity coefficient models for electrolyte and non-electrolyte solutions. 

The simulations can be made to reproduce the initial ratio of fits in Equation 9.27 using the measured 7j (ps) and fitting the distance r (nm), which is the only adjustable parameter. For canthaxan-thin and BI the experimental fits and the integrated values showed the best match in a very narrow temperature range ( 10 K) in the vicinity of the maximum enhancement in the relaxation rate. The distances obtained from the curve fits were similar to those determined from l/rM - 1/7 M0 difference, namely, 13.0 2.0 A for canthaxanthin and 10.0 2.0 A for BI. It was found for canthaxanthin, which shows no prominent peak in the relaxation rate, that the distance does not depend on l/rM -l/rM0. Using the ratio of curve fits, we can estimate the value of r for canthaxanthin as 9. 0+3.0 A in TiMCM-41 in the temperature range of 110-130K. 

---