Sinusoidal fluctuation

Also considered in this chapter are oscillating reactions. These are a class of reactions, still not too numerous, in which the concentrations of the intermediates and the buildup rate of a product fluctuate over time. That is, there are sinusoidal fluctuations in rate and concentration with time. We shall see how these can arise from straightforward, albeit complicated, schemes, often involving the catalysis of one step by a product of another. 

For Reynolds numbers of 500-300,000, based on the external diameter of the pitot tube, an error of not more than 1 per cent is obtained with this instrument. A Reynolds number of 500 with the standard 7.94 mm pitot tube corresponds to a water velocity of 0.070 m/s or an air velocity of 0.91 m/s. Sinusoidal fluctuations in the flowrate up lo... 

This result could be improved by assuming a more appropriate distribution function of T instead of a simple sinusoidal fluctuation however, this example—even with its assumptions—usefully illustrates the problem. Normally, probability distribution functions are chosen. If the concentrations and temperatures are correlated, the rate expression becomes very complicated. Bilger [47] has presented a form of a two-component mean-reaction rate when it is expanded about the mean states, as follows ... 

Sinusoidal fluctuation shows clear periodic behavior in the systematic error domain. 

Figure 5-1 shows the behavior of both the quasi steady-state solution, Eq. [27], for t 100 d, and the unsteady-state solution, Eq. [21], for t = 1, 10, 20 and 100 d. Equations [A24] and [A25] have been incorporated into Eq. [21] to develop the concentration curves. The values of the parameters are presented on Fig. 5-1, except co = 1/24 hr1 and R = 1. For t 100 d, the quasi steady-state solution, Eq. [27], gives the same results as are given by Eq. [21], the unsteady-state solution. Obviously, 100 d is a long enough time for the unsteady-state effects to disappear from the 500 cm length of column in Fig. 5-1. Dispersion acts to reduce the height of the peaks and fill in the valleys of the concentration vs>distance curve. Although the sinusoidal fluctuations in concentration amplitude are) dampened by dispersion, they are still visible after the solute travels 500 cmN- ...

The peristaltic pump is by far the most common propulsion device in flow analysis, due to its low cost and easy incorporation into multichannel flow systems. It delivers a pulsating flow (Fig. 3.4b) as a result of the vector sum of two effects. The first is the main action of the roller, which accounts for the main constant flow, whereas the second relates to the roller lift-off from the platen, which accounts for the slight sinusoidal fluctuations (ripple) on the main flow. 

An early survey of the possible sources for the hypersensitivity concluded that the most likely candidate was a mechanism based on the inhomogeneities of the dielectric surrounding the rare-earth or actinide ion (4). It runs as follows. The radiation field induces sinusoidally fluctuating dipole moments in the ligands surrounding the ion. These induced dipoles necessarily radiate, and the emitted fields impinge on the rare-earth or actinide ion. Because of the proximity of source and receiver, the plane-wave condition no longer applies the wave fronts are sufficiently distorted to produce substantial quadrupole components. 

Obviously, Up - [ -represents the slip velocity. Table 7-1 gives allowable seed particle sizes for a 99% amplitude response to sinusoidal fluctuations at 1 kHz and 10 kHz. 

Crystal rotation in a thermally asymmetric field. That is, if the crystal is not rotated in a thermally balanced field while it is being pulled, any point on the soUd-Uquidus interface can experience a sinusoidal fluctuations in growth rate. These fluctuations cause growth striations in the crystal and are the source of "lensing" mentioned above. If the fluctuations exceed R, more severe defects occur in the crystal, as we have already shown. 

In this expression Aoo is the nominal aperture size to deliver at the design flow rate based on the constant set input flow rate. The second term in the parenthetical expression is the product of a proportionality constant K and the difference between the set point level and the actual level as a function of time. We substitute this for Ao in Torricelli s Law and also in the equation describing a system with sinusoidally fluctuating input flow ... 

A discussion of this solution of the kinetic problem of spinodal decomposition leads to the following conclusions Every (approximately sinusoidal) fluctuation in composition of the supersaturated crystal will continue its spontaneous growth if the terms in the bracket of eq. (7-58) yield a negative value. If this value is positive, the fluctuations cannot grow but disappear. Setting the terms in the bracket equal to zero, one can calculate the minimum wavelength of the fluctuation which is capable of growth as ... 

Study of the evolution of this same material system under conditions of condensation—evaporation for the case when Xv = leads to the same evolution equations for amplitude and mean height, except that the characteristic time for amplitude evolution is determined by different material parameters. Derivation of these evolution equations is left as an exercise. If Xv > then the sinusoidal fluctuation in surface shape is superimposed on the mean surface speed h = —cs(flm — Xv)-... 

Figure 6.2 shows the time variation of the amplitude of sinusoidal fluctuations of the composition. The amplitude increases with time and, when the system decomposes into two final phases, the amplitude value becomes formally infinite. From Figure 6.2, in particular, a veiy important consequence ensues concerning the mechanism of formation of transition layers in systems decomposing through the spinodal mechanism. If Co is the starting concentration of one of the components in the system, then the developing phases are characterized by compositions determined by the composition fluctuation amplitudes. 

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