Limit cycle simulated

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

To study a class of mechanisms for isothermal heterogeneous catalysis in a CSTR, Morton and Goodman (1981-1) analyzed the stability and bifurcation of simple models. The limit cycle solutions of the governing mass balance equations were shown to exist. An elementary step model with the stoichiometry of CO oxidation was shown to exhibit oscillations at suitable parameter values. By computer simulation limit cycles were obtained. 

Figure 1. Simulated limit cycles for model la with = exp...

Yegneswaran et al. (1991) used a Monte Carlo method and CTD similar to those of Namdev et al. (1991) to investigate the effects of dissolved oxygen on a culture of antibiotic producing Streptomyces clavuligerus. They found that the yield of cephamycin C was suppressed by almost 44% due to the Monte Carlo simulation as compared to constant period cycling. One limitation in the... 

A complete model for the description of plasma deposition of a-Si H should include the kinetic properties of ion, electron, and neutral fluxes towards the substrate and walls. The particle-in-cell/Monte Carlo (PIC/MC) model is known to provide a suitable way to study the electron and ion kinetics. Essentially, the method consists in the simulation of a (limited) number of computer particles, each of which represents a large number of physical particles (ions and electrons). The movement of the particles is simply calculated from Newton s laws of motion. Within the PIC method the movement of the particles and the evolution of the electric field are followed in finite time steps. In each calculation cycle, first the forces on each particle due to the electric field are determined. Then the... 

The calculated conversions presented in Table VIII used Eq. (57). They are quite remarkable. They reproduce experimental trends of lower conversion and higher peak bed temperature as the S02 content in the feed increases. Bunimovich et al. (1995) compared simulated and experimental conversion and peak bed temperature data for full-scale commercial plants and large-scale pilot plants using the model given in Table IX and the steady-state kinetic model [Eq. (57)]. Although the time-average plant performance was predicted closely, limiting cycle period predicted by the... 

As an illustration, we briefly discuss the SCC-DFTB/MM simulations of carbonic anhydrase II (CAII), which is a zinc-enzyme that catalyzes the interconversion of CO2 and HCO [86], The rate-limiting step of the catalytic cycle is a proton transfer between a zinc-bound water/hydroxide and the neutral/protonated His64 residue close to the protein/solvent interface. Since this proton transfer spans at least 8-10 A depending on the orientation of the His 64 sidechain ( in vs. out , both observed in the X-ray study [87]), the transfer is believed to be mediated by the water molecules in the active site (see Figure 7-1). To carry out meaningful simulations for the proton transfer in CAII, therefore, it is crucial to be able to describe the water structure in the active site and the sidechain flexibility of His 64 in a satisfactory manner. 

Whilst this Chapter is primarily concerned with the methods of determining the free energies of tautomeric or ionisation equilibria via computer simulation of free energy differences, many of the issues raised relate also to the determination of other molecular properties upon which behaviour of the molecule within the body may depend, such as the redox potential or the partition coefficient.6 In the next section, we shall give a brief explanation of the methods used to calculate these free energy differences -namely the use of a thermodynamic cycle in conjunction with ab initio and free energy perturbation (FEP) methods. This enables an explicit representation of the solvent environment to be used. In depth descriptions of the various simulation protocols, or the accuracy limiting factors of the simulations and methods of validation, have not been included. These are... 

Figure 8 shows a point P inside the lobe, and Figure 9 shows the simulation results when the inlet stream concentration and temperature are inside the lobe a trajectory approaches a closed curve or limit cycle and remains there. This behavior has been corroborated in an industrial environment. 

If p, u > 0 —A > 0 and —A/p > 1 the equilibrium point is unstable, and a Shilnikov orbit may appear. For the reactor, with a value of X50 > 1 and X6max x6max)M (see Figure 15), by simulation it is possible to verify the presence of a homoclinic orbit to the equilibrium point. Figure 17 shows the homoclinic orbit for the model and R, when the steady state has been reached. Note that the Shilnikov orbit appear when the coolant flow rate is constrained. If there is no limitation of the coolant flow rate, a limit cycle is obtained both in models R and R, by simulation. 

Molecular models for circadian rhythms were initially proposed [107] for circadian oscillations of the PER protein and its mRNA in Drosophila, the first organism for which detailed information on the oscillatory mechanism became available [100]. The case of circadian rhythms in Drosophila illustrates how the need to incorporate experimental advances leads to a progressive increase in the complexity of theoretical models. A first model governed by a set of five kinetic equations is shown in Fig. 3A it is based on the negative control exerted by the PER protein on the expression of the per gene [107]. Numerical simulations show that for appropriate parameter values, the steady state becomes unstable and limit cycle oscillations appear (Fig. 1). 

Only deterministic models for cellular rhythms have been discussed so far. Do such models remain valid when the numbers of molecules involved are small, as may occur in cellular conditions Barkai and Leibler [127] stressed that in the presence of small amounts of mRNA or protein molecules, the effect of molecular noise on circadian rhythms may become significant and may compromise the emergence of coherent periodic oscillations. The way to assess the influence of molecular noise on circadian rhythms is to resort to stochastic simulations [127-129]. Stochastic simulations of the models schematized in Fig. 3A,B show that the dynamic behavior predicted by the corresponding deterministic equations remains valid as long as the maximum numbers of mRNA and protein molecules involved in the circadian clock mechanism are of the order of a few tens and hundreds, respectively [128]. In the presence of molecular noise, the trajectory in the phase space transforms into a cloud of points surrounding the deterministic limit cycle. 

Simulation of the process with analytical models can be used to evaluate the effects of changes in process parameters, providing the limiting assumptions of the model are noted [16]. These parametric studies can then be used to select critical experiments for selecting a cure cycle or to establish rules for process-cycle development [17]. If the simulation is true enough to the actual behavior of the material and processing vessel and provides the necessary predictions of material quality, it can even be used to select a cure cycle [15,18]. 

The simulation continues until the temperature requirement exceeds a specified end-of-cycle limit. 

Since the models are so simple, a whole test cycle can be simulated in a very short time. The disadvantage of such a model is that it is not based on the underlying chemical processes and so the range of applicability is likely to be limited. 

To study the influence of N02 on SCR efficiency, ESC and ETC test cycles with 0% and 50% N02/NOx ratio in front of SCR catalyst were simulated. However, during transient test cycle operating conditions, a constant supply of optimum 50% N02/NOx ratio will be difficult to achieve. NO to N02 conversion over a DOC is dependent on exhaust temperature, space velocity and exhaust composition. Because of transient operating conditions, it becomes furthermore a function of time. Exploiting the fast SCR reaction is thus limited by the realistically achievable N02/NOx ratio in front of the SCR catalyst. To investigate this, ESC and ETC test cycles were also simulated for a combined system of DOC and SCR, where the N02/NOx ratio in front of the SCR is defined by the N02 conversion over the DOC (Fig. 52). 

To find out routes of the transition from limit cycles to this type of irregular pulsation, we performed hydrodynamic simulations for a series of models the luminosity log(L/L ) = 3.505, and Te = 5300 Kwith a narrow range of the mass, 1.4 M M 1.5 M by using the hydrodynamic code, TGRID (Simon and Aikawa, 1986). 

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