Fitting Dynamic Models to Experimental Data

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 

Transfer function models are linear, but chemical processes are known to exhibit nonlinear behavior. One could use the same type of optimization objective as given in Eq. (8-27) to determine parameters in nonlinear first-principles models, such as Eq. (8-3) presented earlier. Also, nonlinear empirical models, such as neural network models, have recently been proposed for process applications. The key to the use of these nonlinear empirical models is to have high- 

Siality process data, which allows the important nonlinearities to be entified. 

---

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. 

The measurement of correlation times in molten salts and ionic liquids has recently been reviewed [11] (for more recent references refer to Carper et al. [12]). We have measured the spin-lattice relaxation rates l/Tj and nuclear Overhauser factors p in temperature ranges in and outside the extreme narrowing region for the neat ionic liquid [BMIM][PFg], in order to observe the temperature dependence of the spectral density. Subsequently, the models for the description of the reorientation-al dynamics introduced in the theoretical section (Section 4.5.3) were fitted to the experimental relaxation data. The nuclei of the aliphatic chains can be assumed to relax only through the dipolar mechanism. This is in contrast to the aromatic nuclei, which can also relax to some extent through the chemical-shift anisotropy mechanism. The latter mechanism has to be taken into account to fit the models to the experimental relaxation data (cf [1] or [3] for more details). Preliminary results are shown in Figures 4.5-1 and 4.5-2, together with the curves for the fitted functions. 

GP 1] [R 1] A kinetic model for the oxidation of ammonia was coupled to a hydro-dynamic description and analysis of heat evolution [98], Via regression analysis and adjustment to experimental data, reaction parameters were derived which allow a quantitative description of reaction rates and selectivity for all products trader equilibrium conditions. The predictions of the model fit experimentally derived data well. 

In the past decade, vibronic coupling models have been used extensively and successfully to explain the short-time excited-state dynamics of small to medium-sized molecules [200-202]. In many cases, these models were used in conjunction with the MCTDH method [203-207] and the comparison to experimental data (typically electronic absorption spectra) validated both the MCTDH method and the model potentials, which were obtained by fitting high-level quantum chemistry calculations. In certain cases the ab initio-determined parameters were modified to agree with experimental results (e.g., excitation energies). The MCTDH method assumes the existence of factorizable parameterized PESs and is thus very different from AIMS. However, it does scale more favorably with system size than other numerically exact quantum... 

Dynamic models, fitting to experimental data, 20 689-691 Dynamic process control models, 20 687-688... 

Figure 3.17 Comparison between experiment (dashed curve) and calculations combining the polarizable continuum model for solute electronic structure and continuum dielectric theory of solvation dynamics in water. SRF(t) stands for S(t) in our notation. The calculations are for a cavity based on a space-filling model of Cl53, while the experiments are for C343. The two sets of theoretical results correspond to using water e(o>) from simulation (full curve) of SPC/E water and from a fit to experimental data (dash-dotted curve). (Reprinted from F. Ingrosso, A. Tani andJ. Tomasi, J. Mol. Liq., 1117, 85-92. Copyright (2005), with permission from Elsevier).

Figure 3.10 Predictions of the temporary network model [Eq. (3-24)] (lines) compared to experimental data (symbols) for start-up of uniaxial extension of Melt 1, a long-chain branched polyethylene, using a relaxation spectrum fit to linear viscoelastic data for this melt. (From Bird et al. Dynamics of Polymeric Liquids. Vol. 1 Fluid Mechanics, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

The dissolved gas concentration at the bubble/melt interface, can be related to the bubble pressure P, through Henry s law. Gogos compared his model s predictions with the experimental data produced by Spence. The model predictions fit very well with experimental data when selecting a degree of saturation close to 100% (91.9-99.6%). An alternative approach has been proposed by Kontopoulou and Vlachopoulos, who modeled the dynamics of bubble dissolution into the melt using conservation of mass and momentum.f ... 

The past and current series of molecular dynamics simulations, either those predominantly discussed here or the many others using other potential forms, have provided useful insight into molecular behavior in glass and crystalline systems. However, newer and better techniques will come forward with the advance of faster computers and more accurate interatomic potentials. While ab initio techniques will also advance in kind, the use of the more simplified models that incorporate the most important features of a system of interest will enable reasonably accurate simulations of much larger systems (0(10 -10 )) or longer time frames than currently available. Such large scale calculations will then fit much more closely to the experimental world and provide better links to experimental data and, more importantly, data interpretation. 

In the ANN approach to modelling it is not necessary to formulate an analytical description of the process i.e. presentation of an explicit form of the mapping functions q( ), r( ) or S( ). The network is fit to experimental data which characterize the process. This is accomplished by the procedure known as network training, which seeks for the best approximation of an unknown mapping function describing the dynamic system by selecting the best set of weights in the network. 

---