Fitting the Model to Experimental Data

D5mamic strain softening data on SBR/60 phr black cpds 

GVst= G stabie- viscous mod-ulus due to chemical cross-links+stable rubber-filler interactions 

Gv n=G ns,abie viscous modulus due to unstable rubber-filler interactions [=f(strain)] 

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After a reaction model has been constructed, numerical values must be assigned to the model parameters. In the case of non-fundamental models, including empirical, pseudo-kinetic and molecular reaction models they can only be estimated by fitting the model to experimental data since the parameters have no fundamental meaning. In the case... 

It must be stated that up to now models that can describe detailed styrene polymerization including all kinds of initiation step are rare. The work of Dhib ei al. [31] is so far the most comprehensive in this respect. It is a common practice tc fit the model to experimental data under different reaction process conditions. 

Work has been done to infer differential equation models of cellular networks from time-series data. As we explained in the previous section, the general form of the differential equation model is deceit = f(Cj, c2,. cN), where J] describes how each element of the network affects the concentration rate of the network element. If the functions f are known, that is, the individual reaction and interaction mechanisms in the network are available, a wealth of techniques can be used to fit the model to experimental data and estimate the unknown parameters [Mendes 2002]. In many cases, however, the functions f are unknown, nonlinear functions. A common approach for reverse engineering ordinary differential equations is to linearize the functions f around the equilibrium [Stark, Callard, and Hubank 2003] and obtain... 

The structure of this chapter is as follows. First, the results obtained for individual surfactant solutions are presented, which are preceded, for the sake of convenience, by a summary of the main equations corresponding to the various theoretical models discussed in Chapter 2. The software used for fitting the models to experimental data will be described in Chapter 7. Then, surfactant mixtures are considered with a number of experimental and theoretical examples. A summary of the main theoretical equations for mixtures of ionic and non-ionic surfactants (the corresponding fitting software again is described in Chapter 7). Also some approximate theoretical models for the mixture of two or more surfactants is presented and compared with experimental results. 

First, an experimental drying apparatus is used. In such an apparatus, the air passes through the drying material and the air humidity, temperature, and velocity are controlled, whereas the material moisture content and, eventually, the material temperature are monitored versus time. Second, a mathematical model that takes into account the controlling mechanisms of heat and mass transfer is considered. This model includes the heat and mass transport properties as model parameters or, even more, includes the functional dependence of the relevant factors on the transport properties. Third, a regression analysis procedure is used to obtain the transport properties as model parameters by fitting the model to experimental data of material moisture content and temperature. 

The parameters of a model can be estimated by fitting the model to experimental data [182,183]. Using the model of Section 4.7.3, two regression analysis procedures can be applied [43] transport properties estimation and transport properties equations estimation. 

Also a thermodynamic model based on the coupled Equation of State model and Flory-Huggins theory for polymer solutions was developed. The model parameters such as solubility-parameter of asphaltenes, molecular weight of asphaltenes, and molar volume of asphaltenes were obtained by fitting the model to experimental data. 

Macroscale cell-level models are able to provide a great amount of insight into the operation and performance of SOFCs. With the newer mesoscale electrochemistry models, information about the conditions within the SOFC electrodes and electrolytes can even be resolved. However, due to the continuum-scale treatment of the SOFC, these models stiU rely on effective parameters, which need to be determined through smaller scale modehng or by fitting the models to experimental data. 

The parameters of a model can be estimated by fitting the model to experimental data [181,182]. Using the... 

Increasing the number of interconnected spring and dashpot elements in building viscoelastic models will increase the degrees of freedom in fitting the models to experimental data. Generalized models based on an infinite number of single elements will match the continuum mechanics approach of solid- and fluid dynamics. 

Least-squares methods are usually used for fitting a model to experimental data. They may be used for functions consisting of square sums of nonlinear functions. The well-known Gauss-Newton method often leads to instabilities in the minimization process since the steps are too large. The Marquardt algorithm [9 1 is better in this respect but it is computationally expensive. 

Fitting Dynamic Models to Experimental Data In developing empiricaTtransfer functions, it is necessary to identify model parameters from experimental data. There are a number of approaches to process identification that have been published. The simplest approach involves introducing a step test into the process and recording the response of the process, as illustrated in Fig. 8-21. The xs in the figure represent the recorded data. For purposes of illustration, the process under study will be assumed to be first-order with dead time and have the transfer function... 

Thus, Tis a linear function of the new independent variables, X, X2,. Linear regression analysis is used to fit linear models to experimental data. The case of three independent variables will be used for illustrative purposes, although there can be any number of independent variables provided the model remains linear. The dependent variable Y can be directly measured or it can be a mathematical transformation of a directly measured variable. If transformed variables are used, the fitting procedure minimizes the sum-of-squares for the differences... 

It is noteworthy that most comparisons and fits of models to experimental data deal only with cr (ft)) response. An advantage of this procedure is that cr (< ) and ef (o) = are the only ones of the eight real and imaginary parts of the four... 

Figure 5. Fit of model (-) to experimental data (o) citral hydrogenation in the monolith reactor system.

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