Example The characteristic time

The characteristic time defined in (9.21) establishes a time scale for surface evolution of the kind discussed in the preceding section. Its definition depends on a number of parameter values that are not measurable and, therefore, are not known with any certainty. To get some idea of its magnitude, estimate the value of the characteristic time for the particular case of a Si surface with a mismatch strain of Cin = 0.008 at a temperature of T = 600 °C. Base the estimate on the unit cell dimension of a = 0.5431 nm for the diamond cubic crystal structure, and on the following values of macroscopic material parameters an elastic modulus of E = 130 GPa, a Poisson ratio of = 0.25, a mass density oi p = 2328kg/m, and the surface energy of 70 = 2J/m. Assume that 10% of the surface atoms are involved in the mass transport process at any instant so that = 0.1. 

By combining (8.60) and (9.7) with (9.21), the expression for the characteristic time becomes 

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For some reactions the rate constant kj can be very large, leading potentially to very rapid transients in the species concentrations (e.g., [A]). Of course, other species may be governed by reactions that have relatively slow rates. Chemical kinetics, especially for systems like combustion, is characterized by enormous disparities in the characteristic time scales for the response of different species. In a flame, for example, the characteristic time scales for free-radical species (e.g., H atoms) are extremely short, while the characteristic time scales for other species (e.g., NO) are quite long. It is this huge time-scale disparity that leads to a numerical (computational) property called stiffness. 

The most difficult problem is the adequate determination of the "abnormally (i.e. unexpected) slow transition process. For this purpose, we must imagine the simplest system of expectations. It is based on the hypotheses about reaction mechanisms. A catalytic reaction is represented as a combination of elementary steps (see Chap. 3). We admit some hypotheses concerning values of the corresponding rate coefficients. Typical concentrations of gas-phase substances and of surface compounds are also assumed to be known. One can also introduce a concept of the characteristic time of a step. For example, the characteristic time for the step A B can be determined as 1 l(k+ + k ), where k and k are the rate constants for the direct and reverse reactions. For the reaction A + B - C one can introduce two characteristic times 1 [kCA and 1 /kCB, where CA and CB are the characteristic concentrations of A and B. 

In the following ptu-agraphs this approach will be applied to production of an alkaloid from plant cells cultured in a bioreactor. Regime analysis can be performed by comparison of characteristic parameters of the mechanisms involved in the process. Here the characteristic time concept will be used. The characteristic time is a measure for the rate of a mechanism. A fast mechanism has a short characteristic time. Other terms used are relaxation time, process time, or time constant. A time constant is formally only defined for first-order linear processes. Not all mechanisms involved in a plant cell production process are first order, therefore the term characteristic time is used. The characteristic time is defined as the ratio of a capacity and a flow for example, the characteristic time for oxygen transfer to the liquid phase in a aerated bioreactor q.l becomes... 

Reaction rates are often compared using a characteristic time. For example, the characteristic time might be defined as the time needed for the destruction of 50% of the reactants (a = p = 0.5). This time is called the reaction s half-life, ty,. When t = ty a= p = 0.5. The half-life for an unopposed reaction where 1 is found by setting m = 0.5 in Eq. (3.18). 

In order to exemplify the potential of micro-channel reactors for thermal control, consider the oxidation of citraconic anhydride, which, for a specific catalyst material, has a pseudo-homogeneous reaction rate of 1.62 s at a temperature of 300 °C, corresponding to a reaction time-scale of 0.61 s. In a micro channel of 300 pm diameter filled with a mixture composed of N2/02/anhydride (79.9 20 0.1), the characteristic time-scale for heat exchange is 1.4 lO" s. In spite of an adiabatic temperature rise of 60 K related to such a reaction, the temperature increases by less than 0.5 K in the micro channel. Examples such as this show that micro reactors allow one to define temperature conditions very precisely due to fast removal and, in the case of endothermic reactions, addition of heat. On the one hand, this results in an increase in process safety, as discussed above. On the other hand, it allows a better definition of reaction conditions than with macroscopic equipment, thus allowing for a higher selectivity in chemical processes. 

The electron-transfer mechanism for electrophilic aromatic nitration as presented in Scheme 19 is consistent with the CIDNP observation in related systems, in which the life-time of the radical pair [cf. (87)] is of particular concern (Kaptein, 1975 Clemens et al., 1984, 1985 Keumi et al., 1988 Morkovnik, 1988 Olah et al., 1989 Johnston et al., 1991 Ridd, 1991 Rudakov and Lobachev, 1991). As such, other types of experimental evidence for aromatic cation radicals as intermediates in electrophilic aromatic nitration are to be found only when there is significant competition from rate processes on the timescale of r<10 los. For example, the characteristic C-C bond scission of labile cation radicals is observed only during the electrophilic nitration of aromatic donors such as the dianthracenes and bicumene analogues which produce ArH+- with fragmentation rates of kf> 1010s-1 (Kim et al., 1992a,b). 

O2 consumption rate becomes smaller under 0.7 V, the O2 concentration at the reaction surface recovers, thus leading to an increase in the cell current density. The current rise time corresponds well with the characteristic time scale of gas phase transport as analyzed above. The rise in the cell current, however, experiences an overshoot because the polymer membrane still maintains a higher water content corresponding to 0.6 V. It then takes about 15 s for the membrane to adjust its water content at the steady state corresponding to 0.7 V. This numerical example clearly illustrates the profound impact of water management on transient dynamics of low humidity PEFC engines where the polymer membrane relies on reaction water for hydration or dehydration. 

The characteristic time-scales mentioned above take into account some but not all practical considerations. For example, really intense stirring (rpm > 500) in the CSTR is not recommended for in situ studies since a deep vortex ivill be formed in the liquid, gas ivill be entrained, and tivo-phase flow w ill occur in the recycle line. Also, two-phase flow will generally cause cavitation in a mechanical pump (possibly stopping flow) and induce irreproducible spectroscopic measurements. 

Table 3 Case B. Parameter values for the characteristic time 240 s for the second example from Taylor s paper...

Schwartz and Freiberg (1981) have calculated the rates of these processes for S02 and expressed them in terms of characteristic times t, which for Step 5, chemical reaction, is the natural lifetime discussed in Section 5.A.I.C. For Steps 1-4, the characteristic time is the time to establish the appropriate steady state or equilibrium for the process involved for example, for Step 1, it is the time to establish a steady-state concentration of the gas in the air surrounding the droplet. Seinfeld (1986) discusses in detail calculation procedures for these characteristic times. A brief summary of the results of Schwartz and Freiberg (1981) for Steps... 

For the three stirrer types treated in this example, the mixing time characteristics are presented in Figure 13. 

For the two flow regimes of River G discussed in Illustrative Example 24.1, calculate (a) the characteristic time and length scale for vertical mixing (b) the characteristic time and length scale for transversal mixing and (c) the dispersion coefficient. 

The coefficient of vertical diffusivity is calculated from Eq. 24-32 and from a evaluated in Illustrative Example 24.2. The characteristic time and length scales for vertical mixing are given by Eqs. 24-33 and 24-34. The following table summarizes the results ... 

Sj - S0, the interconversion into the triplet state Tlf is also possible. The characteristic time of the phosphorescence decay of MP, on the other hand, is rather large and amounts to 10-2 s. (For a review of the physical and chemical properties of MP, see, for example, ref. 52.)... 

A more complicated situation emerges in motion along nonintersecting surfaces with variable curvatures. If the distance between these surfaces remains finite everywhere, then the field lines do not expand infinitely in the directions normal to the surfaces. In the absence of dissipation this means that there is no unbounded growth of the normal field component. However, introduction of the finite conductivity yields an equation for the normal component which is not decoupled it contains the contribution of the Laplacian of the remaining components. At the same time, it is possible for all other components to increase exponentially with an increment which depends on the conductivity and vanishes for infinite conductivity. The authors called this mechanism of field amplification a slow dynamo, in contrast to the fast dynamo feasible in the three-dimensional case, i.e., the mechanism related only to infinite expansion of the field lines as, for example, in motion with magnetic field loop doubling. In a fast dynamo the characteristic time of the field increase must be of the same order as the characteristic period of the motion s fundamental scale. 

Discussion. We can now propose a coarse description of the paraffinic medium in a lamellar lyotropic mesophase (potassium laurate-water). Fast translational diffusion, with D 10"6 at 90 °C, occurs while the chain conformation changes. The characteristic times of the chain deformations are distributed up to 3.10"6 sec at 90 °C. Presence of the soap-water interface and of neighboring molecules limits the number of conformations accessible to the chains. These findings confirm the concept of the paraffinic medium as an anisotropic liquid. One must also compare the frequencies of the slowest deformation mode (106 Hz) and of the local diffusive jump (109 Hz). When one molecule wants to slip by the side of another, the way has to be free. If the swinging motions of the molecules, or their slowest deformation modes, were uncorrelated, the molecules would have to wait about 10"6 sec between two diffusive jumps. The rapid diffusion could then be understood if the slow motions were collective motions in the lamellae. In this respect, the slow motions could depend on the macroscopic structure (lamellar or cylindrical, for example)... 

It is clear that a core-hole represents a very interesting example of an unstable state in the continuum. It is, however, also rather complicated [150]. A simpler system with similar characteristics is a doubly excited state in few-body systems, as helium. Here, it is possible [151-153] to simulate the whole sequence of events that take place when the interaction with a short light pulse first creates a wave packet in the continuum, including doubly excited states, and the metastable components subsequently decay on a timescale that is comparable to the characteristic time evolution of the electronic wave packet itself. On the experimental side, techniques for such studies are emerging. Mauritsson et al. [154] studied recently the time evolution of a bound wave packet in He, created by an ultra-short (350 as) pulse and monitored by an IR probe pulse, and Gilbertson et al. [155] demonstrated that they could monitor and control helium autoionization. Below, we describe how a simulation of a possible pump-probe experiment, targeting resonance states in helium, can be made. 

The maximum temperature of synthesis reaction was calculated for the substitution reaction example as a function of the process temperature and with different feed rates corresponding to a feed time of 2, 4, 6, and 8 hours. The straight line (diagonal in Figure 7.11) represents the value for no accumulation, that is, for a fast reaction. This clearly shows that the reactor has to be operated at a sufficiently high temperature to avoid the accumulation of reactant B. But a too high temperature will also result in a runaway due to the high initial level, even if the accumulation is low. In this example, the characteristics of the decomposition reaction... 

The second reason why complete dynamic and conformational information is not available is related to the characteristic time scale of particular experiments. For example, the elegant techniques 2... 

Although a theoretical approach has been desecrated as to how one can apply the generalized coupled master equations to deal with ultrafast radiationless transitions taking place in molecular systems, there are several problems and limitations to the approach. For example, the number of the vibrational modes is limited to less than six for numerical calculations. This is simply just because of the limitation of the computational resources. If the efficient parallelization can be realized to the generalized coupled master equations, the limitation of the number of the modes can be relaxed. In the present approach, the Markov approximation to the interaction between the molecule and the heat bath mode has been employed. If the time scale of the ultrashort measurements becomes close to the characteristic time of the correlation time of the heat bath mode, the Markov approximation cannot be applicable. In this case, the so-called non-Markov treatment should be used. This, in turn, leads to a more computationally demanding task. Thus, it is desirable to develop a new theoretical approach that allows a more efficient algorithm for the computation of the non-Markov kernels. Another problem is related to the modeling of the interaction between the molecule and the heat bath mode. In our model, the heat bath mode is treated as... 

Note, however that the concepts about the lipid membrane as the isotropic, structureless medium are oversimplified. It is well known [19, 190] that the rates and character of the molecular motion in the lateral direction and across the membrane are quite different. This is true for both the molecules inserted in the lipid bilayer and the lipid molecules themselves. Thus, for example, while it still seems possible to characterize the lateral movement of the egg lecithin molecule by the diffusion coefficient D its movement across the membrane seems to be better described by the so-called flip-flop mechanism when two lipid molecules from the inner and outer membrane monolayers of the vesicle synchronously change locations with each other [19]. The value of D, = 1.8 x 10 8 cm2 s 1 [191] corresponds to the time of the lateral diffusion jump of lecithin molecule, Le. about 10 7s. The characteristic time of flip-flop under the same conditions is much longer (about 6.5 hours) [19]. The molecules without long hydrocarbon chains migrate much more rapidly. For example for pyrene D, = 1.4x 10 7 cm2 s1 [192]. 

In the photosynthetic and mitochondrial membranes the components of the transmembrane electron transport chain are not linked with covalent bonds, but fixed in a protein matrix. An example of such an arrangement of the electron transport chain in an artificial system can be found in papers by Tabushi et al. [244, 245], which deal with the dark electron transfer across the lipid membranes containing the dimers of cytochrome c3 from Desulfovibrio vulgaris. The dimer size is about 60 A, i.e. it somewhat exceeds the membrane thickness. This enables electron to move across the membrane via the cytochrome along the chain of hem fragments embedded in the protein. However, the characteristic time of the transmembrane electron transfer by this method is rather long (about 10 s). 

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