Parameter estimations, experimental methods

It is important to point out that to identify the parameters of the model, the experimental research made with physical laboratory models (apparatus) has previously established the experimental working methods that allow the identification of the actual process parameters. These experimental methods tend to be promoted as standardized methods and this reduces the dimension of the problem that is formulated for identifying the parameters of the model to the situations where (Pi,P2, Pi) contains one, two or a maximum of three parameters to be estimated simultaneously. 

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. 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

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. 

A sampling of appHcations of Kamlet-Taft LSERs include the following. (/) The Solvatochromic Parameters for Activity Coefficient Estimation (SPACE) method for infinite dilution activity coefficients where improved predictions over UNIEAC for a database of 1879 critically evaluated experimental data points has been claimed (263). (2) Observation of inverse linear relationship between log 1-octanol—water partition coefficient and Hquid... 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

The experimental results imply that the main reaction (eq. 1) is an equilibrium reaction and first order in nitrogen monoxide and iron chelate. The equilibrium constants at various temperatures were determined by modeling the experimental NO absorption profile using the penetration theory for mass transfer. Parameter estimation using well established numerical methods (Newton-Raphson) allowed detrxmination of the equilibrium constant (Fig. 1) as well as the ratio of the diffusion coefficients of Fe"(EDTA) andNO[3]. 

Parameter estimation. Integral reactor behavior was used for the interpretation of the experimental data, using N2O conversion levels up to 70%. The temperature dependency of the rate parameters was expressed in the Arrhenius form. The apparent rate parameters have been estimated by nonlinear least-squares methods, minimizing the sum of squares of the residual N2O conversion. Transport limitations could be neglected. 

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

Drawing straight lines through data points is a slightly arbitrary procedure. The slope of the straight line does not depend very much on this arbitrariness but the value of the intercept usually depends very much on it. Consequently, the value of the kinetic parameter related to the intercept will be estimated with the accuracy of the eyes capability of finding the best fit between experimental points and those lying on the line drawn. An objective method of parameter estimation consist in evaluation of the minimum of the function ... 

In this example the number of measured variables is less than the number of state variables. Zhu et al. (1997) minimized an unweighted sum of squares of deviations of calculated and experimental concentrations of HPA and PD. They used Marquardt s modification of the Gauss-Newton method and reported the parameter estimates shown in Table 16.24. 

The complete data series is used to calculate the temperature response, but only certain parts of the experimental data are used to calculate the error. An example of a calibration run is given in Figure 53, the final calibrated TRNSYS model run is shown in Figure 54. Using the first part of the data (with constant heat flux) an estimate of ground thermal conductivity of 2.15 was obtained. Yavatzturk s method yielded an estimate of 2.18, while the estimate obtained with the TRNSYS parameter estimation method was 2.10. 

Another way to measure resolution from experimental 2DLC data is to use a computer method to calculate the first and second moments of the zones. For highly fused zones this must be done with a parameter estimation algorithm based on some minimization criteria usually, some form of least-squares method can be utilized to fit the zone shapes with a zone model. 

Shear cell measurements offer several pieces of information that permit a better understanding of the material flow characteristics. Two parameters, the shear index, n, and the tensile strength, S, determined by fitting simplified shear cell data to Eq. (6), are reported in Table 2. Because of the experimental method, only a poor estimate of the tensile strength is obtained in many cases. The shear index estimate, however, is quite reliable based on the standard error of the estimate shown in parenthesis in Table 2. The shear index is a simple measure of the flowability of a material and is used here for comparison purposes because it is reasonably reliable [50] and easy to determine. The effective angle of internal... 

In Section 3.4, traditional methods of obtaining values of rate parameters from experimental data are described. These mostly involve identification of linear forms of the rate expressions (combinations of material balances and rate laws). Such methods are often useful for relatively easy identification of reaction order and Arrhenius parameters, but may not provide the best parameter estimates. In this section, we note methods that do not require linearization. 

In general, parameter estimation by statistical methods from experimental data in which the number of measurements exceeds the number of parameters falls into one of two categories, depending on whether the function to be fitted to the data is linear or nonlinear with respect to the parameters. A function is linear with respect to the parameters, if for, say, a doubling of the values of all the parameters, the value of the function doubles otherwise, it is nonlinear. The right side of equation 3.4-17 is nonlinear. We can put it into linear form by taking logarithms of both sides, as in equation 3.4-4 ... 

The physical properties of -hexane (see Table 3-2) that affect its transport and partitioning in the environment are water solubility of 9.5 mg/L log Kow (octanol/water partition coefficient), estimated as 3.29 Henry s law constant, 1.69 atm-m3 mol vapor pressure, 150 mm Hg at 25 °C and log Koc in the range of 2.90 to 3.61. As with many alkanes, experimental methods for the estimation of the Koc parameter are lacking, so that estimates must be made based on theoretical considerations (Montgomery 1991). 

Unavailable because experimental methods for estimation of this parameter for aldehydes are lacking in the documented literature. However, its miscibility in water suggests its adsorption to soil will be nominal (Lyman et al., 1982). 

---