Time scales mixing

Theoretical calculations for ultrafast neat water spectroscopy are difficult to perform and difficult to interpret (because of the near-resonant OH stretch coupling). One classical calculation of the 2DIR spectrum even preceded the experiments [163] Torii has calculated the anisotropy decay [97], finding reasonable agreement with the experimental time scale. Mixed quantum/ classical calculations of nonlinear spectroscopy for many coupled chromo-phores is a daunting task. We developed the TAA for linear spectroscopy, and Jansen has very recently extended it to nonlinear spectroscopy [164]. We hope that this will allow for mixed quantum/classical calculations of the 2DIR spectrum for neat water and that this will provide the context for a molecular-level interpretation of these complex but fascinating experiments. 

Laser-based pump strategies are generally necessary to study reactions taking place on time scales faster tlian microseconds. Lasers can be used to produce L-jumps on time scales faster tlian microseconds or to initiate reactions tlirough rapid photochemical or photophysical processes. Lasers can also initiate ultrarapid mixing via a wide variety... 

Dimensionless Numbers. With impeller diameter D as length scale and mixer speed N as time scale, common dimensionless numbers encountered in mixing depend on several controlling phenomena (Table 2). These quantities are useful in characterizing hydrodynamics in mixing tanks and when scaling up mixing systems. 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

First, let us consider batch mixing processes, as exemplified by ordinaiy laboratory practice in solution kinetics. A portion of one solution (say, of the substrate) is added by pipet to a second solution (containing the reagent) in a flask, the flask is shaken to achieve homogeneity, and then samples are withdrawn at known times for analysis, or the solution is subjected to continuous observation as a function of time, for example, by spectrophotometry. For reactions on a time scale (measured by the half-life) of hours or even several minutes, the time consumed in these operations is a negligible portion of the reaction time, but as the half-life of the reaction decreases, it becomes necessary to consider these preliminary steps. Let us distinguish three stages ... 

Gas constant (J/mol K) coarse particle radius Specific weight of feed Degree of mixing Time scale (s)... 

Revisions of the continuous-flow method have been made to allow observations along the length of the flow tube rather than at right angles.5 This method, fast continuous flow, eliminates the dead time during which the reaction cannot be observed. Kinetic data can be extracted to a time resolution of nearly 10 p,s, but the mathematics is more complicated in this limit, because the mixing and chemical reaction occur on the same time scale. Rate constants nearly as large as the diffusion-controlled value have been determined in favorable cases.6... 

This process does not lead to net ozone depletion because it is rapidly followed by reaction 2, which regenerates the ozone. Reactions 2 and 3 have, however, another important function, namely the absorption of solar energy as a result, the temperature increases with altitude, and this inverted temperature profile gives rise to the stratosphere (see Figure 1). In the lower layer, the troposphere, the temperature decreases with altitude and vertical mixing occurs on a relatively short time scale. In contrast, the stratosphere is very stable towards vertical mixing because of its inverted temperature profile. 

The spatio-temporal variations of the concentration field in turbulent mixing processes are associated wdth very different conditions for chemical reactions in different parts of a reactor. This scenario usually has a detrimental effect on the selectivity of reactions when the reaction time-scale is small compared with the mixing time-scale. Under the same conditions (slow mixing), the process times are increased considerably. Due to mass transfer inhibitions, the true kinetics of a reaction does not show up instead, the mixing determines the time-scale of a process. This effect is known as mixing masking of reactions [126]. 

Micro reactors show, under certain conditions, low axial flow dispersion reactions with unstable intermediates can be carried out in a fast, stepwise manner on millisecond time-scales. Today s micro mixers mix on a millisecond scale and below [40]. Hence in micro reactors reactions can be carried out in the manner of a quench-flow analysis, used for determination of fast kinetics [93]. 

Table 5.4-24 summarises the various characteristic time constants for reaction and mixing. Instantaneous (very rapid), fast (rapid) and slow (very slow) reactions have been classified based on characteristic time constants (time scales) for mixing and reaction. Denoting the mixing time scale by xm (t99, to, xms, xds, or x ,) reactions can be classified as follows from the viewpoint of competition with individual stages of mixing tm xf => instantaneous TM XR => fast TM XR => slow... 

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