Time-averaged concentration

Intensity of segregation A measure of the difference in concentration between neighboring clumps of fluid, = C /C (l - C ), where C is the time-averaged concentration and C a is the fluctuating component of concentration A. 

C>i (Ensemble) time-averaged concentration for averaging time t, ... 

As an excellent, simple example of how fluctuating parameters can affect a reacting system, one can examine how the mean rate of a reaction would differ from the rate evaluated at the mean properties when there are no correlations among these properties. In flow reactors, time-averaged concentrations and temperatures are usually measured, and then rates are determined from these quantities. Only by optical techniques or very fast response thermocouples could the proper instantaneous rate values be measured, and these would fluctuate with time. 

Figure 5.6 Time-averaged concentration field. The contour values are normalized by the source concentration.

Figure 5.7 Profiles of the time-averaged concentration at four downstream locations. The profiles are self-similar and the solid line corresponds to a Gaussian profile shape.

As described above, the plume becomes wider and more dilute as it evolves in the streamwise direction, thus ccenteriine and a are changing with x. The decrease of the time-averaged concentration along the centerline of the plume follows a v 1 profile for x/H > 2 (Fig. 5.8). This power law decrease agrees well with the time-averaged concentration field predicted by modeling efforts that assume... 

Figure 5.8 Time-averaged concentration along the plume centerline. Also shown is a power law curve.

Figure 5.10 Time-averaged concentration as a function of sampling period at an arbitrary location in the flow. The total time record of 600 s was divided into numerous shorter periods to demonstrate the slow convergence of the time-averaged concentration calculation. (Adapted from data in Webster and Weissburg [4].)...

The release location influences the vertical distribution of the time-averaged concentration and fluctuations. For a bed-level release, vertical profiles of the time-averaged concentration are self-similar and agreed well with gradient diffusion theory [26], In contrast, the vertical profiles for an elevated release have a peak value above the bed and are not self-similar because the distance from the source to the bed introduces a finite length scale [3, 25, 37], Additionally, it is clear that the size and relative velocity of the chemical release affects both the mean and fluctuating concentration [4], The orientation of the release also appears to influence the plume structure. The shape of the profiles of the standard deviation of the concentration fluctuations is different in the study of Crimaldi et al. [29] compared with those of Fackrell and Robins [25] and Bara et al. [26], Crimaldi et al. [29] attributed the difference to the release orientation, which was vertically upward from a flush-mounted orifice at the bed in their study. 

Nonlinearity of a reaction rate is a reason for improvement of the simple irreversible reaction A" — B. Low-frequency temperature oscillation around the average value T increases the reaction rate compared to a steady-state calculated at this average temperature. A positive effect can also be obtained due to concentration variation if n > 1 and a time average concentration is restricted. 

Fig 16. NSC for protein and mRNA copies for different size perturbations in kr when time-averaged concentrations are used. First-order (forward or backward) and central finite difference approximations are used. The deterministic NSC for both species is 1 (vertical dashed line). 

Equation 6.71 can be integrated term by term to yield the time average concentration of the sample in the eluent between any two times, fi and fa, giving the following relationship... 

Thus, the maximum raffinate concentration obtained with a given SMB is smaller than that produced by an equivalent TMB while the opposite is true for the maximum extract concentration. However, the differences between these concentrations are small when j6 is close to 1. The concentrations of the product streams are equal in the two processes in the limit case when = 1. Fiuthermore, it is possible to show that the time averaged concentrations of both the raffinate and the extract in SMB are equal to the constant concentrations of these streams obtained in TMB. This is required for mass conservation. Finally, the main difference between these two processes is that the steady state can be obtained only in an asymptotic sense in SMB, which means that it will take a large number of periods, hence a long time to reach it. 

For KjpCpD — 1, the relation between concentration and temperature. (9.9), is independent of the nature of the flow, either laminar or turbulent. It applies to both the instantaneous and time-averaged concentration and temperature fields, but only in regions in which condensation has not yet occurred. When the equations of transport for the jet flow are reduced to the form used in turbulent flow, the molecular diffusivity and thermal diffusivity are usually neglected in comparison with the turbulent diffusivities. This is acceptable for studies of gross transport and the time-averaged composition and temperature. However, this frequently made assumption is not correct for molecular scale processes like nucleation and condensation, which depend locally on the molecular transport properties. 

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