Averaging Procedures

An important point for all these studies is the possible variability of the single molecule or single particle studies. It is not possible, a priori, to exclude bad particles from the averaging procedure. It is clear, however, that high structural resolution can only be obtained from a very homogeneous ensemble. Various classification and analysis schemes are used to extract such homogeneous data, even from sets of mixed states [69]. In general, a typical resolution of the order of 1-3 mn is obtained today. 

Because of the variation in potential energy with the angle of rotation, not all locations on the rim of the cones in Fig. 1.7b are equally favored. The probability of a particular angular position depends on the potential energy at that location, V, and an averaging procedure which considers this angular variation must be used to modify Eq. (1.60). The result of this procedure is... 

An average crystal size can be used to characterize a CSD. However, the average can be determined on any of several bases, and the basis selected must be specified for the average to be usehil. More than 20 different averaging procedures have been proposed, yet none is generally satisfactory or preferred (5). 

Again this averaging procedure can only be expected to work when the reactions are sufficiently similar . This is difficult to quantify a priori. The Marcus equation is therefore more a conceptual tool for explaining trends, than for deriving quantitative results. 

According to Eq. (86), the averaging procedure mentioned above leads to the following total reaction current J,... 

Then, after solving Eq. (91), according to the averaging procedure of Eq. (92), the fluctuation is averaged. At the initial stage, the following fluctuation-diffusion current equation is obtained,... 

When perturbation is fast enough in comparison with the molecular response to it, the averaging procedure proposed in [91] is justified. After substitution of a formal solution of Eq. (2.20)... 

This averaging procedure has the effect of improving the approximation. Alternatively, each segment could be treated as a well-mixed tank with the outflow variables equal to the tank values. This simpler approach would require a greater number of segments for the same accuracy. 

Given the additional fact that the reading at infinite time is 12.16, use a numerical averaging procedure to determine the reaction rate constant. 

The time A between readings is thus greater than the reaction half-life, and the numerical averaging procedure leading to equation 3.3.24 could be used for the analysis of these data if an estimate of the reading at infinite time can be obtained. In order to manipulate this equation to a form that can make use of the available data, several points must be noted. 

We will use a numerical averaging procedure to evaluate the ratio of rate constants. Thus... 

It is also common practice to integrate the pressures and impulses, over the surface areas, to obtain average values, rather than try and compute structural response to spatially-varying, as well as time-varying loads. But, this averaging procedure should be used cautiously for long walls or ceilings, because it can lead to serious underprediction of shock loads for part of the surface. 

After reducing this TBF to an effective, density dependent, two-body force by the averaging procedure described earlier, the resulting effective two-nucleon potential assumes a simple structure,... 

The leading parameter that characterizes the distributions of randomly branched samples is an exponent Tthat is defined in the next section. The average procedures for the z-average of the mean square radius of gyration and the weight average molar mass results in the relationship [7]... 

This allows for a high amount of integrations or averaging procedures, leading to a substantial reduction in the signal-to-noise ratio. On the other hand, one disadvantage of OMAs is their spectral resolution, which is lower than that provided by monochromators. 

It appears that Schumann [60] was one of the first to conduct LES of a reacting flow. However, the assumption made in this work to neglect the contribution of the SGS scalar fluctuations to the filtered reaction rate needs to be justified for general applications. As indicated in section 9.2, the importance of such fluctuations is well recognized in Reynolds averaged procedures. Therefore, it is natural to believe that these fluctuations are also important in LES. This issue has been the subject of broad investigation [61-76]. 

Using the Favre averaging procedure [2, 5] one obtains the following set of equations for the gas phase in a multiphase flow [6] (the averaging bars are removed for simplicity) ... 

Figure 6 The block-averaging procedure considers a full range of block sizes. The upper panel shows the time series for the squared cosine of the central dihedral of butane, with two different block sizes annotated. The lower panel shows the block-averaged standard error for that times series, as a function of block size.

By analogy to single-phase flow, the two-phase frictional pressure-drop can be expressed by the conventional Fanning equation, and thereby a friction factor is defined. These friction factors may be based on liquid properties, gas properties, or on a fictitious single fluid of mean properties obtained by some averaging procedure. Typical definitions, such as those shown in equation (32), have been given and discussed recently by Govier and Omer (G4) ... 

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