Induction time-derivative

During the vacuum fractional distillation of bulked residues (7.2 t containing 30-40% of the bis(hydroxyethyl) derivative, and up to 900 ppm of iron) at 210-225°C/445-55 mbar in a mild steel still, a runaway decomposition set in and accelerated to explosion. Laboratory work on the material charged showed that exothermic decomposition on the large scale would be expected to set in around 210-230°C, and that the induction time at 215°C of 12-19 h fell to 6-9 h in presence of mild steel. Quantitative work in sealed tubes showed a maximum rate of pressure rise of 45 bar/s, to a maximum developed pressure of 200 bar. The thermally induced decomposition produced primary amine, hydrogen chloride, ethylene, methane, carbon monoxide and carbon dioxide. 

The pressure dependence, as before, is derived not only from the perfect gas law for p, but from the density-pressure relationship in Z as well. Also, the effect of the stoichiometry of a reacting gas mixture would be in Z. But the mole fraction terms would be in the logarithm, and therefore have only a mild effect on the induction time. For hydrocarbon-air mixtures, the overall order is approximately 2, so Eq. (7.46) becomes... 

The value of the critical nuclearity allowing the transfer from the monitor depends on the redox potential of this selected donor S . The induction time and the donor decay rate both depend on the initial concentrations of metal atoms and of the donor [31,62]. The critical nuclearity corresponding to the potential threshold imposed by the donor and the transfer rate constant value, which is supposed to be independent of n, are derived from the fitting between the kinetics of the experimental donor decay rates under various conditions and numerical simulations through adjusted parameters (Fig. 5) [54]. By changing the reference potential in a series of redox monitors, the dependence of the silver cluster potential on the nuclearity was obtained (Fig. 6 and Table 5) [26,63]. 

Due to the high barrier, it is safe to assume that the induction time is much shorter (by a factor of e P ) than the reaction time (1/T) so that the time dependence on the right hand side of Eq. 13 may be ignored. Then, noting that the derivative of a step function is a Dirac delta function, and using detailed balance one finds the desired formula ... 

Nauk (UkrainRSR) 1966(7), 871-74 CA 65, 19919 (1966) "Criterion of Uni-dimensional Instability of Gas Detonations (The criterion was derived by using Zel dovich-Von Neumann model, which represents a detonation wave in an ideal gas as a stationary complex consisting of a shock wave and the front of an instantaneously occurring reaction with a characteristic induction time that follows the shock wave at a definite distance. The results showed that the criterion assumes the form dependent... 

Using the value for p just derived and the data from Table 2.1, we predict an induction time of 1625 s ( 27 min). 

The buildup of the H2 concentration, for any given depth x, starts with all its time derivatives zero at t = 0, increases gradually, and after a depth-dependent induction time becomes linear in t. The unbounded growth can be truncated by allowing the molecules either to dissociate or to diffuse. Dissociation will of course modify the development of the H° distribution molecular diffusion will not. As regards dissociation, there are to date no time-dependent solutions for this problem available presumably if the molecules are immobile, they would show an approach to a flat thermal-equilibrium distribution, which would extend to deeper depths at longer times. The case of diffusion without dissociation will be taken up in the paragraphs to follow. 

Zel dovich theory — The theory determines the time dependence of the nucleation rate 7(f) = d N (f )/df and of the number N(t) of nuclei and derives a theoretical expression for the induction time T needed to establish a stationary state in the supersaturated system. The -> Zel dovich approach [i] (see also [ii]) consists in expressing the time dependence of the number Z(n,t) of the n-atomic clusters in the supersaturated parent phase by means of a partial differential equation ... 

The Bloch equations can be solved analytically under the condition of slow passage, for which the time derivatives of Eq. 2.48 are assumed to be zero to create a steady state. The nuclear induction can be shown to consist of two components, absorption, which is 90° out of phase with B, and has a Lorentzian line shape, and dispersion, which is in phase with B,. The shapes of these signals are shown in Fig. 2.10. By appropriate electronic means (see Section 3.3), we can select either of these two signals, usually the absorption mode. 

This parameter (find) is defined as the time elapsed between the creation of supersaturation and the appearance of crystals, and decreases as supersaturation inoreases. Mathematioal equations for the induotion time that hold for all nuclei forming and growing in a saturated solution have been reported [138], The induction time is usually determined from oonduotivity measurements. Thus, the formation of crystals is signaled by a drop in the solution oonductivity. The crystallization time is taken to be the time where the derivative of the oonductivity with respect to time becomes negative. 

Schott and Kinsey in 1958 [102] were the first to use induction time measurements in shocked H2 02 Ar mixtures in order to derive kinetic... 

Hoffman and Friedman [106] proposed a mechanism for the induction time in development based on heterogeneous electrochemical theories of corrosion. The model explains the empirical relation derived by Pontius and Willis. In this case the two half-cell reactions of silver and developer give rise to a mixed or corrosion potential, Ecorr, which if electron transfer is rate-limiting can be described by Eq. (74),... 

Polymerization and Cyclization of Alkynes (163). In 1940 Reppe and Schweckendieck 163) discovered that the substitution derivatives of tetracarbonylnickel with phosphines, and particularly those of the type Ni(CO)2(PR3)2, catalyze the linear polymerization and the cyclization to benzene derivatives of alkynes. Schrauzer 167) more recently showed that stoichiometric amounts of bisacrylonitrilenickel give the cyclization reaction. The reaction becomes, on the contrary, catalytic in the presence of a tertiary phosphine. Meriwether and co-workers (57, 136, 137, I40) studied the possible mechanism of Reppe s reaction. They showed that both the polymerization and the cyclization reactions are particularly effective with monosubstituted acetylenes, and that the cyclization reaction is infiuenced by steric and electronic effects. They then proposed a mechanism by which the induction time is ascribed to this rather slow reaction ... 

In deriving Eq. (36), and hence Eq. (38), it was assumed that the induction time is dominated by the time required to accumulate a critical areal concentration of cation vacancies (i.e., the formation time of the vacancy condensate). However, as evident from Fig. 33, the induction time may exceed the vacancy condensation time by the amount required for the film to dissolve locally and rupture. We previously lumped these effects into the relaxation time, r, and for the pitting of metals such as iron, copper, nickel, and stainless steels, which form very thin barrier layers, this approximation appears to be reasonable, in that Eq. (36) accounts for the experimental data very well. It is possible, however, to envision a case where the time... 

The statement is frequently made that one zeolite synthesis is faster than another but the measurement criteria may be extremely loosely defined. Often, there is no distinction between the induction time and the growth period (section 6), so that it is impossible to tell whether a reaction is slow because it takes a long time to nucleate or because the crystals grow slowly in the given circumstances. A comparison of some hypothetical synthesis reactions is shown in Table 2. The data are derived from a computer simulation of zeolite growth based on a very simple kinetic crystal growth model [50,73], i.e. 

To derive the mathematical equation for the adiabatic induction time the imsteady adiabatic heat balance of the BR is required. If the thermal inertia of the system is neglected, the balance may be obtained fi om Equ. (4-56), formally setting the heat exchange term equal to zero. 

The induction time in homogeneous nucleation of a or p polymorphs of quizalofop-ethyl, i.e. the time the crystals first appeared in a supersaturated solution of quizalofop-e yl-ethanol mixtures was measured at temperatures between 293 and 31 IK in a batch stirred crystallizer. Experimental equations on the induction time have been derived semiquantitatively on the basis of an equation of classical nucleation rate. In addition, the concentration of supersaturated... 

Classically, this law is written with a time derivation and corresponds in that case to Faraday s law of induction. Here, one has written a generalized law valid independently of any definition of time. This law is also customarily written with a minus sign because of the choice of the generator convention for orienting current and potential difference. For harmonization, it is preferable to use the receptor convention used in most domains. (This point will be made clearer when a solenoid will be associated to other different dipoles as in Chapter 9.)... 

At the level of a surface point, a localized capacitive potential is dehned as the contrarotational of the electric held and a localized inductive potential is dehned as the time derivative of the electromagnetic induction ... 

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