Transient modeling

In the work presented here, a slightly different two-parameter transient model has been used. Instead of specifying a center frequency b and the bandwidth parameter a of the amplitude function A(t) = 6 , a simple band pass signal with lower and upper cut off frequencies and fup was employed. This implicitly defined a center frequency / and amplitude function A t). An example of a transient prototype both in the time and frequency domain is found in Figure 1. 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Classification Process simulation refers to the activity in which mathematical models of chemical processes and refineries are modeled with equations, usually on the computer. The usual distinction must be made between steady-state models and transient models, following the ideas presented in the introduction to this sec tion. In a chemical process, of course, the process is nearly always in a transient mode, at some level of precision, but when the time-dependent fluctuations are below some value, a steady-state model can be formulated. This subsection presents briefly the ideas behind steady-state process simulation (also called flowsheeting), which are embodied in commercial codes. The transient simulations are important for designing startup of plants and are especially useful for the operating of chemical plants. 

This equation provides us with a model-based rule as to how the manipulated variable should be adjusted when we either change the set point or face with a change in the load variable. Eq. (10-5) is the basis of what we call dynamic feedforward control because (10-4) has to be derived from a time-domain differential equation (a transient model). 3... 

Lichtner, P. C., E. H. Oelkers and H. C. Helgeson, 1986, Interdiffusion with multiple precipitation/dissolution reactions transient model and the steady-state limit. Geochimica et Cosmochimica Acta 50, 1951-1966. 

In this paper we will first describe a fast-response infrared reactor system which is capable of operating at high temperatures and pressures. We will discuss the reactor cell, the feed system which allows concentration step changes or cycling, and the modifications necessary for converting a commercial infrared spectrophotometer to a high-speed instrument. This modified infrared spectroscopic reactor system was then used to study the dynamics of CO adsorption and desorption over a Pt-alumina catalyst at 723 K (450°C). The measured step responses were analyzed using a transient model which accounts for the kinetics of CO adsorption and desorption, extra- and intrapellet diffusion resistances, surface accumulation of CO, and the dynamics of the infrared cell. Finally, we will briefly discuss some of the transient response (i.e., step and cycled) characteristics of the catalyst under reaction conditions (i.e.,... 

For porous catalyst pellets with practical loadings, this quantity is typically much larger than the pellet void fraction e, indicating that the dynamic behavior of supported catalysts il dominated by the relaxation of surface phenomena (e.g., 35, 36). This implies that a quasi-static approximation for Equation (1) (i.e., e = 0) can often be safely invoked in the transient modeling of porous catalyst pellets. The calculations showed that the quasi-static approximation is indeed valid in our case the model predicted virtually the same step responses, even when the value of tp was reduced by a factor of 10. 

Other synonyms for steady state are time-invariant, static, or stationary. These terms refer to a process in which the values of the dependent variables remain constant with respect to time. Unsteady state processes are also called nonsteady state, transient, or dynamic and represent the situation when the process-dependent variables change with time. A typical example of an unsteady state process is the operation of a batch distillation column, which would exhibit a time-varying product composition. A transient model reduces to a steady state model when d/dt = 0. Most optimization problems treated in this book are based on steady state models. Optimization problems involving dynamic models usually pertain to optimal control or real-time optimization problems (see Chapter 16)... 

Corwin, D., Rhoades, J., Simunek, J. Leaching requirement for soil salinity control Steady-state versus transient models. Agric. Water Manag. 90(3), 165-180 (2007)... 

The other effect considered in this section deals with transients in a single fuel cell. The transient models examine step changes in potential and related phenomena (e.g., gas flow rates, water production, and current density). Hence, they are aimed at examining how a fuel-cell system handles different load requirements, which may occur during automotive operation or start up and shut down. They are not trying to model slow degradation processes that lead to failure or the transients associated with impedance experiments (i.e., potential or current oscillations). These types of models are discussed in section 7. 

Figure 4.47 Simulated turn-on of SG1524 buck regulator using transient model.

Figure 4.50 Average versus transient model step load response of inductor current.

Figure 4.53 SIMetrix results transient model, step load response, inductor current...

The SPICE transient model for this circuit is shown in Fig. 5.2. Note the additional 10 mf2 resistor (R14) that exists in the SPICE model but... 

Integration of a time-dependent thermal-capillary model for CZ growth (150, 152) also has illuminated the idea of dynamic stability. Derby and Brown (150) first constructed a time-dependent TCM that included the transients associated with conduction in each phase, the evolution of the crystal shape in time, and the decrease in the melt level caused by the conservation of volume. However, the model idealized radiation to be to a uniform ambient. The technique for implicit numerical integration of the transient model was built around the finite-element-Newton method used for the QSSM. Linear and nonlinear stability calculations for the solutions of the QSSM (if the batchwise transient is neglected) showed that the CZ method is dynamically stable small perturbations in the system at fixed operating parameters decayed with time, and changes in the parameters caused the process to evolve to the expected new solutions of the QSSM. The stability of the CZ process has been verified experimentally, at least... 

S.H. He, M.M. Mench, One-dimensional transient model for frost heave in polymer electrolyte fuel cells. J. Electrochem. Soc. 153, A1724—A1731 (2006)... 

Nehter, P., 2-Dimensional transient model of a cascaded micro tubular Solid Oxide Fuel Cell fed with methane, Journal of Power Sources 157, 2006, 325-334. 

Shelton M., Celik I., Liese E., Tucker D., Lawson L. (2005) A transient model of a hybrid fuel cell, gas turbine test facility using simulink. In Proceedings of the ASME Turbo Expo, Reno-Tahoe, NV, 2005-GT-68467. 

The next level of transient modeling improves the spatial accuracy beyond the lumped analysis given in prior sections by breaking up the original control volume that previously spanned the entire cell into numerous additional control volumes, or nodes (see Figure 9.8). However, forpurposes of computational efficiency, it is noted that once the number of computational nodes becomes greater than 10, additional... 

Yakabe, H., Hishinuma, M. and Yasuda, I., Static and transient model analysis on expansion behavior of LaCiO under an oxygen potential gradient, Journal of The Electrochemical Society 147, 2000, 4071. 

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