Quasi-stationary process

At high temperatures or in the presence of catalysts, hydroperoxide decomposes at a high rate, so that, after t t, inhibited oxidation becomes a quasi-stationary process with balanced rates of ROOH formation and decomposition. In this case, kdr 1, where kd is the overall specific rate of ROOH decomposition with allowance made for its decomposition... 

We consider now the situation that component A and B are not volatile and that the volume of the stagnant film is small compared with the bulk of the liquid. This means that the bulk concentration of component A and B can be assumed to be constant. In fact we consider a quasi stationary process. Then the boundary conditions of equations (23), (24) and (25) are ... 

The applicability of Maxwell s equation is limited in describing particle growth or depletion by mass transfer. Strictly speaking, mass transfer to a small droplet cannot be a steady process because the radius changes, causing a change in the transfer rate. However, when the difference between vapor concentration far from the droplet and at the droplet surface is small, the transport rate given by Maxwell s equation holds at any instant. That is, the diffusional transport process proceeds as a quasi-stationary process. 

During the extraction phase the whole mass transfer may be regarded as a quasi-stationary process. Scale-up rules therefore take account of external mass transfer from the solid surface to the supercritical fluid only. If pilot and production plants are required to display the same mass transfer properties, then... 

Mathematical difficulties forced Kramers to restrict his discussion. to the case in which the barrier height Q = EMt is large compared to the mean thermal energy of the molecules kT and in which the diffusion over the barrier can be treated as a quasi-stationary process. Kramers showed that under these conditions the calculated reaction rate is very close to the equilibrium rate, as given by absolute rate theory, and that for E/kT > 10 the rate calculated from his model agrees with the equilibrium rate to within about 10 per cent over a rather wide range of rj. 

Once again, other dimensionless numbers are used to represent the shifting boundary conditions. The periodic behavior of SMB processes is induced by the stepwise switching of the inlet and outlet ports. To describe this operation as a quasi-stationary process comparable to the TMB process the following flow rates are introduced ... 

Here the value of the reaction rate constant ratio in the quasi-stationary process, has the dimension of concentration and is called Michaelis-Menten constant. It describes conditions, at which the biodegradation rate does not change in time. 

The use of the differential recurrence relations to calculate the mean first passage time is based on the observation that if in Eq. (5.48) one ignores the term sY(x, s) (which is tantamount to assuming that the process is quasi-stationary, i.e., all characteristic frequencies associated with it are very small), then one has... 

Some simulation results for trilobic particles (citral hydrogenation) are provided by Fig. 2. As the figure reveals, the process is heavily diffusion-limited, not only by hydrogen diffusion but also that of the organic educts and products. The effectiviness factor is typically within the range 0.03-1. In case of lower stirrer rates, the role of external diffusion limitation becomes more profound. Furthermore, the quasi-stationary concentration fronts move inside the catalyst pellet, as the catalyst deactivation proceeds. 

If a system is disturbed by periodical variation of an external parameter such as temperature (92), pressure, concentration of a reactant (41,48,65), or the absolute configuration of a probe molecule (54,59), then all the species in the system that are affected by this parameter will also change periodically at the same frequency as the stimulation, or harmonics thereof (91). Figure 24 shows schematically the relationship between stimulation and response. A phase lag <)) between stimulation and response occurs if the time constant of the process giving rise to some signal is of the order of the time constant Inim of the excitation. The shape of the response may be different from the one of the stimulation if the system response is non-linear. At the beginning of the modulation, the system relaxes to a new quasi-stationary state, about which it oscillates at frequency cu, as depicted in Fig. 24. In this quasi-stationary state, the absorbance variations A(v, t) are followed by measuring spectra... 

The increased temperature results in an increased rate of destruction of the branching intermediate (methyl hydroperoxide) with a consequent further increase of the rate, but also a decreased rate of formation of fresh hydroperoxide since Equilibrium 5 is displaced to the left, and the alternative reactions of methylperoxy increase in rate faster than that leading to formation of hydroperoxide. Consequently the quasi-stationary concentration of methyl hydroperoxide falls, and the rate of reaction declines since the new product of methyl oxidation—formaldehyde— cannot bring about branching at these temperatures. The temperature of the reaction mixture falls (because the rate has fallen), and when it has fallen sufficiently, provided sufficient of the reactants remain, the whole process may be repeated, and several further flames may be observed. 

The resulting rate can be estimated as logT 4>q(G/Gq)x If o < 1, this reduces to log T 4>o(G/Go)Ith/If- In the opposite limit, the estimation for the rate reads log r 4>o(G/Gq ) l< (It J h), F being a dimensionless function 1. It is important to note that these expressions match the quantum tunneling rate log Jr Uqt/K (G/Gq)< provided eVr h. Therefore the quasi-stationary approximation is valid when the quantum tunneling rate is negligible and the third factor mentioned in the introduction is not relevant. For equilibrium systems, the situation corresponds to the well-known crossover between thermally activated and quantum processes at k Tr h [9]. 

The thermal and kinetic models discussed above are the basis for determining the processing conditions for reactive processing by ionic polymerization,29 addition polymerization, vulcanization of rubbers and radical polymerization, although in the latter case additional assumptions of a constant initiation rate and a quasi-stationary concentration of radicals are made.89 These models can also be used to solve optimization problems to improve the performance and properties of end-products. 

It is reasonable to assume that in the vast majority of cases encountered in reactive processing Re < < 1 and (H/L)Re < < 1. Thus, we can consider the flow to be quasi-stationary and that temperature changes occur quickly after alterations in the temperature and degree of conversion distributions.202 Now we can rewrite the system of balance equations in the following dimensionless form ... 

Measurements of the environmental parameters in the monitoring regime provide sets of series of quantitative characteristics for the system of data processing, which cannot be analyzed because of their stationarity. There are many ways to overcome time dependence and thereby remove the contradiction between the applicability of statistical methods and the level of observational data stationarity. One such way consists in partitioning a series of noise-loaded measurements into quasi-stationary parts (Borodin et al., 1996 Krapivin el al., 2004). 

Quasi-stationary models assume that, besides their spatial distribution, the temperature of the fast electrons also does not evolve during the ion acceleration process. These issues do not seem to provide a major problem when determining the maximum ion energy, since the acceleration of these ions (which will be the most energetic) takes place over a time scale over which the temperature and the entire distribution do not vary appreciably [93]. 

To illustrate the concepts of determining, non-determining and negligible processes, the mechanism of the pyrolysis of neopentane will be discussed briefly here. Neopentane pyrolysis has been chosen because it has been studied by various techniques batch reactor [105— 108], continuous flow stirred tank reactor [74, 109], tubular reactor [110], very low pressure pyrolysis [111], wall-less reactor [112, 113], non-quasi-stationary state pyrolysis [114, 115], single pulse shock tube [93, 116] amongst others, and over a large range of temperature, from... 

For free radicals which are not involved in termination processes, i.e. those radicals which are the most reactive and, accordingly, the least concentrated, the QSSA can be applied even during the true induction period of the reaction. This is so for chain carrier radicals not involved in termination processes the concentrations of these radicals are not at all constant or slowly varying during the induction period however, the QSSA may be applied to them. For this reason, this special kind of QSSA will be termed pseudo-stationary state approximation (PSSA). As a consequence of the PSSA, the observation of a non-quasi-stationary behaviour for a radical concentration does not necessarily mean that the QSSA cannot be applied. This fact has probably played a role in the criticism of the QSSA. 

The treated systems should be stationary or quasi-stationary in the sense that the change of their characteristic parameters should be much slower than the internal relaxation processes and the external measurement, otherwise, the maximal entropy is not attained. 

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