Numerical Simulation Results Part

In the examples presented here, unless otherwise specified, the system s parameter values are those listed in Table 6.2. 

For the parameter values and the initial conditions selected, aU simulation results satisfy v — 1 and N 0 conditions. As a result, the simplified averaged system equation given by (6.40) or (6.43) is used. Computationally, it is much more efficient to use (6.40) instead of the infinite sum of (6.43). 

It must be noted that, the steady-sUding vibration amplitude in Figs.6.9 and 6.10 are still predicted very accurately by the averaged equation for the parameter values given in Table 6.2. 

Hopf bifurcation of the original system [69]. The unstable branch, shown by the dotted line, determines the domain of attraction of the trivial or steady-shding equilibrium point. As discussed by Remark 6.6, for c Cst the steady-shding state is stable and no limit cycles exist. 

---

The analytical models developed in this part of the study describe the performance of the basic system and allow one to predict the output signal produced by the system when its operational parameters are known. Unlike previous work [76-86], these models explicitly take into account the operational mode of the system (i.e., the reactor type in which the reactions involved take place). This approach was taken in order not only to use these analytical models for numerical simulations, but also to allow us to interpret the experimental results obtained using real systems (Section 4.3) and to assess the validity of the analytical models employed. The models developed are based on mass balances of the components involved and on the characteristics related to the particular reactor used. Unless otherwise indicated, the simulations described below were carried out using these types of input signals with variations of the parameters defined above. 

The basic system was designed to operate as an information-processing unit. As such, the output signal should differ from the input signal in at least one property type, cycle time, or amplitude. As part of this research, many numerical simulations have been carried out. The results presented here are only a fraction of what was done. They were chosen as being representative of the abilities of the systems considered to perform informationprocessing tasks, and also to reveal the main parameters that affect the system achievements. 

As a specific example to study the characteristics of the controller, the problem involving four modes of longitudinal oscillations is considered herein. The natural radian frequency of the fundamental mode, normalized with respect to 7ra/L, is taken to be unity. The nominal linear parameters Dni and Eni in Eq. (22.12) are taken from [1], representing a typical situation encountered in several practical combustion chambers. An integrated research project comprising laser-based experimental diagnostics and comprehensive numerical simulation is currently conducted to provide direct insight into the combustion dynamics in a laboratory dump combustor [27]. Included as part of the results are the system and actuator parameters under feedback actions, which can... 

The most important difference between wall excitation and free stream excitation is the direction of propagation of the disturbance field with respect to the local mean flow. For low amplitude wall excitation, disturbances propagate downstream only for unseparated flows. In contrast, when the shear layer is excited by train of vortices in the freestream, then a part of the disturbance held travels upstream- in addition to the downstream propagating wall mode. While these observations are for two-dimensional flow held, it is noted that very low-frequency wall disturbances create only three-dimensional disturbances -the Klebanoff mode. Thus, when a shear layer is excited by periodic freestream sources, the resultant flow field is a mixture of two- and three-dimensional disturbances and their mutual interaction as they travel upstream and downstream with respect to the excitation source. These elements were noted in the Direct Numerical Simulation (DNS) results of Wu et al (1999) and Jacobs Durbin (2001). In the former, the... 

Various issues in the development of a flow model and its numerical simulation have been already discussed in the previous section. It will be useful to make a few comments on the validation of the simulated results and their use in reactor engineering. More details are discussed in Part III and Part IV. Even before validation, it is necessary to carry out a systematic error analysis of the generated computer simulations. The influence of numerical issues on the predicted results and errors in integral balances must be checked to ensure that they are within the acceptable tolerances. The simulated results must be examined and analyzed using the available post-processing tools. The results must be checked to verify whether the model has captured the major qualitative features of the flow such as shear layers and trailing vortices. 

This chapter starts with an introduction to modeling of chromatographic separation processes, including discussion of different models for the column and plant peripherals. After a short explanation of numerical solution methods, the next main part is devoted to the consistent determination of the parameters for a suitable model, especially those for the isotherms. These are key issues towards achieving accurate simulation results. Methods of different complexity and experimental effort are presented that allow a variation of the desired accuracy on the one hand and the time needed on the other hand. Appropriate models are shown to simulate experimental data within the accuracy of measurement, which permits its use for further process design (Chapter 7). Finally, it is shown how this approach can be used to successfully simulate even complex chromatographic operation modes. 

The worked examples and simulation results in Chapter 14 were, for the most part, obtained with a simulation program known as ChemSep (Kooijman and Taylor, 1992). You will need to obtain this program (or something equivalent) in order to carry out the numerical exercises for this chapter. Contact R. Taylor for more information on the availability of ChemSep. 

An alternative approach is to use direct numerical simulation (DNS). Numerical results can offer considerable detail and allow access to the complete fluid flow. For example, how big is the role of disturbances generated in the atomizer Does the Rayleigh-Taylor instability matter in primary atomization Do ijewly formed droplets immediately collide with ligaments and previously formed droplets How useful are small-perturbation analyses for primary atomization The disadvantage is that DNS is capable of simulating only a small part of the spray. 

Fig. 5.17. Top Control domains in tbe K,R) parameter plane for diagonal control of the unstable periodic orbit with period r = 7.389. Large dots successful control in the numerical simulation, small grey dots no control, dotted lines analytical result for the boundary of the control domain according to Ref. 87. Bottom Leading real parts A of the Floquet spectrum for diagonal control in dependence on K (fJ = (1). 47]...

In the specific case considered in figs. 6.21 and 6.22, the mixing of 5% of periodic cells with 95% of chaotic cells results in periodic oscillations of cAMP (lower part of figures) the strange attractor associated with chaos thus transforms into a limit cycle. The periodic oscillations in the mixed suspension correspond to the behaviour predicted by the bifurcation diagram of fig. 6.3 for the effective value of parameter given by eqn (6.10). Qualitatively similar results are also obtained by numerical simulations when the two populations differ only by their intracellular supply of ATP, namely, VjV2-... 

An alternative way to obtain the spectral density is by numerical simulation. It is possible, at least in principle, to include the intramolecular modes in this case, although it is rarely done [198]. A standard approach [33-36,41] utilizes molecular dynamics (MD) trajectories to compute the classical real time correlation function of the reaction coordinate from which the spectral density is calculated by the cosine transformation [classical limit of Eq. (9.3)]. The correspondence between the quantum and the classical densities of states via J(co) is a key for the evaluation of the quantum rate constant, that is, one can use the quantum expression for /Cj2 with the classically computed J(co). This is true only for a purely harmonic system [199]. Real solvent modes are anharmonic, although the response may well be linear. The spectral density of the harmonic system is temperature independent. For real nonlinear systems, J co) can strongly depend on temperature [200]. Thus, in a classical simulation one cannot assess equilibrium quantum populations correctly, which may result in serious errors in the computed high-frequency part of the spectrum. Song and Marcus [37] compared the results of several simulations for water available at that time in the literature [34,201] with experimental data [190]. The comparison was not in favor of those simulations. In particular, they failed to predict... 

---