Time modeling

Markov model A mathematical model used in reliabihty analysis. For many safety apphcations, a discrete-state (e.g., working or failed), continuous-time model is used. The failed state may or may not be repairable. 

Fig. 19-13. Three-parameter averaging-time model fitted through the arithmetic mean and the second highest 3-hr and 24-hr SOj concentrations measured in 1972 a few miles from a coal-burning power plant. Source From Larsen (21).

Larsen (18-21) has developed averaging time models for use in analysis and interpretation of air quality data. For urban areas where concentrations for a given averaging time tend to be lognormally distributed, that is, where a plot of the log of concentration versus the cumulative frequency of occurrence on a normal frequency distribution scale is nearly linear,... 

Ziomas, I. C. et al., 1989, Design of a System for Real Time Modeling of the Dispersion of Hazardous Gas Releases from Industrial Plants, J. Loss Prevention 2 October. 

I. Residence-Time Model for Total Mass Transfer with and without Chemical... 

Fig. 11. Change in concentration with the radius of the residence-time model for con-stant gas holdup and varying contact time and reaction rate [after Gal-Or and Resnick G2,G6)].

The dimensionless variance has been used extensively, perhaps excessively, to characterize mixing. For piston flow, a = 0 and for a CSTR, a = l. Most turbulent flow systems have dimensionless variances that lie between zero and 1, and cr can then be used to fit a variety of residence time models as will be discussed in Section 15.2. The dimensionless variance is generally unsatisfactory for characterizing laminar flows where > 1 is normal in liquid systems. 

This section opens the black box in order to derive residence time models for common flow systems. The box is closed again in Section 15.3, where the predictions can be based on either models or measurements. 

This section describes residence time models that are based on a hydrodynamic description of the process. The theory is simplified but the resulting models still have substantial utility as conceptual tools and for describing some real flow systems. 

Reasoning in Time Modeling, Analysis, and Pattern Recognition of Temporal Process Trends... 

REASONING IN TIME MODELING, ANALYSIS, AND PATTERN RECOGNITION OF TEMPORAL PROCESS TRENDS... 

When experimental data are collected over time or distance there is always a chance of having autocorrelated residuals. Box et al. (1994) provide an extensive treatment of correlated disturbances in discrete time models. The structure of the disturbance term is often moving average or autoregressive models. Detection of autocorrelation in the residuals can be established either from a time series plot of the residuals versus time (or experiment number) or from a lag plot. If we can see a pattern in the residuals over time, it probably means that there is correlation between the disturbances. 

Recursive estimation methods are routinely used in many applications where process measurements become available continuously and we wish to re-estimate or better update on-line the various process or controller parameters as the data become available. Let us consider the linear discrete-time model having the general structure ... 

Fig. 3.6.8 Comparison of estimated permeability for 35 sandstones. Left the log mean relaxation time model. Right the Coats-Timur model [33].

The main challenge in short-term scheduling emanates from time domain representation, which eventually influences the number of binary variables and accuracy of the model. Contrary to continuous-time formulations, discrete-time formulations tend to be inaccurate and result in an explosive binary dimension. This justifies recent efforts in developing continuous-time models that are amenable to industrial size problems. 

The models developed to take the PIS operational philosophy into account are detailed in this chapter. The models are based on the SSN and continuous time model developed by Majozi and Zhu (2001), as such their model is presented in full. Following this the additional constraints required to take the PIS operational philosophy into account are presented, after which, the necessary changes to constraints developed by Majozi and Zhu (2001) are presented. In order to test the scheduling implications of the developed model, two solution algorithms are developed and applied to an illustrative example. The final subsection of the chapter details the use of the PIS operational philosophy as the basis of operation to design batch facilities. This model is then applied to an illustrative example. All models were solved on an Intel Core 2 CPU, T7200 2 GHz processor with 1 GB of RAM, unless specifically stated. 

The literature example results for the scenario without heat integration are summarized in Table 10.2. The discrete-time model proposed by Papageorgiou et al. 

What if the behavior of the system cannot be reliably predicted In the combustion of air-fuel mixtures in a car engine or a power station, the number of distinct chemical reactions taking place may exceed one thousand real-time modeling of such a system is not feasible. 

Relationship Between Strength and Load Rate Derived from the Eyring Reduced Time Model... 

Fig. 63 The Eyring reduced time model involves the activated site model for plastic and viscoelastic shear deformation of adjacent chains...

Fig. 64 Creep and creep failure can be modelled by the time-dependent shear deformation as described by the Eyring reduced time model...

The Eyring reduced time model provides the framework for the derivation of the creep equation of polymer fibres [10]. The creep shear strain of a domain... 

The proposed model for creep rupture based on the condition of maximum shear strain and the Eyring reduced time model explain the observed relations concerning the lifetime of aramid, polyamide 66 and polyacrylonitrile fibres. However, with increasing temperatures, in particular above 300 °C, chemical degradation of PpPTA also determines the lifetime. Furthermore, the model... 

In spectroscopy we may distinguish two types of process, adiabatic and vertical. Adiabatic excitation energies are by definition thermodynamic ones, and they are usually further defined to refer to at 0° K. In practice, at least for electronic spectroscopy, one is more likely to observe vertical processes, because of the Franck-Condon principle. The simplest principle for understandings solvation effects on vertical electronic transitions is the two-response-time model in which the solvent is assumed to have a fast response time associated with electronic polarization and a slow response time associated with translational, librational, and vibrational motions of the nuclei.92 One assumes that electronic excitation is slow compared with electronic response but fast compared with nuclear response. The latter assumption is quite reasonable, but the former is questionable since the time scale of electronic excitation is quite comparable to solvent electronic polarization (consider, e.g., the excitation of a 4.5 eV n — n carbonyl transition in a solvent whose frequency response is centered at 10 eV the corresponding time scales are 10 15 s and 2 x 10 15 s respectively). A theory that takes account of the similarity of these time scales would be very difficult, involving explicit electron correlation between the solute and the macroscopic solvent. One can, however, treat the limit where the solvent electronic response is fast compared to solute electronic transitions this is called the direct reaction field (DRF). 49,93 The accurate answer must lie somewhere between the SCRF and DRF limits 94 nevertheless one can obtain very useful results with a two-time-scale version of the more manageable SCRF limit, as illustrated by a very successful recent treatment... 

J. Li, C. J. Cramer, and D. G. Truhlar, A two-response-time model based on CM2/INDO/S2 electrostatic potentials for the dielectric polarization component of solvatochromic shifts on vertical excitation energies, Int. J. Quan. Chem. 77 264 (2000). 

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