Characteristic Time Scale Analysis

Under the quasisteady formulation of Eqs. 3.1-3.8 and 3.11, the characteristic time scales for heat conduction in the sohd substrate must be suliiciently longer than the characteristic convective, diflhisive, and chemical time scales of the reacting flow inside the channel. This ensures that at every time step the gaseous flow and chemistry equilibrate to the imposed, at every time step, solid wall temperature. Accordingly, the time step At must be longer than the characteristic convective, diflhisive and chemical time scales of the reacting flow, but short enough to accurately resolve the transient heat response of the solid. 

For the axial convection inside the microreactor, the characteristic time (g X UUju (not accounting for flow acceleration of the gas due to combustion) was estimated between 6 and 33 ms. Diflhisive transport time scales of the gas in 

In a further analysis, the same characteristic time scale analysis was performed as before, but with the inclusion of gas-phase reactions. The calculated characteristic chemical time scales are presented in Fig. 8.4. Important intermediate species of lean CH4/air combustion such as formaldehyde (CH2O) and acetylene (C2H2) now impose further restrictions on the required time step At, as they have characteristic times of the same magnitude as total oxidation products CO2 and H2O. However, a significant drop of the chemical times to below 50 ms is evident in Fig. 8.4 for all species at — 850 K. This further justifies the use of the aforementioned combination of At and Tj for all subsequent numerical simulations. It is finally emphasized that the analysis in Figs. 8.3 and 8.4 is quite strict when applied to the channel in Fig. 8.1, since as the solid starts heating above the initial temperature T x, t — 0) — 850 K, the chemical time scales shorten substantially already at 900 K, the chemical times are a factor of 2.2-3.5 shorter than the ones at 850 K shown in Figs. 8.3 and 8.4. 

Tin = 800 K, dashed bars Tin = 850 K. Dashed line time step At = 50 ms used in subsequent channel-flow simulations 

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In the following sections, the impact of microreactor operating conditions and wall material on the large variation of characteristic start-up times (see Table 8.2) will be addressed, along with a characteristic time scale analysis identifying relevant time scales. 

Here the lattice positions i and j should be adjacent and the -function assures that one of the two lattice positions is occupied and the other one is free, r/j is a characteristic time scale for a diffusion jump. The time-dependence of the average si) is calculated by approximating the higher moments (siSj) [49]. In practice the analysis is rather involved, so we do not give further details here. An important result, for example, is the correction to the Wilson-Frenkel rate (33) at high temperatures ... 

In computational science and engineering, researchers frequently speak of multiple time-scale problems. These are problems ivhere the characteristic times for events span many orders of magnitude. More often than not, the full range of desired time-scales is not experimentally accessible. The combined experimental and numerical approach must settle for analysis of a narrow ivindoiv of time-scales. For most of our purposes, the half-life of an event w ill prove to be a useful definition of characteristic time-scale... 

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. 

Separation of characteristic time scales of two-stage growth allows simplifying of dynamics analysis. Ratio of characteristic times of growth of cells of the first zx and the second z2 groups depends on parameter pi = b/p. 

Setup and principle of FCS and FCCS have been reviewed extensively previously [12,13]. The technique is based on the statistical analysis of equilibrium fluorescence fluctuations induced by, e.g., the d3mamics of fluorescent molecules in a tiny observation volume. By correlating these fluctuations with itself at a later time r, an autocorrelation curve is obtained, which can be fitted to an appropriate model function to extract the characteristic time scales of the system. The two basic parameters of a FCS autocorrelation curve are the decay time, reflecting time scales of molecules dynamics, and the amplitude, indicating the average number of particles in the detection volume. 

Classify the fixed points of the logistic equation, using linear stability analysis, and find the characteristic time scale in each case. 

Ludwig et al. (1978) proposed and analyzed an elegant model of the interaction between budworms and the forest. They simplified the problem by exploiting a separation of time scales the budworm population evolves on a fast time scale (they can increase their density fivefold in a year, so they have a characteristic time scale of months), whereas the trees grow and die on a slow time scale (they can completely replace their foliage in about 7-10 years, and their life span in the absence of budworms is 100-150 years.) Thus, as far as the budworm dynamics are concerned, the forest variables may be treated as constants. At the end of the analysis, we will allow the forest variables to drift very slowly—this drift ultimately triggers an outbreak. 

The exploration of ultrafast molecular and cluster dynamics addressed herein unveiled novel facets of the analysis and control of ultrafast processes in clusters, which prevail on the femtosecond time scale of nuclear motion. Have we reached the temporal boarders of fundamental processes in chemical physics Ultrafast molecular and cluster dynamics is not limited on the time scale of the motion of nuclei, but is currently extended to the realm of electron dynamics [321]. Characteristic time scales for electron dynamics roughly involve the period of electron motion in atomic or molecular systems, which is characterized by x 1 a.u. (of time) = 24 attoseconds. Accordingly, the time scales for molecular and cluster dynamics are reduced (again ) by about three orders of magnitude from femtosecond nuclear dynamics to attosecond electron dynamics. Novel developments in the realm of electron dynamics of molecules in molecular clusters pertain to the coupling of clusters to ultraintense laser fields (peak intensity I = lO -lO W cm [322], where intracluster fragmentation and response of a nanoplasma occurs on the time scale of 100 attoseconds to femtoseconds [323]. 

Xia et al. (1992) applied this signal analysis method to study the oscillatory behavior of light output signals in a fast fluidized bed. Figure 4-23 shows the typical power spectral density of optic output signals in the fast fluidized bed. The oscillatory behavior of the optic output signals has no characteristic time scale, or a deterministic frequency response, but forms fractal time characteristics. 

Analysis of the three characteristic time scales - Tev (residence time in discharge zone), Tp (effective reaction time in after-glow), and tvt (vibrational relaxation time) - shows that only the first is determined by ionization degree n jno, whereas the two others are connected (Tp < Tvt). The analysis determines three ionization degree regimes of the process stimulated by vibrational excitation. 

We focus on the case, most relevant in applications, that the period T of Dy t) is much longer than the characteristic time scale of the kinetics, which implies that P T 1. Analysis of (11.25), (11.26), and (11.27) leads to the conclusion that there exists a /r > 1, i.e., the uniform steady state is unstable to nonuniform perturbations if either (i) P > r or (ii) [F -I- (P ) - (P ) ] < 4r (P ). Somewhat lengthy further calculations show that, to leading order for large T, the uniform steady state will be driven unstable by diffusion if and only if either... 

The ideal characteristics of instrumentation to monitor fluidized beds can be obtained by a relatively simple dimensional analysis. Assuming that the dimensions of structures to be imaged are in the centimetre range and that characteristic flow velocities are between 1 and 10 ms one can calculate characteristic time scale as 10 and 10 of a second. Furthermore, it... 

The analysis of expression (2.110) shows that there are two characteristic time scales T = /kF is the characteristic time of the activated molecules addition to aggregates, and Z2=yp is the characteristic time of molecules activation. 

For a classical diffusion process, Fickian is often the term used to describe the kinetics of transport. In polymer-penetrant systems where the diffusion is concentration-dependent, the term Fickian warrants clarification. The result of a sorption experiment is usually presented on a normalized time scale, i.e., by plotting M,/M versus tll2/L. This is called the reduced sorption curve. The features of the Fickian sorption process, based on Crank s extensive mathematical analysis of Eq. (3) with various functional dependencies of D(c0, are discussed in detail by Crank [5], The major characteristics are... 

From the slope of the straight line, the effective mixing cell volume was calculated to be 30.1 cm, with a 50% relaxation time of about 0.08 s. Similar mixing characteristics were observed following a step decrease (i.e., CO + N ), giving an effective mixing cell volume of 31.8 cm and a 50% relaxation time of 0.09 s. Since these response times of the reactor are not much faster than the time scale of the adsorption process (a halfscale relaxation time of about 0.2 s), the transients of the reactor cell were included in our analysis. For our simulations, the mixing cell volume was taken to be 31 cm. ... 

HasweU and Barclay [3] have described a microwave system coupled to an atomic absorption detection system for the analysis of sludges and soils. A major constraint at the present time is that the preferred operation of these types of systems is for sample matrices to be closely matched. A widely varying sample, which exhibits different heating characteristics, wiU either show up as an invaHd result or the time required to cope with this procedure for aU the samples wiU greatly extend the on-Hne analyses time scales. As more of these instrumental systems become Hnked to laboratory information management systems, it wiU become feasible to interact between the control database and the instrumentation so that each sample is treated in an appropriate manner and the optimum time frame is selected for each sample type. When new samples are analysed, the steps could be monitored so that the required time scales are obtained and then stored for future reference. 

We have established that the volume change kinetics of responsive gels are usually diffusion-controlled processes. Even when the diffusion analysis failed, the rates were comparable to or slower than a classical diffusive process. The implications of this for practical applications are quite negative, since diffusive processes are quite slow. A gel slab 1 mm thick with a diffusion coefficient of 10-7 cm2/s will take over an hour to reach 50% of equilibrium and more than six hours to reach 90% of equilibrium in response to a stimulus. This is far too slow for almost all potential applications of these materials. Since diffusion times scale with the square of dimension, decreasing the characteristic dimension of a sample will increase the rates dramatically. Thus if an application can make use of submillimeter size gels, millisecond response times become possible. Unfortunately, it may not always practical to use gels of such small dimension. 

Thermodynamics only identify whether a particular reaction mixture has a tendency to form products, but do not indicate whether that tendency will ever occur in a biologically appropriate time scale. To have a real idea if that reaction occurs, it would be necessary to know the rate of the chemical reactions that is, making a kinetic analysis. For a given reaction 13, the reaction rate (v) is proportional to the molar concentration of the reactants (A, B) raised to a simple power (a, (1). These values are called partial orders with respect to each of the species participating in the reaction. The rate constant (k) is essentially defined by the thermodynamic characteristics of the species under reaction and... 

Hereafter we put /ig = 1. Below we express our results in terms of the statistical properties (correlators) of the environment s noise, X(t). Depending on the physical situation at hand, one can choose to model the environment via a bath of harmonic oscillators [6, 3]. In this case the generalized coordinate of the reservoir is defined as X = ]T)Awhere xi are the coordinate operators of the oscillators and Aj are the respective couplings. Eq. 2 is then referred to as the spin-boson Hamiltonian [8]. Another example of a reservoir could be a spin bath [11] 5. However, in our analysis below we do not specify the type of the environment. We will only assume that the reservoir gives rise to markovian evolution on the time scales of interest. More specifically, the evolution is markovian at time scales longer than a certain characteristic time rc, determined by the environment 6. We assume that rc is shorter than the dissipative time scales introduced by the environment, such as the dephasing or relaxation times and the inverse Lamb shift (the scale of the shortest of which we denote as Tdiss, tc [Pg.14]

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