Nonlinear Dynamic Simulation

To demonstrate the superior dynamic controllability of high-conversion and low-temperature designs, the nonlinear differential equations are numerically integrated for the four different design cases. Disturbances in feed flowrate, temperature controller set-point, and overall heat-transfer coefficient are made, and the peak deviations in reactor 

I Anti reset windup (only integrate ettcr if op signal is between 0 and 1 

These dynamic results provide graphic and quantitative proof that CSTRs designed for high conversions and/or low temperature are easier to control and provide tighter temperature control than are CSTRs that are designed for low conversion and/or high temperature. 

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In the next chapter we take a quantitative look at the dynamics of these CSTR systems using primarily rigorous nonlinear dynamic simulations (time-domain analysis). However, some of the powerful linear Laplace and frequency-domain techniques will be used to gain insight into the dynamics of these systems. 

Linear analysis predicted an ultimate gain of 8.54 (Table 3.2), but the nonlinear dynamic simulations indicate that a higher gain can be used. If a gain of 10 is used, the response is stable, but performance is still worse in this 85% conversion case than in the 95% conversion case as shown in Figure 3.12. 

All of the dynamic simulations discussed in this book use pressure-driven flows. The alternative of using a flow-driven simulation is more simple, but not at all realistic of the actual situation in a real physical process. The plumbing in the real process has to be set up so that water flows downhill. Pumps, compressors, and valves must be used in the appropriate locations to make the hydraulics of the system operate. If valves are not designed with sufficient pressure drop under steady-state conditions, they may not be able to provide the required increase in flow even when wide open. So valve saturation must be included in the rigorous nonlinear dynamic simulation. It is much better to simulate a realistic system by using a pressure-driven simulation. 

Nonlinear Dynamic Simulation The nonlinear ordinary differential equations are numerically integrated in the Matlab program given in Figure 4.2. A simple Euler integration algorithm is used with a step size of 2 s. The effects of several equipment and operating parameters are explored below. 

The choice of where production rate is set (Step 4) is often a pivotal decision, but it frequently is determined externally by a business objective. This removes another degree of freedom that cannot be used. If we are free to choose the handle for production rate, then Steps 5 through 7 are the priority order. However, at Step 7 we may determine that the choice will not work in light of other plantwide control considerations, in which case we would return to Step 4 and select a different variable to set production rate. Determining the best choice at Step 4 can only be done via nonlinear dynamic simulation of disturbances with a complete control strategy. 

An effective base-level regulatory control system has been developed and tested using a rigorous, nonlinear dynamic simulation of the entire system tubular reactor, heat exchangers, and three distillation columns. 

We showed through nonlinear dynamic simulations how the process reacts to various disturbances and changes in operating conditions. We have not shown any attempts to optimize process performance, to improve the process design, or to apply any advanced control techniques (model-based, nonlinear, feedforward, valve-position, etc.). These would be the natural next steps after the base-level regulatory control system had been developed to keep the process at a stable desired operating point. 

Nonlinear dynamic simulations of plantwide systems can be performed using a variety of software packages and computer platforms. We can write our own program to integrate numerically the differential equations describing the system. We also can use one of the commercial modeling programs that are now available. 

Certainly the bio-process with heat integration considered here would be a challenging test for many of these approaches in terms of the size and scope of what must be considered, the number of potential design alternatives, and the type of control objectives and disturbances that must be considered. A typical industrial approach would be to work through systematically all of the control objectives using a nonlinear dynamic simulation of the process to assess alternatives and to analyze performance (Fig. 11). 

SchlickT, Mandziuk M, Skeel R D and Srinivas K 1998 Nonlinear resonance artifacts in molecular dynamics simulations J. Comput. Phys. 140 1-29... 

T. Schlick, M. Mandziuk, R.D. Skeel, and K. Srinivas. Nonlinear resonance artifacts in molecular dynamics simulations. J. Comp. Phys., 139 1-29, 1998. 

The first reaction filmed by X-rays was the recombination of photodisso-ciated iodine in a CCI4 solution [18, 19, 49]. As this reaction is considered a prototype chemical reaction, a considerable effort was made to study it. Experimental techniques such as linear [50-52] and nonlinear [53-55] spectroscopy were used, as well as theoretical methods such as quantum chemistry [56] and molecular dynamics simulation [57]. A fair understanding of the dissociation and recombination dynamics resulted. However, a fascinating challenge remained to film atomic motions during the reaction. This was done in the following way. 

Off-line analysis, controller design, and optimization are now performed in the area of dynamics. The largest dynamic simulation has been about 100,000 differential algebraic equations (DAEs) for analysis of control systems. Simulations formulated with process models having over 10,000 DAEs are considered frequently. Also, detailed training simulators have models with over 10,000 DAEs. On-line model predictive control (MPC) and nonlinear MPC using first-principle models are seeing a number of industrial applications, particularly in polymeric reactions and processes. At this point, systems with over 100 DAEs have been implemented for on-line dynamic optimization and control. 

In terms of nonlinear dynamical systems, the second waveguide of the junction can be considered as a system that is initially more or less far from its stable point. The global dynamics of the system is directly related to the spatial transfomation of the total field behind the plane of junction. In structure A, the initial linear mode transforms into a nonlinear mode of the waveguide with the same width and refractive index. In the structure B, the initial filed distribution corresponds to a nonlinear mode of the first waveguide it differs from nonlinear mode of the second waveguide, however. The dynamics in both cases is complicated and involves nonlinear modes as well as radiation. Global dynamics of a non-integrable system usually requires numerical simulations. For the junctions, the Cauchy problem also cannot be solved analytically. 

Research Areas Modeling, Simulation and Optimization of Chemical and Biological Processes, Clean Fuels (Hydrogen, Biodiesel and Ethanol), Fixed and Fluidized Bed Catalytic Reactors, Nonlinear Dynamics, Bifurcation and Chaos,... 

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

The Duffing Equation 14.4 seems to be a model in order to describe the nonlinear behavior of the resonant system. A better agreement between experimentally recorded and calculated phase portraits can be obtained by consideration of nonlinear effects of higher order in the dielectric properties and of nonlinear losses (e.g. [6], [7]). In order to construct the effective thermodynamic potential near the structural phase transition the phase portraits were recorded at different temperatures above and below the phase transition. The coefficients in the Duffing Equation 14.4 were derived by the fitted computer simulation. Figure 14.6 shows the effective thermodynamic potential of a TGS-crystal with the transition from a one minimum potential to a double-well potential. So the tools of the nonlinear dynamics provide a new approach to the study of structural phase transitions. 

K. Ando and S. Kato, Dielectric relaxation dynamics of water and methanol solutions associated with the ionization of /V,/V-dimcltiylanilinc theoretical analyses, J. Chem. Phys., 95 (1991) 5966-82 D. K. Phelps, M. J. Weaver and B. M. Ladanyi, Solvent dynamic effects in electron transfer molecular dynamics simulations of reactions in methanol, Chem. Phys., 176 (1993) 575-88 M. S. Skaf and B. M. Ladanyi, Molecular dynamics simulation of solvation dynamics in methanol-water mixtures, J. Phys. Chem., 100 (1996) 18258-68 D. Aheme, V. Tran and B. J. Schwartz, Nonlinear, nonpolar solvation dynamics in water the roles of elec-trostriction and solvent translation in the breakdown of linear response, J. Phys. Chem. B, 104 (2000) 5382-94. 

The simulation shown in Fig. 6 demonstrates that a close to 1 1 ratio of homo- and heterochiral species can be reproduced, even under conditions in which the thermodynamic stability of the heterochiral dimers is considered higher than that of the homo chiral ones. This effect results from the nonlinear dynamics of the reaction and from mirror-symmetry breaking. 

Figure 13.4 illustrates three aspects of the basal ganglia network. First, a reciprocal control exists between GPe and STN. Second the SNc acts not only on the striatum but also on the cortex, the STN and the GPi. Finally the location of the STN at the intersection between vertical and horizontal feedback loops is crucial. Reference [50] concludes that the BG can no longer be considered as a unidirectional linear system that transfers information based solely on a firing-rate code and must rather be seen as a highly organized network with operational characteristics that simulate a nonlinear dynamical system (Fig. 13.4). 

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