Modeling steady-state

Mohindra, S., and Clark, P. A., A distributed fault diagnosis method based on digraph models Steady-state analysis, Comput. Chem. Eng. 17, 193 (1993). 

In connection with practical situations where CO oxidation is important, we must also consider the perennial question of how to connect the low pressure results onto those at high pressure. Qualitatively this has been done for the CO oxidation reaction but it would still be worthwhile to attempt a numerical prediction of high pressure results based on low-pressure rate parameters. A very nice paper modeling steady-state CO oxidation data over a supported Pt catalyst at CO and O2 pressures of several torr has very recently appeared (.25). Extension of this work to other systems in warranted and, even though unresolved questions continue to exist, every indication is that the high and low pressure data can be reliably modeled with the same rate parameters if no adsorption - desorption equilibria are assumed. 

Kee, R. J., Grear, J. F., Smooke, M. D., and Miller, J. A., A Fortran program for modeling steady state laminar one-dimensional premixed flames, Sandia Report (1985). 

Fig. 10. A model of the mechanism of activation and turnover in P. aeruginosa CCP after Foote et al. (.62). In order to successfully model steady-state turnover, Foote et al. used the following experimentally derived rate constants h = 1 = 0.2...

Application of the theory of Markov chains to model steady-state parameters of complex circuits... 

G. Veser, J. Frauhammer, Modeling steady state and ignition during catalytic methane oxidation in a monolith reactor. Chem. Eng. Sci.,... 

Figure 22. (a) Calculated cyclic voltammograms and stationary I/U curve for the formic acid oxidation model [Eq. (15)]. The anodic and cathodic scans are indicated by arrows. The dashed line shows the portion of the stationary state curve that corresponds to unstable steady states. The triangle at U = 0.6 V marks the location of the Hopf bifurcation, (b) Calculated one-parameter bifurcation diagram of the formic acid model steady-state coverage of OH, 6 oH. vs. applied voltage U. Solid line indicates a stable steady state (SS), the dashed line an unstable steady state, and the dot-dashed line shows the maximum amplitude of stable oscillations. (Reprinted with permission from P. Strasser, M. Eiswirth and G. Ertl, J. Chem. Phys. 107, 991-1003, 1997. Copyright 1997 American Institute of Physics.)... 

Equation 6.14 provides a formal connection between creep crack growth and the kinetics of creep deformation in that the steady-state crack growth rates can be predicted from the data on uniaxial creep deformation. Such a comparison was made by Yin et al. [3] and is reconstructed here to correct for the previously described discrepancies in the location of the crack-tip coordinates (from dr/2 to dr) with respect to the microstructural features, and in the fracture and crack growth models. Steady-state creep deformation and crack growth rate data on an AlSl 4340 steel (tempered at 477 K), obtained by Landes and Wei [2] at 297, 353, and 413 K, were used. (AU of these temperatures were below the homologous temperature of about 450 K.) The sensitivity of the model to ys, N, and cr is assessed. 

In 1924 Lewis and Whitman 1 suggested that the film theory model could be applied to both die gas and liquid phases during gas absorption. This two-film theory has hed extensive use in modeling steady-state transport between two phases. Transferor species A occurring between a gas phase and liquid phase, each of which may be in turbulent flow, can be described by the individual rate expressions bstween the bulk of each phase and the interface. 

Figure 15. Experimental and model steady-state pressure drop versus liquid velocity. Gas-phase velocity is held constant. Symbols are experimental data solid lines are model predictions. Error bars are shown.

Tdie most common approach to modeling steady-state hot flow stress, o, is the Sellars-Tegart law (Ref 53), combining the dependence on temperature, T, and strain rate, e, via the Zener-Hollomon parameter, Z ... 

Approximate models Steady-state distributions and partuneters ace known for many stochastic processes e.g., queueing, inventory, Markov chains. These results ctm be used to approximate the simulation model. For example, a service system can be approximated by a Markovian queue to determine the expected number of customers in the system. This value can be used to set the initial number of customers in the system for the simulation, rather them using the (convenient) initial condition of an empty system. Chapter 81 of the Handbook is a good source of approximations. Even cruder approximations, such as replacing a random quantity by its expectation, can also be used. 

A number of viscoelastic (i.e., rheological) models have been proposed to model steady-state creep in soils. A selection of fom of these models is presented in Figure 8.47. A model incorporating spring constants, and E2 a slider element of resistance x and a dash-pot with viscosity, v, was proposed by Murayama and Shibata (1964), which is shown in Figure 8.47a. The time-dependent deformation is controlled by the slider element Xq. Deformation will only occur for applied stresses in excess of Xq. 

In the chemical master equation, the steady-state probability distribution of the equUihrium steady state is a Poisson distribution. For Schlogl s model steady-state probability distributions become... 

Table 6.5 Idealized flow and axial dispersion models (steady state).

Sklyar, O., Trauble, M., Zhao, C.A. and Wittstock, G. (2006) Modeling steady-state experiments with a scanning electrochemical microscope involving several independent diffusing species using the boundary element method. Journal of Physical Chemistry B, 110, 15869-15877. 

Alternatively, we could attempt to obtain the feedforward gains (AO empirically by plant testing, providing that we can introduce a disturbance into feed enthalpy. We may be able to determine K from analysis of historical data but if these were collected while tray temperature (or some other composition) control was in service then it will only be possible to model steady state behaviour. Similarly we could identify K from steady state simulation. Dynamic compensation would then have to be tuned by trial and error. 

PBPK approaches are classified into three categories [82] quasiequilibiium models, steady-state models, and dynamic models. The classification of these models is based upon their dependence on spatial and temporal variables. The quasiequi-librium models, which are independent of spatial and temporal variables, include the p/f-partition hypothesis and absorption potential concept. The steady-state models are limited to prediction of the extent but not the rate of oral drag absorption. The dynamic models consider spatial and temporal variables and can predict both the rate and extent of oral drag absorption. The dynamic models include dispersion... 

All of these types of chemical models - steady-state, pseudo-time-dependent, and fully time-dependent - are themselves dependent on rate coefficients which are often highly uncertain and which need to be determined b experimental and/or theoretical techniques before the models can be put on a sound basis. Tlie approach in this review will be to... 

Elimination Kinetics. Determination of the rate of elimination is a useful exercise that can be used to calculate the half-life ty and determine the persistence of PAHs in tissue, in addition to modeling steady-state tissue burdens. The balance between uptake and elimination will determine the bioconcentration or bioaccumulation factor, which can be compared to an expected value (for example, see BCFpred in Appendix). Computation of half-life is also a good benchmark for interspecific comparison of the per-... 

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