Distributed time constant

The impedance models developed in Chapters 9,10,11, and 12 are based on the assumption that the electrode behaves as a uniformly active surface where each physical phenomenon or reaction has a single-valued time constant. The assumption of a uniformly active electrode is generally not valid. Time-constant dispersion can be observed due to variation along the electrode surface of reactivity or of current and potential. Such a variation is described in Section 13.1.1 as resulting in a 2-dimensional distribution. Time-constant dispersion can also be caused by a distribution of time constants that reflect a local property of the electrode, resulting in a 3-dimensional distribution. 

In both equations (13.1) and (13.2), the parameters a and Q are independent of frequency. When x —, Q has units of a capacitance, i.e., F/cm, and represents the capacity of the interface. When a 7 1, Q has units of s /Ocm and the system shows behavior that has been attributed to surface heterogeneity or to continuously distributed time constants for charge-transfer reactions. " Independent of the cause of CPE behavior, the phase angle associated with a CPE is independent of frequency. 

The effective CPE coefficient representation in Figures 17.6 and 17.13 yields, for a = 1, information concerning the high-frequency capacitance of the system. In the case that a < 1, Figures 17.6 and 17.13 yield an effective CPE coefficient Qeg that can be related to the film capacitance through a model of the distributed time constants following Brug et al. ° ... 

Thus the Voigt circuit can provide an adequate description of impedance data influenced by mass transfer or by distributed-time-constant phenomena such as is described in Chapter 13. In addition, inductive loops can be fitted by a Voigt circuit by using a negative resistance and capacitance in an element. Such an element will have a positive RC time constant. The Voigt circuit serves as a convenient generalized measurement model. 

This is the typical result in TDS yielding the time constant x, and the relaxation strength A. In contrast to a three element circuit, more complicated objects yield several time constants over a wide frequency range. More sophisticated is the calculation of distributed time constant as it would be required for the Cole model. 

But Jonscher was apparently not aware that Cole already in 1928 used the circle segment analysis in the Wessel plane and found many circular arcs with suppressed circle centers. The concept of CPE was introduced, and in Cole and Cole (1941), the idea was introduced that a dielectric could have a distribution of many relaxation time constants. The Debye model with ideal components presupposed one single relaxation time constant and therefore a complete semicircle. However, the Cole—Cole model implied that the distributed time constants do not correspond to one exponential, but a fractional power law. It seems that Jonscher (1996) did not accept the flieory of distributed... 

FIG. 8-48 Temperature leaving a heat exchanger responds as a distributed lag, the gain and time constant of which vary inversely with flow. 

WASH-1400 treated the probability of failure with time as being exponentially distributed with constant X. It treated X itself as being lognormally distributed. There are better a priori reasons... 

Up to now it has been tacitly assumed that each molecular motion can be described by a single correlation time. On the other hand, it is well-known, e.g., from dielectric and mechanical relaxation studies as well as from photon correlation spectroscopy and NMR relaxation times that in polymers one often deals with a distribution of correlation times60 65), in particular in glassy systems. Although the phenomenon as such is well established, little is known about the nature of this distribution. In particular, most techniques employed in this area do not allow a distinction of a heterogeneous distribution, where spatially separed groups move with different time constants and a homogeneous distribution, where each monomer unit shows essentially the same non-exponential relaxation. Even worse, relaxation... 

Constant RTD control can be applied in reverse to startup a vessel while minimizing olf-specification materials. For this form of startup, a near steady state is first achieved with a minimum level of material and thus with minimum throughput. When the product is satisfactory, the operating level is gradually increased by lowering the discharge flow while applying Equation (14.8) to the inlet flow. The vessel Alls, the flow rate increases, but the residence time distribution is constant. 

In our opinion, the interesting photoresponses described by Dvorak et al. were incorrectly interpreted by the spurious definition of the photoinduced charge transfer impedance [157]. Formally, the impedance under illumination is determined by the AC admittance under constant illumination associated with a sinusoidal potential perturbation, i.e., under short-circuit conditions. From a simple phenomenological model, the dynamics of photoinduced charge transfer affect the charge distribution across the interface, thus according to the frequency of potential perturbation, the time constants associated with the various rate constants can be obtained [156,159-163]. It can be concluded from the magnitude of the photoeffects observed in the systems studied by Dvorak et al., that the impedance of the system is mostly determined by the time constant. 

However, T2 is sensitive to the molecular interactions of spins and dependent on the molecular environment [60]. Thus, T2 may overlap for different components in certain materials and this technique alone may not be sufficient to identify the components. The relaxation time distributions are often broad, e.g., in meat [21], thus making it more difficult to associate the relaxation time constants with the components. 

It should be noted that the decomposition shown in Eq. 3.7.2 is not necessarily a subdivision of separate sets of spins, as all spins in general are subject to both relaxation and diffusion. Rather, it is a classification of different components of the overall decay according to their time constant. In particular cases, the spectrum of amplitudes an represents the populations of a set of pore types, each encoded with a modulation determined by its internal gradient. However, in the case of stronger encoding, the initial magnetization distribution within a single pore type may contain multiple modes (j)n. In this case the interpretation could become more complex [49]. 

From these data, aquatic fate models construct outputs delineating exposure, fate, and persistence of the compound. In general, exposure can be determined as a time-course of chemical concentrations, as ultimate (steady-state) concentration distributions, or as statistical summaries of computed time-series. Fate of chemicals may mean either the distribution of the chemical among subsystems (e.g., fraction captured by benthic sediments), or a fractionation among transformation processes. The latter data can be used in sensitivity analyses to determine relative needs for accuracy and precision in chemical measurements. Persistence of the compound can be estimated from the time constants of the response of the system to chemical loadings. 

To minimize absorption from the solution, optical thin layer cells have been designed. The working electrode has the shape of a disc, and is mounted closely behind an IR-transparent window. For experiments in aqueous solutions the intervening layer is about 0.2 to 2 ftm thick. Since the solution layer in front of the working electrode is thin, its resistance is high this increases the time required for double-layer charging - time constants of the order of a few milliseconds or longer are common - and may create problems with a nonuniform potential distribution. 

Relaxation is an inherent property of all nuclear spins. There are two predominant types of relaxation processes in NMR of liquids. These relaxation processes are denoted by the longitudinal (Ti) and transverse (T2) relaxation time constants. When a sample is excited from its thermal equihbrium with an RF pulse, its tendency is to relax back to its Boltzmann distribution. The amount of time to re-equilibrate is typically on the order of seconds to minutes. T, and T2 relaxation processes operate simultaneously. The recovery of magnetization to the equilibrium state along the z-axis is longitudinal or the 7 relaxation time. The loss of coherence of the ensemble of excited spins (uniform distribution) in the x-, y-plane following the completion of a pulse is transverse or T2... 

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