Time-dependent phenomena/processes

It can be stated that time correlation functions have done for the theory of time-dependent processes what partition functions have done for equilibrium theory. The time-dependent problem has become well defined, but no easier to solve. One now knows which correlation function corresponds to a given time-dependent phenomenon. Nevertheless, it is still extremely difficult to compute the correlation function. This is analogous to equilibrium theory where one knows that to compute any equilibrium property... 

Spin relaxation in a nucleus is induced by random fluctuations of local magnetic fields. These result from time-dependent modulation of the coupling energy of the resonating nuclear spin with nearby nuclear spins, electron spins, quadrupole moments, etc. Any time-dependent phenomenon able to modulate these couplings can contribute to nuclear relaxation. The distribution of the frequencies contained in these time-dependent phenomena is described by a correlation function, characterized by a parameter Tc, the correlation time. Its reciprocal can be considered as the maximum frequency produced by the fluctuations in the vicinity of the nuclear spin. If more than one process modulates the coupling between the nuclear spin and its surroundings, the reciprocal of the effective correlation time is the sum of the reciprocals of the various contributions... 

Almost all cross-flow filtration processes are inherently susceptible to flux decline due to membrane fouling (a time-dependent phenomenon) and concentration polarization effects which reflect concentration buildup on the membrane surface. This means lower flux (i.e., product output) which could drive the capital costs higher due to the requirement of a larger surface area to realize the desired production rate. In some situations, the lower flux could also result in lower selectivity which means reduced recoveries and/or incomplete removal of impurities from the filtrate. For example, removal of inhibitory metabolites such as lactic acid bacterial or separation of cells from broth while maximizing recovery of soluble products. 1 1... 

Membrane fouling A phenomenon characteristic of all membrane-based filtration processes in which the membrane adsorbs or interacts with feed components. Membrane fouling causes a flux decrease and may also increase the retention of certain components in the feed. Membrane fouling is typically a time-dependent phenomenon and often independent of concentration. In some situations a partial dependence on concentration may be observed. 

Many explosives, which have solid carbon as a detonation product, exhibit behavior that is not described adequately without including some time-dependent phenomenon, such as diffusion-controlled carbon deposition or some other kinetic behavior of the detonation products. A time-dependent carbon deposition is the only process known that could account for the large energy deficits required by the build-up model. The observed velocity constancy and large C-J pressure variations can be reproduced by the time-dependent carbon deposition mechanism. 

The development of the theory of heat-flow calorimetry (Section VI) has demonstrated that the response of a calorimeter of this type is, because of the thermal inertia of the instrument, a distorted representation of the time-dependence of the evolution of heat produced, in the calorimeter cell, by the phenomenon under investigation. This is evidently the basic feature of heat-flow calorimetry. It is therefore particularly important to profit from this characteristic and to correct the calorimetric data in order to gain information on the thermokinetics of the process taking place in a heat-flow calorimeter. 

Both, the mechanism and the extent of particle degradation depend not only on the process type but also on properties of the solid material, and to a large extent on the process conditions. Clift (1996) has stated that attrition is a triple-level problem, i.e., one is dealing with phenomena on three different length and time scales the processing equipment, the individual particles, and the sub-particle phenomenon such as fracture which leads to the formation of fines. The appearance of attrition can, therefore, differ very much between the various applications. For that reason, the following section deals with the various modes of attrition and the factors affecting them. 

In a further development of the continuous chain model it has been shown that the viscoelastic and plastic behaviour, as manifested by the yielding phenomenon, creep and stress relaxation, can be satisfactorily described by the Eyring reduced time (ERT) model [10]. Creep in polymer fibres is brought about by the time-dependent shear deformation, resulting in a mutual displacement of adjacent chains [7-10]. As will be shown in Sect. 4, this process can be described by activated shear transitions with a distribution of activation energies. The ERT model will be used to derive the relationship that describes the strength of a polymer fibre as a function of the time and the temperature. 

The excited state of a molecule can last for some time or there can be an immediate return to the ground state. One useful way to think of this phenomenon is as a time-dependent statistical one. Most people are familiar with the Gaussian distribution used in describing errors in measurement. There is no time dependence implied in that distribution. A time-dependent statistical argument is more related to If I wait long enough it will happen view of a process. Fluorescence decay is not the only chemically important, time-dependent process, of course. Other examples are chemical reactions and radioactive decay. 

In order to determine whether a given physical phenomenon can be described by a Markov process, it is not sufficient to study equations for the time dependence of px(x, /), such as Eqs. (22) and (16). It is necessary to investigate the equation for dw2/dt. We4 have undertaken this investigation for w2(p, t/Po> to)> the conditional probability density for a particle to have momentum between p and p + dp at time t if it had momentum p0 at time t0, in the low density limit. We find that w2 does obey the SCK... 

For times between these two extremes, the creep mismatch ratio is time dependent and can be determined using Eqns. (4) and (5) and an iterative process. Since, in general, CMR +1, time-dependent load redistribution between the fibers and matrix is a general phenomenon that occurs in all fiber-reinforced ceramics. 

---